## Universality proof question

For discussion of other cellular automata.
period54
Posts: 20
Joined: November 12th, 2014, 12:08 pm

### Universality proof question

I have found a CA that can emulate any logic circuits using glider guns. I can make sliding block memory, but I haven't found a way to build a gun that produces stream of gliders needed to move a block.
Do I need to find a explicit way to build a block-moving gun to prove computational universality?
Also, does lack of kickback reaction imply lack of universal constructor?

dvgrn
Moderator
Posts: 6986
Joined: May 17th, 2009, 11:00 pm
Contact:

### Re: Universality proof question

period54 wrote:I have found a CA that can emulate any logic circuits using glider guns. I can make sliding block memory, but I haven't found a way to build a gun that produces stream of gliders needed to move a block.
Do I need to find a explicit way to build a block-moving gun to prove computational universality?
No, that's certainly not a requirement. When you say you can make a sliding block memory, does that mean you can hook up logic circuits to it to push, pull or check for zero location of the block? If so, doesn't that imply that you can make a block-moving gun by stringing together a few logic circuits to repeatedly send a 'push' signal to the sliding-block memory?
period54 wrote:Also, does lack of kickback reaction imply lack of universal constructor?
No, you can certainly design a U.C. that doesn't use kickback reactions. The obvious counterexample is that there's nothing like a kickback reaction in von Neumann's original 29-state CA.

It's necessary to stretch the definition of "universal constructor" a bit to get it to fit a CA with two states instead of 29. Mainly, in von-Neumann-inspired rules supporting self-replicators, it's generally trivial to prove that any pattern of quiescent cells (i.e., any still life) can be constructed by feeding the right program to some U.C. -- precisely because those multi-state rules were specifically designed to make such constructions trivial.

In Life-like rules, still lifes in general are much more difficult to construct because they can be very delicately balanced. Just at the moment people are still working on solving the last 18-bit still life in Conway's Life, for example. A general solution for constructing all possible B3/S23 still lifes seems theoretically possible but a very long way off in practice.

But Conway's original proof of B3/S23 construction universality didn't worry about whether all possible still lifes could be constructed. All that mattered was the ability to construct the objects (still lifes or oscillators) used in a universal set of logic circuitry. If you can prove that you can close the loop by designing construction circuitry that's capable of constructing itself and/or other arbitrarily complex logic circuitry, then that's a universal constructor in the same sense as Conway's original U.C. design.

Conway's proof did make extensive use of kickback reactions, but that was just the best known way in 1970 to move gliders arbitrary distances and get them lined up to do complex constructions. Things are much simpler now -- for example, the GoL propagator duplicates its circuitry and glider stream without using a single kickback reaction. There aren't any two-glider collisions at all, come to think of it, unless you count the ones where two gliders on parallel paths collide with various other objects to manipulate the construction-arm "elbow".

I suspect that the big problem for proving construction universality for your rule -- what is the rule, by the way? -- will be to show how to collide gliders to build your various logic circuits. Especially if they're made out of the usual guns and oscillators, then there tend to be a lot of tricky timing problems to solve.

If you get that far, one idea that hasn't been explored much yet is Serizawa-style "armless constructors", which seem to be adaptable enough to work in a wide range of rules. Two-arm construction along the lines of the Gemini spaceship could also be fairly efficient.

period54
Posts: 20
Joined: November 12th, 2014, 12:08 pm

### Re: Universality proof question

dvgrn wrote: When you say you can make a sliding block memory, does that mean you can hook up logic circuits to it to push, pull or check for zero location of the block?
No, that's my problem. Only way to push block I've found requires two gliders to be very close when colliding with it. I haven't found a way to produce gliders at that distance with guns. Pull and check for zero are very simple, though.
dvgrn wrote:I suspect that the big problem for proving construction universality for your rule -- what is the rule, by the way? -- will be to show how to collide gliders to build your various logic circuits. Especially if they're made out of the usual guns and oscillators, then there tend to be a lot of tricky timing problems to solve.
My rule is from family that is similar to outer totalistic rules, except that it has negative cells and negative neighbor counts. If cell has two positive and three negative neighbors, it's neighbor count is -1.

I made a reddit post about it few days ago: http://www.reddit.com/r/cellular_automa ... ive_found/

The rule itself is this:

Code: Select all

--+---00000+---++
------00000++++++
--+++-00000+++-++
Each line describes new cell for certain value of current cell and sum of it's neighbors. First line is for negative cells, second for dead and third for positive. To get a value of cell in next generation, character with index of cell's neighbors count + 8 is looked up in line corresponding to this cell. Zero means cell dies, plus means it turns positive and minus means negative.

Sorry for weird notation, couldn't think of anything better.

Here is the rule in golly format:

Code: Select all

@RULE BTCA1

@COLORS
0 0 0 0
1 32 64 255
2 255 64 32

@TABLE
# rules: 203
#
# Golly rule-table format.
# Each rule: C,N,NE,E,SE,S,SW,W,NW,C'
# N.B. Where the same variable appears multiple times in a transition,
# it takes the same value each time.
#
# Default for transitions not listed: no change
#
n_states:3
neighborhood:Moore
symmetries:rotate8reflect
var a={0,1}
var b={0,1}
var c={0,1}
var d={0,1}
var e={0,1}
var f={0,2}
var g={0,2}
var h={0,2}
var i={0,2}
var j={0,2}
var k={1,2}
var l={0,1,2}
var m={0,1,2}
var n={1,2}
var o={1,2}
0,a,b,c,d,e,1,1,1,1
0,f,g,h,i,j,2,2,2,2
0,0,a,b,c,1,0,1,1,1
0,0,f,g,h,2,0,2,2,2
0,0,0,a,1,0,0,1,1,1
0,0,a,0,1,0,1,0,1,1
0,a,b,c,1,1,1,1,2,1
0,a,b,0,1,1,1,2,1,1
0,0,a,0,1,1,2,1,1,1
0,f,g,h,1,2,2,2,2,2
0,0,0,f,2,0,0,2,2,2
0,0,f,0,2,0,2,0,2,2
0,0,f,g,2,1,2,2,2,2
0,0,f,0,2,2,1,2,2,2
f,0,0,1,0,0,1,0,1,1
0,0,a,1,0,1,1,1,2,1
0,0,a,1,0,1,1,2,1,1
f,0,0,1,0,1,2,1,1,1
f,0,0,1,0,2,1,1,1,1
0,f,g,1,0,2,2,2,2,2
0,0,a,1,1,0,1,1,2,1
0,0,a,1,1,0,1,2,1,1
f,0,0,1,1,0,2,1,1,1
0,0,a,1,1,1,0,1,2,1
f,0,0,1,1,1,0,2,1,1
0,0,a,1,1,1,1,0,2,1
0,0,f,1,2,0,2,2,2,2
0,f,g,1,2,2,0,2,2,2
0,f,0,1,2,2,2,0,2,2
a,0,0,2,0,0,2,0,2,2
a,0,0,2,0,1,2,2,2,2
a,0,0,2,0,2,1,2,2,2
a,0,0,2,0,2,2,1,2,2
a,0,0,2,1,0,2,2,2,2
a,0,0,2,1,2,0,2,2,2
a,0,0,2,2,0,1,2,2,2
f,0,1,0,1,0,1,1,2,1
f,0,1,0,1,0,1,2,1,1
f,0,1,0,1,1,0,1,2,1
f,0,1,0,1,1,0,2,1,1
f,0,1,0,1,1,1,0,2,1
a,0,1,0,2,0,2,2,2,2
a,0,1,0,2,2,0,2,2,2
f,0,1,1,0,1,1,0,2,1
0,a,1,1,1,1,1,2,2,1
0,a,1,1,1,1,2,1,2,1
f,0,1,1,1,1,2,2,1,1
0,a,1,1,1,2,1,1,2,1
f,0,1,1,1,2,1,2,1,1
f,0,1,1,1,2,2,1,1,1
0,a,1,1,2,1,1,1,2,1
f,0,1,1,2,1,1,2,1,1
f,0,1,1,2,1,2,1,1,1
0,f,1,1,2,2,2,2,2,2
a,0,1,2,0,2,0,2,2,2
a,0,1,2,0,2,2,0,2,2
f,0,1,2,1,1,1,1,2,1
f,0,1,2,1,1,1,2,1,1
0,f,1,2,1,2,2,2,2,2
a,0,1,2,2,0,2,0,2,2
0,f,1,2,2,1,2,2,2,2
0,f,1,2,2,2,1,2,2,2
a,0,1,2,2,2,2,1,2,2
a,0,1,2,2,2,2,2,1,2
a,0,2,0,2,0,2,1,2,2
f,0,2,1,1,1,1,1,2,1
a,0,2,1,1,2,2,2,2,2
a,0,2,1,2,1,2,2,2,2
a,0,2,1,2,2,1,2,2,2
a,0,2,1,2,2,2,1,2,2
a,0,2,2,1,1,2,2,2,2
a,0,2,2,1,2,1,2,2,2
k,0,0,0,0,0,0,l,m,0
k,0,0,0,0,f,1,g,l,0
k,f,0,0,0,a,l,1,2,0
k,0,0,0,a,b,l,0,2,0
1,0,0,0,0,0,2,2,2,2
k,0,0,0,0,1,f,g,n,0
k,l,0,0,0,1,a,f,2,0
k,f,0,0,g,1,l,2,1,0
k,0,0,0,a,f,b,c,2,0
1,0,0,0,0,2,0,2,2,2
k,0,a,b,f,2,c,1,2,0
k,0,0,0,1,0,f,g,n,0
k,0,0,0,n,0,l,1,2,0
k,f,0,0,1,g,l,2,1,0
k,0,0,f,l,g,2,1,1,0
k,0,0,0,n,1,0,o,2,0
k,0,0,a,1,2,b,2,n,0
k,a,0,b,1,2,l,0,2,0
1,0,0,0,1,2,2,2,2,2
k,0,0,a,2,b,c,d,2,0
1,0,0,0,2,0,0,2,2,2
k,0,a,b,f,c,1,2,2,0
1,0,0,0,2,0,2,0,2,2
k,0,0,f,g,1,l,1,2,0
1,0,0,0,2,1,2,2,2,2
1,0,0,0,2,2,1,2,2,2
k,0,0,1,0,0,n,l,2,0
k,f,0,1,0,g,1,2,l,0
k,0,0,1,f,1,g,l,2,0
k,0,0,1,f,n,g,2,1,0
k,0,0,n,0,1,a,f,2,0
k,0,0,1,0,n,2,0,o,0
k,0,0,1,a,2,b,l,2,0
k,0,0,n,f,2,g,1,1,0
1,0,0,1,0,2,2,2,2,2
k,f,0,1,1,0,g,l,2,0
k,0,0,n,1,f,1,g,2,0
k,0,0,n,1,1,f,g,2,0
1,0,0,1,1,1,1,1,1,2
k,f,0,1,a,1,l,2,2,0
k,f,0,1,a,n,2,1,2,0
k,f,g,1,1,n,2,2,1,0
k,0,0,1,1,2,2,1,1,0
k,0,0,1,2,0,0,n,2,0
1,0,0,1,2,0,0,2,1,0
k,0,a,1,2,b,f,c,2,0
1,0,0,1,2,0,2,2,2,2
k,0,f,l,2,1,a,2,1,0
k,a,b,1,2,2,c,d,2,0
1,0,0,1,2,2,0,2,2,2
k,a,0,1,f,n,1,2,2,0
1,0,0,1,2,2,2,0,2,2
k,0,0,2,0,0,2,1,2,0
1,0,0,2,0,1,0,2,2,0
k,f,g,2,a,b,1,1,2,0
k,0,a,n,0,1,2,b,2,0
k,a,b,2,c,2,d,1,2,0
k,a,b,2,2,1,1,2,2,0
k,0,1,0,1,f,l,g,2,0
k,0,1,0,1,l,2,m,2,0
k,f,1,0,1,l,a,2,2,0
1,0,1,0,1,1,1,1,1,2
k,0,n,a,1,b,2,0,2,0
k,f,1,g,1,l,2,2,1,0
k,0,1,0,1,2,a,1,2,0
k,0,n,f,1,2,a,2,1,0
k,0,1,0,2,a,l,b,2,0
k,0,1,a,2,1,f,n,2,0
1,0,1,0,2,2,1,2,2,0
1,0,1,1,0,1,1,1,1,2
k,f,1,1,0,n,2,a,2,0
k,0,n,a,f,1,2,2,1,0
1,0,1,1,1,0,1,1,1,2
k,0,1,1,n,0,o,2,2,0
k,0,1,1,1,0,2,a,2,0
k,0,1,1,1,1,0,2,2,0
k,0,n,o,1,2,0,1,2,0
k,f,1,1,a,2,0,2,n,0
k,0,1,1,1,2,1,0,2,0
k,0,1,1,2,0,1,1,2,0
k,0,1,1,2,0,1,2,1,0
k,f,1,n,2,g,2,1,1,0
k,0,1,1,2,1,1,0,2,0
1,0,1,1,2,2,2,0,2,0
1,0,1,1,2,2,2,2,2,2
1,0,1,2,0,1,2,2,2,0
k,l,1,2,a,2,b,2,1,0
k,0,1,2,1,0,1,2,1,0
k,0,1,2,1,0,2,2,2,0
k,a,1,2,1,2,0,2,2,0
k,a,1,2,1,2,2,b,n,0
1,0,1,2,1,2,2,2,2,2
k,0,1,2,2,0,1,2,2,0
k,0,1,2,2,0,2,1,2,0
k,a,1,2,2,0,2,2,1,0
k,0,1,2,2,1,0,2,2,0
k,a,1,2,2,1,2,0,2,0
1,0,1,2,2,1,2,2,2,2
k,0,1,2,2,2,1,0,2,0
1,0,1,2,2,2,1,2,2,2
k,0,2,0,2,1,1,2,2,0
k,0,2,1,1,2,0,2,2,0
k,0,2,1,2,0,2,1,2,0
1,f,2,2,2,2,2,2,2,2
1,1,1,1,1,1,1,1,2,2
k,1,1,1,n,o,2,2,2,0
1,1,1,1,2,1,2,1,2,0
k,1,2,1,2,2,1,2,2,0
2,0,0,0,0,0,1,1,1,1
2,0,0,0,0,1,0,1,1,1
2,0,0,0,1,0,0,1,1,1
2,0,0,0,1,0,1,0,1,1
2,0,0,0,1,1,1,1,2,1
2,0,0,0,1,1,1,2,1,1
2,0,0,0,1,1,2,1,1,1
2,0,0,1,0,1,1,1,2,1
2,0,0,1,0,1,1,2,1,1
2,0,0,1,1,0,1,1,2,1
2,0,0,1,1,0,1,2,1,1
2,0,0,1,1,1,0,1,2,1
2,0,0,1,1,1,1,0,2,1
2,0,0,2,2,2,2,2,2,1
2,a,1,1,1,1,1,1,1,1
2,0,1,1,1,1,1,2,2,1
2,0,1,1,1,1,2,1,2,1
2,0,1,1,1,2,1,1,2,1
2,0,1,1,2,1,1,1,2,1
2,0,2,0,2,2,2,2,2,1
2,0,2,2,0,2,2,2,2,1
2,0,2,2,2,0,2,2,2,1
2,1,2,2,2,2,2,2,2,1

And here are almost all patterns discovered in this rule (because of symmetry, it doesn't change behavior if whole pattern is negated):

Code: Select all

[M2] (golly 2.5)
#R BTCA1
1 0 1 0 1
1 1 0 1 0
2 1 2 0 0
3 0 0 3 0
4 0 0 0 4
5 0 5 0 0
1 1 1 1 1
2 0 7 0 0
1 0 2 2 2
1 2 0 2 2
2 0 0 9 10
2 0 0 0 7
1 2 2 0 2
1 2 2 2 0
2 13 14 0 0
3 8 11 12 15
2 7 0 0 0
1 0 0 1 1
2 0 0 0 18
2 0 0 7 0
1 1 1 0 0
2 0 21 0 0
3 17 19 20 22
4 0 0 16 23
3 0 0 19 0
1 0 2 0 2
1 2 2 2 2
2 0 21 26 27
1 2 0 2 0
2 0 0 29 18
2 26 27 0 18
2 29 21 0 0
3 28 30 31 32
4 25 0 33 0
3 22 0 0 0
4 35 0 0 0
5 24 34 0 36
6 0 0 6 37
7 0 0 0 38
1 0 0 2 2
1 2 2 0 0
2 40 0 41 0
2 0 1 0 0
1 1 1 1 2
2 44 7 0 0
3 0 42 43 45
1 0 0 0 2
1 0 2 0 0
2 0 47 0 48
1 0 0 2 0
1 2 0 0 0
2 50 47 51 48
1 1 1 2 1
2 53 44 0 0
3 49 52 8 54
4 46 55 0 0
2 50 0 51 0
1 0 0 0 1
2 40 0 41 58
1 2 1 1 1
2 0 7 0 60
3 57 59 17 61
1 0 0 1 2
2 0 0 63 18
1 0 0 2 1
1 0 0 1 0
2 0 0 65 66
1 2 0 1 0
2 44 7 68 0
1 0 2 0 1
1 1 2 1 1
2 53 7 70 71
3 64 67 69 72
2 0 53 0 7
1 0 1 0 0
2 40 75 41 0
3 0 74 0 76
1 1 0 2 0
2 78 0 71 7
1 0 1 0 2
2 80 44 60 7
1 1 2 0 0
2 82 21 0 0
1 2 1 0 0
1 1 0 0 0
2 84 85 0 0
3 79 81 83 86
4 62 73 77 87
5 56 88 0 0
3 42 59 0 61
2 0 0 65 63
3 64 91 69 54
2 0 60 0 53
3 0 74 42 93
2 78 0 2 0
2 0 0 0 70
2 68 0 78 0
1 0 1 2 1
2 0 98 0 7
3 95 96 97 99
4 90 92 94 100
2 0 0 18 0
2 7 50 98 2
3 102 0 103 0
2 0 0 18 60
1 1 1 2 0
2 47 18 7 106
2 53 82 7 68
3 105 107 108 0
2 7 51 44 0
2 0 0 0 47
2 2 0 68 0
2 0 1 0 60
3 110 111 112 113
2 60 78 7 0
1 1 2 1 0
2 116 0 78 0
3 115 0 117 0
4 104 109 114 118
3 0 61 0 22
2 2 0 68 65
1 2 1 0 1
1 0 2 1 1
2 0 122 123 7
2 21 51 0 0
3 121 124 54 125
4 120 126 0 0
2 0 53 0 1
3 95 128 0 0
2 63 18 44 7
2 65 63 53 44
3 130 131 0 0
4 129 132 0 0
5 101 119 127 133
6 0 0 89 134
2 50 0 7 2
2 60 68 53 78
3 136 0 137 0
2 7 2 60 68
2 53 78 7 2
3 139 0 140 0
4 138 0 141 0
1 1 0 0 2
2 60 68 7 143
2 0 0 65 40
1 1 2 2 2
2 0 27 0 146
1 2 2 2 1
2 148 27 78 0
3 144 145 147 149
2 0 0 63 65
2 0 0 40 63
1 2 2 1 2
2 153 148 0 0
2 27 153 0 0
3 151 152 154 155
2 0 153 0 27
2 68 0 29 0
2 0 146 0 41
2 78 63 153 148
3 157 158 159 160
1 0 1 2 2
2 162 27 41 85
2 148 27 0 0
3 0 151 163 164
4 150 156 161 165
1 0 2 1 2
2 148 27 0 167
2 66 0 29 0
3 145 0 168 169
2 0 27 80 148
2 85 0 0 0
2 153 29 0 0
3 171 172 173 0
4 170 0 174 0
5 142 0 166 175
1 0 1 1 1
2 0 177 0 75
2 66 0 51 0
3 0 0 178 179
2 0 0 0 58
2 0 0 177 0
2 0 177 0 0
1 1 2 2 1
2 184 85 85 0
3 181 182 183 185
4 0 0 180 186
2 0 0 0 26
3 0 0 0 188
1 0 2 1 0
2 58 190 7 53
1 0 1 2 0
2 40 192 27 0
1 1 1 1 0
2 71 194 75 58
3 181 191 193 195
4 0 189 189 196
2 0 48 0 0
3 0 198 0 0
4 199 0 0 0
5 187 197 0 200
6 0 0 176 201
7 0 0 135 202
8 39 203 0 0
9 0 204 0 0
2 0 0 50 0
3 0 0 0 206
4 0 0 207 0
5 0 0 0 208
2 40 50 27 0
3 0 0 210 0
2 0 0 66 0
2 58 0 0 0
3 212 0 213 0
2 47 27 26 146
3 0 0 0 215
4 211 0 214 216
1 2 1 2 2
2 27 50 218 29
3 0 0 219 0
4 0 0 220 0
4 0 199 0 0
2 51 0 0 0
3 223 0 0 0
4 224 0 0 0
5 217 221 222 225
2 0 13 0 190
2 10 0 75 10
1 1 1 0 1
1 2 1 1 2
2 229 230 75 229
2 190 13 0 0
3 227 228 231 232
3 206 0 0 0
4 233 234 0 0
2 0 13 0 48
2 48 13 0 0
3 236 228 0 237
4 238 234 0 0
5 208 235 239 0
6 0 209 226 240
2 47 14 0 0
2 0 18 0 0
3 0 181 242 243
4 0 0 0 244
2 66 0 0 0
1 0 1 1 0
2 0 247 0 0
3 0 181 246 248
2 0 0 18 66
2 1 0 0 0
3 0 250 172 251
4 0 0 249 252
5 245 253 0 0
2 0 58 0 0
2 18 0 66 0
3 0 212 255 256
2 0 26 0 48
2 41 0 0 0
3 0 0 258 259
4 0 0 257 260
2 0 14 41 51
3 0 0 262 223
2 18 2 1 21
3 0 0 264 198
4 0 0 263 265
5 261 266 0 0
6 0 0 254 267
7 0 0 241 268
2 13 0 48 13
3 0 0 270 0
2 0 0 40 29
1 2 2 0 1
2 26 273 0 47
2 50 0 27 51
3 272 0 274 275
4 0 0 271 276
2 0 0 0 48
3 0 0 0 278
2 0 0 26 40
2 146 68 13 0
3 280 0 281 0
4 0 0 279 282
3 198 0 0 0
2 48 0 0 0
3 0 285 0 0
4 284 286 0 0
5 277 283 287 0
2 0 48 0 1
3 0 111 0 289
2 51 0 7 247
3 206 0 291 0
4 0 0 290 292
1 1 0 1 2
2 0 294 75 21
3 0 278 0 295
2 10 50 13 0
2 2 0 21 0
3 297 0 298 0
4 0 0 296 299
2 0 75 0 0
3 0 301 0 0
4 302 0 0 0
2 14 0 51 0
3 304 0 0 0
4 199 305 0 0
5 293 300 303 306
6 0 0 288 307
1 1 0 1 1
2 58 309 18 53
3 0 0 0 310
2 78 85 44 85
3 212 0 312 0
4 0 0 311 313
2 0 0 1 2
1 1 2 1 2
1 2 1 2 1
2 40 40 316 317
3 0 315 188 318
2 50 0 29 0
3 0 0 320 0
4 0 0 319 321
2 58 1 0 0
3 0 323 0 0
4 324 224 0 0
2 41 41 1 2
3 198 326 0 0
3 0 255 0 0
4 327 328 0 0
5 314 322 325 329
2 0 0 0 2
1 0 2 2 1
2 0 332 48 146
2 40 21 148 50
3 0 331 333 334
3 0 181 172 43
4 0 0 335 336
2 75 50 47 332
1 1 2 2 0
1 2 0 0 1
2 29 0 339 340
3 0 0 338 341
4 0 0 342 0
2 18 41 1 0
2 339 0 0 0
3 344 345 0 0
4 346 0 0 0
2 0 26 0 0
3 348 183 0 0
2 194 0 85 0
3 350 0 0 0
4 349 351 0 0
5 337 343 347 352
6 0 0 330 353
7 0 0 308 354
8 269 355 0 0
2 1 66 85 26
2 0 0 27 1
3 0 0 357 358
2 66 0 309 0
3 0 0 360 0
4 0 0 359 361
2 0 48 0 177
2 41 75 194 0
3 363 364 0 172
2 0 332 0 7
3 0 366 0 0
4 365 367 0 0
2 7 0 339 0
3 369 0 0 0
2 0 18 0 1
3 371 298 0 0
4 370 372 0 0
5 362 0 368 373
2 0 0 10 50
2 58 0 18 66
3 0 0 375 376
4 0 0 377 0
2 47 0 332 27
2 2 0 0 0
3 19 212 379 380
2 58 66 229 230
3 0 0 0 382
4 0 0 381 383
2 13 29 0 0
3 385 213 0 0
2 0 58 0 1
3 0 387 0 0
4 386 388 0 0
2 0 339 21 0
3 390 223 0 0
2 0 229 41 0
3 258 392 0 0
4 391 393 0 0
5 378 384 389 394
6 0 0 374 395
2 0 1 0 1
3 0 0 212 397
2 0 0 229 7
2 0 0 7 85
2 44 317 44 53
1 1 1 2 2
2 218 7 402 7
3 399 400 401 403
4 0 0 398 404
2 0 58 75 7
2 0 85 10 0
3 0 406 111 407
4 0 0 0 408
3 172 301 0 0
2 229 60 21 21
2 60 21 21 85
3 411 412 0 0
4 410 413 0 0
1 2 1 0 2
2 0 415 0 48
2 27 0 0 21
3 416 417 0 301
4 0 418 0 0
5 405 409 414 419
3 212 0 0 0
3 0 0 0 371
4 0 0 421 422
3 0 0 246 0
4 0 0 424 0
2 0 0 85 0
2 0 26 0 13
3 426 427 0 0
2 40 40 27 27
1 0 2 2 0
2 430 58 41 177
2 48 48 0 0
3 429 431 432 0
4 428 433 0 0
2 0 9 0 0
3 111 0 435 304
4 421 436 0 0
5 423 425 434 437
6 0 0 420 438
7 0 0 396 439
2 0 0 66 13
2 0 63 58 309
2 309 190 68 0
3 0 441 442 443
2 50 0 10 0
3 445 0 0 0
4 0 0 444 446
2 0 0 58 0
2 18 66 0 0
3 0 448 0 449
2 0 0 47 9
3 0 0 451 0
4 0 0 450 452
2 0 85 0 0
3 454 0 0 0
4 455 0 0 0
1 2 1 1 0
2 9 457 48 340
2 51 0 18 0
3 458 459 0 246
4 0 460 0 0
5 447 453 456 461
2 0 0 47 40
3 0 0 0 463
3 0 0 0 375
2 26 0 0 0
2 0 50 13 10
3 0 466 0 467
2 13 0 0 0
2 0 0 58 18
3 198 469 0 470
4 464 465 468 471
3 0 0 0 467
3 0 0 0 19
4 0 0 473 474
2 1 7 0 0
3 0 476 0 375
3 198 469 0 0
4 286 477 0 478
2 0 229 0 58
3 0 480 0 8
4 286 481 0 0
5 472 475 479 482
6 0 0 462 483
3 0 0 181 212
2 0 0 0 66
3 0 0 0 486
4 0 0 485 487
3 0 0 0 212
2 0 229 0 75
2 309 0 0 0
3 490 491 0 0
4 489 0 492 0
1 1 0 2 1
2 494 53 53 494
2 85 229 0 0
3 495 172 496 380
2 58 7 0 75
3 0 498 0 0
4 497 499 0 0
2 0 50 9 153
2 0 9 0 48
1 2 2 1 1
1 1 2 0 1
2 503 504 27 316
3 172 501 502 505
2 153 50 14 0
3 206 0 507 0
2 48 14 0 0
3 0 509 0 0
4 506 508 510 0
5 488 493 500 511
3 0 0 181 0
2 75 7 0 85
3 514 246 0 0
4 0 513 0 515
5 0 0 0 516
6 0 0 512 517
7 0 0 484 518
8 440 519 0 0
9 356 520 0 0
10 205 521 0 0
2 0 0 0 1
3 0 0 523 102
4 0 0 524 0
5 0 0 0 525
2 0 0 58 177
3 0 0 0 527
2 0 194 0 0
3 0 529 0 0
4 0 528 0 530
3 22 380 0 0
2 26 27 0 0
3 172 463 0 533
4 532 0 534 0
2 0 0 0 309
2 75 229 0 0
3 536 212 537 0
2 0 0 0 27
2 0 41 0 0
3 0 539 0 540
2 0 0 29 0
3 542 0 223 0
4 538 0 541 543
5 531 535 544 0
6 526 0 545 0
7 0 0 546 0
8 547 0 0 0
9 548 0 0 0
10 549 0 0 0
11 0 0 522 550


The collisions that move block forward and back:

Code: Select all

x = 492, y = 352, rule = BTCA1
3.B$.3B$.3B$3B39$45.B$43.3B$43.3B$42.3B11$59.B$57.3B$57.3B$56.3B$63.B
$61.3B$61.3B$60.3B27$93.B$91.3B$91.3B$90.3B39$135.B$133.3B$133.3B$132.3B11$149.B$147.3B$147.3B$146.3B$153.B$151.3B$151.3B$150.3B27$183.
B$181.3B$181.3B$180.3B39$225.B$223.3B$223.3B$222.3B11$239.B$237.3B$
237.3B$236.3B$243.B$241.3B$241.3B$240.3B27$273.B$271.3B$271.3B$270.3B 39$315.B$313.3B$313.3B$312.3B11$329.B$327.3B$327.3B$326.3B$333.B139.B
$331.3B137.3B$331.3B137.3B$330.3B137.3B17$350.2B138.2B$350.2B138.2B!  The construction of logic gates with guns is trivial, this website describes it pretty well: https://sucs.org/~pwb/report/lifecomp.html This can be used as eater, but gliders must be of opposite color: Code: Select all x = 3, y = 3, rule = BTCA1 .A$.A$3A!  I'll be searching for a way to construct gun with gliders, it doesn't seem to be extremely hard because it's built from relatively common patterns. period54 Posts: 20 Joined: November 12th, 2014, 12:08 pm ### Re: Universality proof question A 23 cell pattern in this rule that takes 62087 generations to stabilize. Code: Select all x = 25, y = 10, rule = BTCA1 13B6$22.3B$21.3B$21.3B$21.B!  A for awesome Posts: 2076 Joined: September 13th, 2014, 5:36 pm Location: Pembina University, Home of the Gliders Contact: ### Re: Universality proof question 28,412-gen 15-cell methuselah: Code: Select all x = 11, y = 7, rule = BTCA1 2.A$2.A$3A2$.A8.B$.A.A4.3B$.A.A6.B!

Edit:

Code: Select all

x = 123, y = 72, rule = BTCA1
121.2B$121.2B67$3B$3B$3B$3.B!  I'm beginning to like this rule . Last edited by A for awesome on November 14th, 2014, 6:02 pm, edited 1 time in total. praosylen#5847 (Discord) x₁=ηx V*_η=c²√(Λη) K=(Λu²)/2 Pₐ=1−1/(∫^∞_t₀(p(t)ˡ⁽ᵗ⁾)dt) $$x_1=\eta x$$ $$V^*_\eta=c^2\sqrt{\Lambda\eta}$$ $$K=\frac{\Lambda u^2}2$$ $$P_a=1-\frac1{\int^\infty_{t_0}p(t)^{l(t)}dt}$$ dvgrn Moderator Posts: 6986 Joined: May 17th, 2009, 11:00 pm Location: Madison, WI Contact: ### Re: Universality proof question period54 wrote: dvgrn wrote: When you say you can make a sliding block memory, does that mean you can hook up logic circuits to it to push, pull or check for zero location of the block? No, that's my problem. Only way to push block I've found requires two gliders to be very close when colliding with it. I haven't found a way to produce gliders at that distance with guns. Yeah, it looks tricky to put one glider that close behind the other. Maybe gliders next to each other would be easier? Code: Select all x = 134, y = 134, rule = BTCA1 10.B$8.3B$8.3B$7.3B4$3.B$.3B$.3B$3B31$51.B$49.3B$49.3B$48.3B4$44.B$
42.3B$42.3B$41.3B31$92.B$90.3B$90.3B$89.3B4$85.B$83.3B$83.3B$82.3B31$133.B$131.3B$131.3B$130.3B4$126.B$124.3B4.2B$124.3B4.2B$123.3B!
I like the matter/antimatter effect of the two different colors, and the way that random patterns tend to develop a predominant color, but that even if you start with all one color, opposite-color patches can appear on occasion, and spread or even take over if they're very lucky.

It's also interesting that almost any monochromatic near-horizontal line will stabilize into a string of striped diamonds, which sits there quietly until something touches it, then explodes into chaos. (You can tell I've been scribbling...)

-- I think that early on I saw one of these diamond-shaped areas growing slowly for a while instead of exploding. There may be a way for a nearby pattern to progressively turn on cells at one corner or edge of a diamond so that the stability is maintained. Has anyone else seen that happening?

c0b0p0
Posts: 645
Joined: February 26th, 2014, 4:48 pm

### Re: Universality proof question

A for awesome wrote:Edit:[RLE for 15735-generation methuselah]
Diagonally symmetric patterns tend to have longer lifespans than other patterns. For example, here is a 25022-generation methuselah with 12 cells.

Code: Select all

x = 22, y = 22, rule = BTCA1
.2A$3A$3A18$20.2A$20.2A!


period54
Posts: 20
Joined: November 12th, 2014, 12:08 pm

### Re: Universality proof question

Another methuselah: 15 cells, stabilizes in 76708 generations. Curiously, it lacks symmetry.

Code: Select all

x = 3, y = 6, rule = BTCA1
.B$3B$3B$3B$BAB$.2B!  It seems to be relatively common. My search program found three random soups that produce it. dvgrn wrote: Yeah, it looks tricky to put one glider that close behind the other. Maybe gliders next to each other would be easier? It seems easier indeed. I should search for 90 degree reflectors. dvgrn wrote:I like the matter/antimatter effect of the two different colors, and the way that random patterns tend to develop a predominant color, but that even if you start with all one color, opposite-color patches can appear on occasion, and spread or even take over if they're very lucky. Metaphor of matter/antimatter is what actually led me to research of this family of rules. And some of things you stated were my actual criteria when I was searching for a rule. dvgrn wrote:It's also interesting that almost any monochromatic near-horizontal line will stabilize into a string of striped diamonds, which sits there quietly until something touches it, then explodes into chaos. (You can tell I've been scribbling...) Yeah, this is one of my favorite things about this rule. It's even weirder when you collide a glider with a huge diamond. Chaos seems to spread very unevenly across it, and even leave triangular or linear patches of stable patterns. The growth you have noticed is probably result of two diamonds merging together: Code: Select all x = 3, y = 26, rule = BTCA1 2.B$2.B$2.B$2.B$2.B$2.B$2.B$2.B$2.B$2.B$2.B$2.B$2.B$B$B$B$B$B$B$B$B$B
$B$B$B$B!

And you can grow a diamond with gliders:

Code: Select all

x = 23, y = 15, rule = BTCA1
16.A2$14.5A2$3.A8.9A$.3A$.3A6.13A$3A$10.13A$4.3A$3.3A6.9A$3.3A$3.A10.
5A2$16.A!  It would require a glider-producing puffer to make a diamond grow infinitely. Also, there is another oscillator that explodes rather chaotically: Code: Select all x = 37, y = 36, rule = BTCA1 30.3B$30.B$28.3B$28.B$26.3B$26.B$24.3B$24.B$22.3B$22.B$20.3B$20.B$18. 3B$18.B$16.3B$16.B$14.3B$14.B$12.3B$12.B$10.3B$10.B$8.3B$8.B$6.3B$6.B
$4.3B$4.B$2.3B$2.B$3B$B$B33.3B$33.3B$33.3B$33.B!

You can extend it by adding more segments.

period54
Posts: 20
Joined: November 12th, 2014, 12:08 pm

### Re: Universality proof question

And another: 10 cells, 59529 generations.

Code: Select all

x = 5, y = 6, rule = BTCA1
2.3B$3.B$.B$.B$2B$.2B!  Sphenocorona Posts: 489 Joined: April 9th, 2013, 11:03 pm ### Re: Universality proof question Since the fate of a cell in the next generation is determined by the sum of the values of its neighbors, the ruletable for Golly could be made far simpler by using 'symmetries:permute' instead of 'symmetries:rotate8reflect'. Permute symmetry is useful if the order of the neighbors around a given cell does not matter. Then you won't need to have, for example, both "0,0,0,f,2,0,0,2,2,2" and "0,0,f,0,2,0,2,0,2,2". Additionally, variable declarations can reference preceding variable declarations, which still cuts down the size but not nearly as much as changing to permute. Pretty cool rule, though bprentice Posts: 686 Joined: September 10th, 2009, 6:20 pm Location: Coos Bay, Oregon ### Re: Universality proof question Wondering what other interesting rules can be found in this rule family, I implemented this rule in Square Cell. Once this was done the Java code defining the rule was inserted into the Golly provided RuleTreeGen.java file and used to create a Golly rule tree. Thus providing another easy way to generate a Golly implementation of the rule. Here is the modified RuleTreeGen.java file and the resulting rule tree file. Notice the rule definition in lines 6 thru 39 of the Java file. Code: Select all import java.util.*; public class RuleTreeGen { /* Put your state count, neighbor count, and function here */ final static int numStates = 3; final static int numNeighbors = 8; final static int ruleTable[][] = { {1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 2}, {1, 1, 2, 1, 1, 1, 0, 0, 0, 0, 0, 2, 1, 1, 1, 2, 2}, {1, 1, 2, 2, 2, 1, 0, 0, 0, 0, 0, 2, 2, 2, 1, 2, 2} }; /* order for nine neighbors is nw, ne, sw, se, n, w, e, s, c */ /* order for five neighbors is n, w, e, s, c */ private int getState(int s) { if (s == 1) return -1; else if (s == 2) return 1; return 0; } int f(int[] a) { int neighborCount = getState(a[0]) + getState(a[1]) + getState(a[2]) + getState(a[3]) + getState(a[4]) + getState(a[5]) + getState(a[6]) + getState(a[7]); return ruleTable[a[8]][neighborCount + 8]; } final static int numParams = numNeighbors + 1; HashMap<String, Integer> world = new HashMap<String, Integer>(); ArrayList<String> r = new ArrayList<String>(); int[] params = new int[numParams]; int nodeSeq = 0; int getNode(String n) { Integer found = world.get(n); if (found == null) { found = nodeSeq++; r.add(n); world.put(n, found); } return found; } int recur(int at) { if (at == 0) return f(params); String n = "" + at; for (int i=0; i<numStates; i++) { params[numParams-at] = i; n += " " + recur(at-1); } return getNode(n); } void writeRuleTree() { System.out.println("num_states=" + numStates); System.out.println("num_neighbors=" + numNeighbors); System.out.println("num_nodes=" + r.size()); for (int i=0; i<r.size(); i++) System.out.println(r.get(i)); } public static void main(String[] args) throws Exception { RuleTreeGen rtg = new RuleTreeGen(); rtg.recur(numParams); rtg.writeRuleTree(); } }  Code: Select all @RULE PlusMinus001 @TREE num_states=3 num_neighbors=8 num_nodes=69 1 0 0 0 2 0 0 0 3 1 1 1 1 1 1 1 2 0 3 0 3 1 4 1 1 2 2 2 2 0 0 6 3 1 1 7 4 2 5 8 1 1 1 2 2 3 10 0 3 4 11 1 4 5 12 2 1 2 1 2 2 6 0 14 3 7 1 15 4 8 2 16 5 9 13 17 2 10 10 3 3 11 19 4 4 12 20 5 5 13 21 9 2 14 6 14 3 15 7 23 4 16 8 24 5 17 9 25 6 18 22 26 1 1 2 2 2 10 28 10 3 19 29 11 4 20 30 12 5 21 31 13 6 22 32 18 1 2 1 1 2 14 14 34 3 23 15 35 4 24 16 36 5 25 17 37 6 26 18 38 7 27 33 39 2 28 3 10 3 29 41 19 4 30 42 20 5 31 43 21 6 32 44 22 7 33 45 27 2 34 14 6 3 35 23 47 4 36 24 48 5 37 25 49 6 38 26 50 7 39 27 51 8 40 46 52 2 3 3 28 3 41 54 29 4 42 55 30 5 43 56 31 6 44 57 32 7 45 58 33 8 46 59 40 2 6 34 6 3 47 35 61 4 48 36 62 5 49 37 63 6 50 38 64 7 51 39 65 8 52 40 66 9 53 60 67  Brian Prentice c0b0p0 Posts: 645 Joined: February 26th, 2014, 4:48 pm ### Re: Universality proof question Although this does not compare to period54's new methuselah, it does have an interesting history and a fourfold symmetry. Code: Select all x = 13, y = 13, rule = BTCA1 11.B$10.3B$11.B8$.B$3B$.B!


bprentice
Posts: 686
Joined: September 10th, 2009, 6:20 pm
Location: Coos Bay, Oregon

### Re: Universality proof question

Not wanting to pollute this thread with other rules, I have started a new thread describing another member of this rule family here:

viewtopic.php?f=11&t=1501

The new rule does not have the elegance of this rule but does seem to have potential.

Brian Prentice

period54
Posts: 20
Joined: November 12th, 2014, 12:08 pm

### Re: Universality proof question

Yet another 10 cell methuselah. Lives 184420 generations, three times more than previous.

Code: Select all

x = 19, y = 16, rule = BTCA1
17.2A$17.2A$17.A$17.A10$A$.2A$.A!


period54
Posts: 20
Joined: November 12th, 2014, 12:08 pm

### Re: Universality proof question

First puffer discovered in this rule!

Code: Select all

x = 12, y = 12, rule = BTCA1
10.B$9.3B$8.3BA$9.BA2$7.2A$6.4A$5.3A2B.A$2.B2.2AB2A$.3B2.AB2A$3BA$.BA
4.A!


period54
Posts: 20
Joined: November 12th, 2014, 12:08 pm

### Re: Universality proof question

And another puffer.

Code: Select all

x = 15, y = 19, rule = BTCA1
8.3B$7.3B$7.3B$7.B$4.3A5.3B$4.3A4.3B$4.3A4.3B$.3B7.B$3B$3B$B$5.3B$4.
3B$4.3B$4.B$9.3B$8.3B$8.3B$8.B!


c0b0p0
Posts: 645
Joined: February 26th, 2014, 4:48 pm

### Re: Universality proof question

period54 wrote:And another puffer.
Here is a way to create puffers of arbitrarily high periods.

Code: Select all

x = 447, y = 443, rule = BTCA1
8.3B$7.3B$7.3B$7.B$4.3A5.3B$4.3A4.3B$4.3A4.3B$.3B7.B$3B13.3B$3B12.3B$
B14.3B$5.3B7.B$4.3B13.3B$4.3B12.3B$4.B14.3B$9.3B7.B$8.3B13.3B$8.3B12. 3B$8.B14.3B$13.3B7.B$12.3B13.3B$12.3B12.3B$12.B14.3B$17.3B7.B$16.3B
13.3B$16.3B12.3B$16.B14.3B$21.3B7.B$20.3B13.3B$20.3B12.3B$20.B14.3B$25.3B7.B$24.3B13.3B$24.3B12.3B$24.B14.3B$29.3B7.B$28.3B13.3B$28.3B12. 3B$28.B14.3B$33.3B7.B$32.3B13.3B$32.3B12.3B$32.B14.3B$37.3B7.B$36.3B
13.3B$36.3B12.3B$36.B14.3B$41.3B7.B$40.3B13.3B$40.3B12.3B$40.B14.3B$45.3B7.B$44.3B13.3B$44.3B12.3B$44.B14.3B$49.3B7.B$48.3B13.3B$48.3B12. 3B$48.B14.3B$53.3B7.B$52.3B13.3B$52.3B12.3B$52.B14.3B$57.3B7.B$56.3B
13.3B$56.3B12.3B$56.B14.3B$61.3B7.B$60.3B13.3B$60.3B12.3B$60.B14.3B$65.3B7.B$64.3B13.3B$64.3B12.3B$64.B14.3B$69.3B7.B$68.3B13.3B$68.3B12. 3B$68.B14.3B$73.3B7.B$72.3B13.3B$72.3B12.3B$72.B14.3B$77.3B7.B$76.3B
13.3B$76.3B12.3B$76.B14.3B$81.3B7.B$80.3B13.3B$80.3B12.3B$80.B14.3B$85.3B7.B$84.3B13.3B$84.3B12.3B$84.B14.3B$89.3B7.B$88.3B13.3B$88.3B12. 3B$88.B14.3B$93.3B7.B$92.3B13.3B$92.3B12.3B$92.B14.3B$97.3B7.B$96.3B
13.3B$96.3B12.3B$96.B14.3B$101.3B7.B$100.3B13.3B$100.3B12.3B$100.B14.
3B$105.3B7.B$104.3B13.3B$104.3B12.3B$104.B14.3B$109.3B7.B$108.3B13.3B
$108.3B12.3B$108.B14.3B$113.3B7.B$112.3B13.3B$112.3B12.3B$112.B14.3B$117.3B7.B$116.3B13.3B$116.3B12.3B$116.B14.3B$121.3B7.B$120.3B13.3B$120.3B12.3B$120.B14.3B$125.3B7.B$124.3B13.3B$124.3B12.3B$124.B14.3B$129.3B7.B$128.3B13.3B$128.3B12.3B$128.B14.3B$133.3B7.B$132.3B13.3B$132.3B12.3B$132.B14.3B$137.3B7.B$136.3B13.3B$136.3B12.3B$136.B14.3B$141.3B7.B$140.3B13.3B$140.3B12.3B$140.B14.3B$145.3B7.B$144.3B13.3B$144.3B12.3B$144.B14.3B$149.3B7.B$148.3B13.3B$148.3B12.3B$148.B14.3B$153.3B7.B$152.3B13.3B$152.3B12.3B$152.B14.3B$157.3B7.B$156.3B13.3B$156.3B12.3B$156.B14.3B$161.3B7.B$160.3B13.3B$160.3B12.3B$160.B14.3B$165.3B7.B$164.3B13.3B$164.3B12.3B$164.B14.3B$169.3B7.B$168.3B13.3B$168.3B12.3B$168.B14.3B$173.3B7.B$172.3B13.3B$172.3B12.3B$172.B14.3B$177.3B7.B$176.3B13.3B$176.3B12.3B$176.B14.3B$181.3B7.B$180.3B13.3B$180.3B12.3B$180.B14.3B$185.3B7.B$184.3B13.3B$184.3B12.3B$184.B14.3B$189.3B7.B$188.3B13.3B$188.3B12.3B$188.B14.3B$193.3B7.B$192.3B13.3B$192.3B12.3B$192.B14.3B$197.3B7.B$196.3B13.3B$196.3B12.3B$196.B14.3B$201.3B7.B$200.3B13.3B$200.3B12.3B$200.B14.3B$205.3B7.B$204.3B13.3B$204.3B12.3B$204.B14.3B$209.3B7.B$208.3B13.3B$208.3B12.3B$208.B14.3B$213.3B7.B$212.3B13.3B$212.3B12.3B$212.B14.3B$217.3B7.B$216.3B13.3B$216.3B12.3B$216.B14.3B$221.3B7.B$220.3B13.3B$220.3B12.3B$220.B14.3B$225.3B7.B$224.3B13.3B$224.3B12.3B$224.B14.3B$229.3B7.B$228.3B13.3B$228.3B12.3B$228.B14.3B$233.3B7.B$232.3B13.3B$232.3B12.3B$232.B14.3B$237.3B7.B$236.3B13.3B$236.3B12.3B$236.B14.3B$241.3B7.B$240.3B13.3B$240.3B12.3B$240.B14.3B$245.3B7.B$244.3B13.3B$244.3B12.3B$244.B14.3B$249.3B7.B$248.3B13.3B$248.3B12.3B$248.B14.3B$253.3B7.B$252.3B13.3B$252.3B12.3B$252.B14.3B$257.3B7.B$256.3B13.3B$256.3B12.3B$256.B14.3B$261.3B7.B$260.3B13.3B$260.3B12.3B$260.B14.3B$265.3B7.B$264.3B13.3B$264.3B12.3B$264.B14.3B$269.3B7.B$268.3B13.3B$268.3B12.3B$268.B14.3B$273.3B7.B$272.3B13.3B$272.3B12.3B$272.B14.3B$277.3B7.B$276.3B13.3B$276.3B12.3B$276.B14.3B$281.3B7.B$280.3B13.3B$280.3B12.3B$280.B14.3B$285.3B7.B$284.3B13.3B$284.3B12.3B$284.B14.3B$289.3B7.B$288.3B13.3B$288.3B12.3B$288.B14.3B$293.3B7.B$292.3B13.3B$292.3B12.3B$292.B14.3B$297.3B7.B$296.3B13.3B$296.3B12.3B$296.B14.3B$301.3B7.B$300.3B13.3B$300.3B12.3B$300.B14.3B$305.3B7.B$304.3B13.3B$304.3B12.3B$304.B14.3B$309.3B7.B$308.3B13.3B$308.3B12.3B$308.B14.3B$313.3B7.B$312.3B13.3B$312.3B12.3B$312.B14.3B$317.3B7.B$316.3B13.3B$316.3B12.3B$316.B14.3B$321.3B7.B$320.3B13.3B$320.3B12.3B$320.B14.3B$325.3B7.B$324.3B13.3B$324.3B12.3B$324.B14.3B$329.3B7.B$328.3B13.3B$328.3B12.3B$328.B14.3B$333.3B7.B$332.3B13.3B$332.3B12.3B$332.B14.3B$337.3B7.B$336.3B13.3B$336.3B12.3B$336.B14.3B$341.3B7.B$340.3B13.3B$340.3B12.3B$340.B14.3B$345.3B7.B$344.3B13.3B$344.3B12.3B$344.B14.3B$349.3B7.B$348.3B13.3B$348.3B12.3B$348.B14.3B$353.3B7.B$352.3B13.3B$352.3B12.3B$352.B14.3B$357.3B7.B$356.3B13.3B$356.3B12.3B$356.B14.3B$361.3B7.B$360.3B13.3B$360.3B12.3B$360.B14.3B$365.3B7.B$364.3B13.3B$364.3B12.3B$364.B14.3B$369.3B7.B$368.3B13.3B$368.3B12.3B$368.B14.3B$373.3B7.B$372.3B13.3B$372.3B12.3B$372.B14.3B$377.3B7.B$376.3B13.3B$376.3B12.3B$376.B14.3B$381.3B7.B$380.3B13.3B$380.3B12.3B$380.B14.3B$385.3B7.B$384.3B13.3B$384.3B12.3B$384.B14.3B$389.3B7.B$388.3B13.3B$388.3B12.3B$388.B14.3B$393.3B7.B$392.3B13.3B$392.3B12.3B$392.B14.3B$397.3B7.B$396.3B13.3B$396.3B12.3B$396.B14.3B$401.3B7.B$400.3B13.3B$400.3B12.3B$400.B14.3B$405.3B7.B$404.3B13.3B$404.3B12.3B$404.B14.3B$409.3B7.B$408.3B13.3B$408.3B12.3B$408.B14.3B$413.3B7.B$412.3B13.3B$412.3B12.3B$412.B14.3B$417.3B7.B$416.3B13.3B$416.3B12.3B$416.B14.3B$421.3B7.B$420.3B13.3B$420.3B12.3B$420.B14.3B$425.3B7.B$424.3B13.3B$424.3B12.3B$424.B14.3B$429.3B7.B$428.3B13.3B$428.3B12.3B$428.B14.3B$433.3B7.B$432.3B$432.3B$
432.B!

period54
Posts: 20
Joined: November 12th, 2014, 12:08 pm

### Re: Universality proof question

Yeah, also you can make huge and extremely messy puffers by combining "engines".

Code: Select all

x = 684, y = 684, rule = BTCA1
673.3B$672.3B$672.3B$672.B$669.3A5.3B$669.3A4.3B$669.3A4.3B$666.3B7.B$665.3B13.3B$665.3B12.3B$665.B14.3B$662.3A5.3B7.B$662.3A4.3B$662.3A4. 3B$659.3B7.B$658.3B$658.3B$658.B$655.3A5.3B$655.3A4.3B$655.3A4.3B$652.3B7.B$651.3B13.3B$651.3B12.3B$651.B14.3B$648.3A5.3B7.B$648.3A4.3B
$648.3A4.3B$645.3B7.B$644.3B$644.3B$644.B$641.3A5.3B$641.3A4.3B$641.
3A4.3B$638.3B7.B$637.3B13.3B$637.3B12.3B$637.B14.3B$634.3A5.3B7.B$
634.3A4.3B$634.3A4.3B$631.3B7.B$630.3B$630.3B$630.B$627.3A5.3B$627.3A 4.3B$627.3A4.3B$624.3B7.B$623.3B13.3B$623.3B12.3B$623.B14.3B$620.3A5. 3B7.B$620.3A4.3B$620.3A4.3B$617.3B7.B$616.3B$616.3B$616.B$613.3A5.3B$613.3A4.3B$613.3A4.3B$610.3B7.B$609.3B13.3B$609.3B12.3B$609.B14.3B$606.3A5.3B7.B$606.3A4.3B$606.3A4.3B$603.3B7.B$602.3B$602.3B$602.B$
599.3A5.3B$599.3A4.3B$599.3A4.3B$596.3B7.B$595.3B13.3B$595.3B12.3B$
595.B14.3B$592.3A5.3B7.B$592.3A4.3B$592.3A4.3B$589.3B7.B$588.3B$588.
3B$588.B$585.3A5.3B$585.3A4.3B$585.3A4.3B$582.3B7.B$581.3B13.3B$581. 3B12.3B$581.B14.3B$578.3A5.3B7.B$578.3A4.3B$578.3A4.3B$575.3B7.B$574. 3B$574.3B$574.B$571.3A5.3B$571.3A4.3B$571.3A4.3B$568.3B7.B$567.3B13.
3B$567.3B12.3B$567.B14.3B$564.3A5.3B7.B$564.3A4.3B$564.3A4.3B$561.3B
7.B$560.3B$560.3B$560.B$557.3A5.3B$557.3A4.3B$557.3A4.3B$554.3B7.B$
553.3B13.3B$553.3B12.3B$553.B14.3B$550.3A5.3B7.B$550.3A4.3B$550.3A4. 3B$547.3B7.B$546.3B$546.3B$546.B$543.3A5.3B$543.3A4.3B$543.3A4.3B$540.3B7.B$539.3B13.3B$539.3B12.3B$539.B14.3B$536.3A5.3B7.B$536.3A4.3B
$536.3A4.3B$533.3B7.B$532.3B$532.3B$532.B$529.3A5.3B$529.3A4.3B$529.
3A4.3B$526.3B7.B$525.3B13.3B$525.3B12.3B$525.B14.3B$522.3A5.3B7.B$
522.3A4.3B$522.3A4.3B$519.3B7.B$518.3B$518.3B$518.B$515.3A5.3B$515.3A 4.3B$515.3A4.3B$512.3B7.B$511.3B13.3B$511.3B12.3B$511.B14.3B$508.3A5. 3B7.B$508.3A4.3B$508.3A4.3B$505.3B7.B$504.3B$504.3B$504.B$501.3A5.3B$501.3A4.3B$501.3A4.3B$498.3B7.B$497.3B13.3B$497.3B12.3B$497.B14.3B$494.3A5.3B7.B$494.3A4.3B$494.3A4.3B$491.3B7.B$490.3B$490.3B$490.B$
487.3A5.3B$487.3A4.3B$487.3A4.3B$484.3B7.B$483.3B13.3B$483.3B12.3B$
483.B14.3B$480.3A5.3B7.B$480.3A4.3B$480.3A4.3B$477.3B7.B$476.3B$476.
3B$476.B$473.3A5.3B$473.3A4.3B$473.3A4.3B$470.3B7.B$469.3B13.3B$469. 3B12.3B$469.B14.3B$466.3A5.3B7.B$466.3A4.3B$466.3A4.3B$463.3B7.B$462. 3B$462.3B$462.B$459.3A5.3B$459.3A4.3B$459.3A4.3B$456.3B7.B$455.3B13.
3B$455.3B12.3B$455.B14.3B$452.3A5.3B7.B$452.3A4.3B$452.3A4.3B$449.3B
7.B$448.3B$448.3B$448.B$445.3A5.3B$445.3A4.3B$445.3A4.3B$442.3B7.B$
441.3B13.3B$441.3B12.3B$441.B14.3B$438.3A5.3B7.B$438.3A4.3B$438.3A4. 3B$435.3B7.B$434.3B$434.3B$434.B$431.3A5.3B$431.3A4.3B$431.3A4.3B$428.3B7.B$427.3B13.3B$427.3B12.3B$427.B14.3B$424.3A5.3B7.B$424.3A4.3B
$424.3A4.3B$421.3B7.B$420.3B$420.3B$420.B$417.3A5.3B$417.3A4.3B$417.
3A4.3B$414.3B7.B$413.3B13.3B$413.3B12.3B$413.B14.3B$410.3A5.3B7.B$
410.3A4.3B$410.3A4.3B$407.3B7.B$406.3B$406.3B$406.B$403.3A5.3B$403.3A 4.3B$403.3A4.3B$400.3B7.B$399.3B13.3B$399.3B12.3B$399.B14.3B$396.3A5. 3B7.B$396.3A4.3B$396.3A4.3B$393.3B7.B$392.3B$392.3B$392.B$389.3A5.3B$389.3A4.3B$389.3A4.3B$386.3B7.B$385.3B13.3B$385.3B12.3B$385.B14.3B$382.3A5.3B7.B$382.3A4.3B$382.3A4.3B$379.3B7.B$378.3B$378.3B$378.B$
375.3A5.3B$375.3A4.3B$375.3A4.3B$372.3B7.B$371.3B13.3B$371.3B12.3B$
371.B14.3B$368.3A5.3B7.B$368.3A4.3B$368.3A4.3B$365.3B7.B$364.3B$364.
3B$364.B$361.3A5.3B$361.3A4.3B$361.3A4.3B$358.3B7.B$357.3B13.3B$357. 3B12.3B$357.B14.3B$354.3A5.3B7.B$354.3A4.3B$354.3A4.3B$351.3B7.B$350. 3B$350.3B$350.B$347.3A5.3B$347.3A4.3B$347.3A4.3B$344.3B7.B$343.3B13.
3B$343.3B12.3B$343.B14.3B$340.3A5.3B7.B$340.3A4.3B$340.3A4.3B$337.3B
7.B$336.3B$336.3B$336.B$333.3A5.3B$333.3A4.3B$333.3A4.3B$330.3B7.B$
329.3B13.3B$329.3B12.3B$329.B14.3B$326.3A5.3B7.B$326.3A4.3B$326.3A4. 3B$323.3B7.B$322.3B$322.3B$322.B$319.3A5.3B$319.3A4.3B$319.3A4.3B$316.3B7.B$315.3B13.3B$315.3B12.3B$315.B14.3B$312.3A5.3B7.B$312.3A4.3B
$312.3A4.3B$309.3B7.B$308.3B$308.3B$308.B$305.3A5.3B$305.3A4.3B$305.
3A4.3B$302.3B7.B$301.3B13.3B$301.3B12.3B$301.B14.3B$298.3A5.3B7.B$
298.3A4.3B$298.3A4.3B$295.3B7.B$294.3B$294.3B$294.B$291.3A5.3B$291.3A 4.3B$291.3A4.3B$288.3B7.B$287.3B13.3B$287.3B12.3B$287.B14.3B$284.3A5. 3B7.B$284.3A4.3B$284.3A4.3B$281.3B7.B$280.3B$280.3B$280.B$277.3A5.3B$277.3A4.3B$277.3A4.3B$274.3B7.B$273.3B13.3B$273.3B12.3B$273.B14.3B$270.3A5.3B7.B$270.3A4.3B$270.3A4.3B$267.3B7.B$266.3B$266.3B$266.B$
263.3A5.3B$263.3A4.3B$263.3A4.3B$260.3B7.B$259.3B13.3B$259.3B12.3B$
259.B14.3B$256.3A5.3B7.B$256.3A4.3B$256.3A4.3B$253.3B7.B$252.3B$252.
3B$252.B$249.3A5.3B$249.3A4.3B$249.3A4.3B$246.3B7.B$245.3B13.3B$245. 3B12.3B$245.B14.3B$242.3A5.3B7.B$242.3A4.3B$242.3A4.3B$239.3B7.B$238. 3B$238.3B$238.B$235.3A5.3B$235.3A4.3B$235.3A4.3B$232.3B7.B$231.3B13.
3B$231.3B12.3B$231.B14.3B$228.3A5.3B7.B$228.3A4.3B$228.3A4.3B$225.3B
7.B$224.3B$224.3B$224.B$221.3A5.3B$221.3A4.3B$221.3A4.3B$218.3B7.B$
217.3B13.3B$217.3B12.3B$217.B14.3B$214.3A5.3B7.B$214.3A4.3B$214.3A4. 3B$211.3B7.B$210.3B$210.3B$210.B$207.3A5.3B$207.3A4.3B$207.3A4.3B$204.3B7.B$203.3B13.3B$203.3B12.3B$203.B14.3B$200.3A5.3B7.B$200.3A4.3B
$200.3A4.3B$197.3B7.B$196.3B$196.3B$196.B$193.3A5.3B$193.3A4.3B$193.
3A4.3B$190.3B7.B$189.3B13.3B$189.3B12.3B$189.B14.3B$186.3A5.3B7.B$
186.3A4.3B$186.3A4.3B$183.3B7.B$182.3B$182.3B$182.B$179.3A5.3B$179.3A 4.3B$179.3A4.3B$176.3B7.B$175.3B13.3B$175.3B12.3B$175.B14.3B$172.3A5. 3B7.B$172.3A4.3B$172.3A4.3B$169.3B7.B$168.3B$168.3B$168.B$165.3A5.3B$165.3A4.3B$165.3A4.3B$162.3B7.B$161.3B13.3B$161.3B12.3B$161.B14.3B$158.3A5.3B7.B$158.3A4.3B$158.3A4.3B$155.3B7.B$154.3B$154.3B$154.B$
151.3A5.3B$151.3A4.3B$151.3A4.3B$148.3B7.B$147.3B13.3B$147.3B12.3B$
147.B14.3B$144.3A5.3B7.B$144.3A4.3B$144.3A4.3B$141.3B7.B$140.3B$140.
3B$140.B$137.3A5.3B$137.3A4.3B$137.3A4.3B$134.3B7.B$133.3B13.3B$133. 3B12.3B$133.B14.3B$130.3A5.3B7.B$130.3A4.3B$130.3A4.3B$127.3B7.B$126. 3B$126.3B$126.B$123.3A5.3B$123.3A4.3B$123.3A4.3B$120.3B7.B$119.3B13.
3B$119.3B12.3B$119.B14.3B$116.3A5.3B7.B$116.3A4.3B$116.3A4.3B$113.3B
7.B$112.3B$112.3B$112.B$109.3A5.3B$109.3A4.3B$109.3A4.3B$106.3B7.B$
105.3B13.3B$105.3B12.3B$105.B14.3B$102.3A5.3B7.B$102.3A4.3B$102.3A4. 3B$99.3B7.B$98.3B$98.3B$98.B$95.3A5.3B$95.3A4.3B$95.3A4.3B$92.3B7.B$
91.3B13.3B$91.3B12.3B$91.B14.3B$88.3A5.3B7.B$88.3A4.3B$88.3A4.3B$85.
3B7.B$84.3B$84.3B$84.B$81.3A5.3B$81.3A4.3B$81.3A4.3B$78.3B7.B$77.3B
13.3B$77.3B12.3B$77.B14.3B$74.3A5.3B7.B$74.3A4.3B$74.3A4.3B$71.3B7.B$70.3B$70.3B$70.B$67.3A5.3B$67.3A4.3B$67.3A4.3B$64.3B7.B$63.3B13.3B$63.3B12.3B$63.B14.3B$60.3A5.3B7.B$60.3A4.3B$60.3A4.3B$57.3B7.B$56.3B$
56.3B$56.B$53.3A5.3B$53.3A4.3B$53.3A4.3B$50.3B7.B$49.3B13.3B$49.3B12. 3B$49.B14.3B$46.3A5.3B7.B$46.3A4.3B$46.3A4.3B$43.3B7.B$42.3B$42.3B$42.B$39.3A5.3B$39.3A4.3B$39.3A4.3B$36.3B7.B$35.3B13.3B$35.3B12.3B$35.
B14.3B$32.3A5.3B7.B$32.3A4.3B$32.3A4.3B$29.3B7.B$28.3B$28.3B$28.B$25.
3A5.3B$25.3A4.3B$25.3A4.3B$22.3B7.B$21.3B13.3B$21.3B12.3B$21.B14.3B$18.3A5.3B7.B$18.3A4.3B$18.3A4.3B$15.3B7.B$14.3B$14.3B$14.B$11.3A5.3B$11.3A4.3B$11.3A4.3B$8.3B7.B$7.3B13.3B$7.3B12.3B$7.B14.3B$4.3A5.3B7.B$
4.3A4.3B$4.3A4.3B$.3B7.B$3B$3B$B$5.3B$4.3B$4.3B$4.B$9.3B$8.3B$8.3B$8. B!  c0b0p0 Posts: 645 Joined: February 26th, 2014, 4:48 pm ### Re: Universality proof question I found a smoking ship that generates a fairly large spark. It is shown below deleting all the reflections and rotations of the seven most common oscillators and still lifes. Code: Select all x = 653, y = 236, rule = BTCA1 225.3B$134.3B87.3B$40.3B90.3B88.3B96.3B$39.3B91.3B88.B97.3B$39.3B91.B 87.3A5.3B90.3B85.3B$39.B56.A33.3A5.3B46.A33.3A4.3B91.B86.3B$36.3A5.3B 47.3A33.3A4.3B47.A.A31.3A4.3B88.3A5.3B79.3B$.A34.3A4.3B49.3A32.3A4.3B
47.A30.3B7.B56.3A31.3A4.3B80.B$.2A33.3A4.3B49.A31.3B7.B79.3B13.3B49.A 33.3A4.3B77.3A5.3B$A32.3B7.B82.3B13.3B72.3B12.3B50.A30.3B7.B44.2A33.
3A4.3B$32.3B13.3B75.3B12.3B73.B14.3B80.3B13.3B36.2A34.3A4.3B91.3B104. 3B$32.3B12.3B76.B14.3B70.3A5.3B7.B82.3B12.3B70.3B7.B92.3B104.3B$32.B 14.3B73.3A5.3B7.B72.3A4.3B13.3B75.B14.3B69.3B13.3B85.3B104.3B$29.3A5.
3B7.B75.3A4.3B13.3B65.3A4.3B12.3B73.3A5.3B7.B71.3B12.3B86.B106.B$29. 3A4.3B13.3B68.3A4.3B12.3B63.3B7.B14.3B73.3A4.3B13.3B64.B14.3B83.3A5. 3B63.A32.3A5.3B$29.3A4.3B12.3B66.3B7.B14.3B62.3B13.3B7.B75.3A4.3B12.
3B62.3A5.3B7.B53.2A30.3A4.3B64.A32.3A4.3B$26.3B7.B14.3B65.3B13.3B7.B 64.3B12.3B13.3B65.3B7.B14.3B62.3A4.3B13.3B46.2A30.3A4.3B63.3A31.3A4. 3B$25.3B13.3B7.B67.3B12.3B13.3B57.B14.3B12.3B65.3B13.3B7.B64.3A4.3B
12.3B76.3B7.B96.3B7.B$25.3B12.3B13.3B60.B14.3B12.3B63.3B7.B14.3B65.3B 12.3B13.3B54.3B7.B14.3B75.3B13.3B88.3B13.3B$25.B14.3B12.3B66.3B7.B14.
3B62.3B13.3B7.B67.B14.3B12.3B54.3B13.3B7.B77.3B12.3B89.3B12.3B$30.3B 7.B14.3B65.3B13.3B7.B64.3B12.3B13.3B65.3B7.B14.3B54.3B12.3B13.3B70.B 14.3B89.B14.3B$29.3B13.3B7.B67.3B12.3B13.3B57.B14.3B12.3B65.3B13.3B7.
B56.B14.3B12.3B68.3A5.3B7.B88.3A5.3B7.B$29.3B12.3B13.3B60.B14.3B12.3B 63.3B7.B14.3B65.3B12.3B13.3B54.3B7.B14.3B68.3A4.3B13.3B81.3A4.3B13.3B$29.B14.3B12.3B66.3B7.B14.3B62.3B13.3B7.B67.B14.3B12.3B54.3B13.3B7.B
70.3A4.3B12.3B82.3A4.3B12.3B$34.3B7.B14.3B65.3B13.3B7.B64.3B12.3B13. 3B65.3B7.B14.3B54.3B12.3B13.3B60.3B7.B14.3B79.3B7.B14.3B$33.3B13.3B7.
B67.3B12.3B13.3B57.B14.3B12.3B65.3B13.3B7.B56.B14.3B12.3B60.3B13.3B7.
B80.3B13.3B7.B$33.3B12.3B13.3B60.B14.3B12.3B63.3B7.B14.3B65.3B12.3B 13.3B54.3B7.B14.3B60.3B12.3B13.3B73.3B12.3B13.3B$33.B14.3B12.3B66.3B
7.B14.3B62.3B13.3B7.B67.B14.3B12.3B54.3B13.3B7.B62.B14.3B12.3B74.B14.
3B12.3B$38.3B7.B14.3B65.3B13.3B7.B64.3B12.3B13.3B65.3B7.B14.3B54.3B 12.3B13.3B60.3B7.B14.3B79.3B7.B14.3B$37.3B13.3B7.B67.3B12.3B13.3B57.B
14.3B12.3B65.3B13.3B7.B56.B14.3B12.3B60.3B13.3B7.B80.3B13.3B7.B$37.3B 12.3B13.3B60.B14.3B12.3B63.3B7.B14.3B65.3B12.3B13.3B54.3B7.B14.3B60. 3B12.3B13.3B73.3B12.3B13.3B$37.B14.3B12.3B66.3B7.B14.3B62.3B13.3B7.B
67.B14.3B12.3B54.3B13.3B7.B62.B14.3B12.3B74.B14.3B12.3B$42.3B7.B14.3B 65.3B13.3B7.B64.3B12.3B13.3B65.3B7.B14.3B54.3B12.3B13.3B60.3B7.B14.3B 79.3B7.B14.3B$41.3B13.3B7.B67.3B12.3B13.3B57.B14.3B12.3B65.3B13.3B7.B
56.B14.3B12.3B60.3B13.3B7.B80.3B13.3B7.B$41.3B12.3B13.3B60.B14.3B12. 3B63.3B7.B14.3B65.3B12.3B13.3B54.3B7.B14.3B60.3B12.3B13.3B73.3B12.3B 13.3B$41.B14.3B12.3B66.3B7.B14.3B62.3B13.3B7.B67.B14.3B12.3B54.3B13.
3B7.B62.B14.3B12.3B74.B14.3B12.3B$46.3B7.B14.3B65.3B13.3B7.B64.3B12. 3B81.3B7.B14.3B54.3B12.3B13.3B60.3B7.B14.3B79.3B7.B14.3B$45.3B13.3B7.
B67.3B12.3B73.B14.3B80.3B13.3B7.B56.B14.3B12.3B60.3B13.3B7.B80.3B13.
3B7.B$45.3B12.3B76.B14.3B78.3B7.B82.3B12.3B70.3B7.B14.3B60.3B12.3B13. 3B73.3B12.3B13.3B$45.B14.3B81.3B7.B79.3B13.3B75.B14.3B69.3B13.3B7.B
62.B14.3B12.3B74.B14.3B12.3B$50.3B7.B82.3B13.3B72.3B12.3B81.3B7.B71. 3B12.3B76.3B7.B14.3B79.3B7.B14.3B$49.3B13.3B75.3B12.3B73.B14.3B80.3B
13.3B64.B14.3B75.3B13.3B7.B80.3B13.3B7.B$49.3B12.3B76.B14.3B78.3B7.B 82.3B12.3B70.3B7.B77.3B12.3B13.3B73.3B12.3B13.3B$49.B14.3B81.3B7.B79.
3B13.3B75.B14.3B69.3B13.3B70.B14.3B12.3B74.B14.3B12.3B$54.3B7.B82.3B 13.3B72.3B12.3B81.3B7.B71.3B12.3B76.3B7.B14.3B79.3B7.B14.3B$53.3B13.
3B75.3B12.3B73.B14.3B80.3B13.3B64.B14.3B75.3B13.3B7.B80.3B13.3B7.B$53.3B12.3B76.B14.3B78.3B7.B82.3B12.3B70.3B7.B77.3B12.3B89.3B12.3B$53.
B14.3B81.3B7.B79.3B91.B14.3B69.3B13.3B70.B14.3B89.B14.3B$58.3B7.B82. 3B88.3B96.3B7.B71.3B12.3B76.3B7.B96.3B7.B$57.3B91.3B88.B97.3B80.B14.
3B75.3B13.3B88.3B13.3B$57.3B91.B188.3B85.3B7.B77.3B12.3B89.3B12.3B$
57.B282.B86.3B86.B14.3B89.B14.3B$427.3B91.3B7.B96.3B7.B$427.B92.3B13.
3B88.3B13.3B$520.3B12.3B89.3B12.3B$520.B14.3B89.B14.3B$525.3B7.B96.3B 7.B$524.3B104.3B$524.3B104.3B$524.B106.B120$47.3B$46.3B182.3B$46.3B 181.3B$46.B183.3B94.3B$43.3A5.3B176.B95.3B$6.A36.3A4.3B174.3A5.3B88.
3B82.3B$7.2A34.3A4.3B174.3A4.3B89.B83.3B$7.A32.3B7.B141.3A32.3A4.3B
86.3A5.3B76.3B205.3B$39.3B13.3B166.3B7.B52.A35.3A4.3B77.B206.3B$39.3B
12.3B136.A29.3B13.3B45.A35.3A4.3B74.3A5.3B199.3B$39.B14.3B166.3B12.3B 46.3A30.3B7.B76.3A4.3B200.B$36.3A5.3B7.B168.B14.3B78.3B13.3B39.A29.3A
4.3B163.A33.3A5.3B$36.3A4.3B13.3B158.3A5.3B7.B80.3B12.3B40.2A25.3B7.B 165.3A31.3A4.3B$36.3A4.3B12.3B159.3A4.3B13.3B73.B14.3B41.A24.3B13.3B
158.A33.3A4.3B$33.3B7.B14.3B159.3A4.3B12.3B71.3A5.3B7.B68.3B12.3B190. 3B7.B$32.3B13.3B7.B158.3B7.B14.3B71.3A4.3B13.3B61.B14.3B189.3B13.3B$32.3B12.3B13.3B150.3B13.3B7.B73.3A4.3B12.3B59.3A5.3B7.B191.3B12.3B$
32.B14.3B12.3B151.3B12.3B13.3B63.3B7.B14.3B59.3A4.3B13.3B184.B14.3B$37.3B7.B14.3B151.B14.3B12.3B63.3B13.3B7.B61.3A4.3B12.3B182.3A5.3B7.B$
36.3B13.3B7.B158.3B7.B14.3B63.3B12.3B13.3B51.3B7.B14.3B182.3A4.3B13.
3B$36.3B12.3B13.3B150.3B13.3B7.B65.B14.3B12.3B51.3B13.3B7.B184.3A4.3B 12.3B$36.B14.3B12.3B151.3B12.3B13.3B63.3B7.B14.3B51.3B12.3B13.3B174.
3B7.B14.3B$41.3B7.B14.3B151.B14.3B12.3B63.3B13.3B7.B53.B14.3B12.3B 174.3B13.3B7.B$40.3B13.3B7.B158.3B7.B14.3B63.3B12.3B13.3B51.3B7.B14.
3B174.3B12.3B13.3B$40.3B12.3B13.3B150.3B13.3B7.B65.B14.3B12.3B51.3B 13.3B7.B176.B14.3B12.3B$40.B14.3B12.3B151.3B12.3B13.3B63.3B7.B14.3B
51.3B12.3B13.3B174.3B7.B14.3B$45.3B7.B14.3B151.B14.3B12.3B63.3B13.3B 7.B53.B14.3B12.3B174.3B13.3B7.B$44.3B13.3B7.B158.3B7.B14.3B63.3B12.3B
13.3B51.3B7.B14.3B174.3B12.3B13.3B$44.3B12.3B13.3B150.3B13.3B7.B65.B 14.3B12.3B51.3B13.3B7.B176.B14.3B12.3B$44.B14.3B12.3B151.3B12.3B13.3B
63.3B7.B14.3B51.3B12.3B13.3B174.3B7.B14.3B$49.3B7.B14.3B151.B14.3B12. 3B63.3B13.3B7.B53.B14.3B12.3B174.3B13.3B7.B$48.3B13.3B7.B158.3B7.B14.
3B63.3B12.3B13.3B51.3B7.B14.3B174.3B12.3B13.3B$48.3B12.3B13.3B150.3B 13.3B7.B65.B14.3B12.3B51.3B13.3B7.B176.B14.3B12.3B$48.B14.3B12.3B151.
3B12.3B13.3B63.3B7.B14.3B51.3B12.3B13.3B174.3B7.B14.3B$53.3B7.B14.3B 151.B14.3B12.3B63.3B13.3B7.B53.B14.3B12.3B174.3B13.3B7.B$52.3B13.3B7.
B158.3B7.B14.3B63.3B12.3B13.3B51.3B7.B14.3B174.3B12.3B13.3B$52.3B12. 3B166.3B13.3B7.B65.B14.3B12.3B51.3B13.3B7.B176.B14.3B12.3B$52.B14.3B
166.3B12.3B79.3B7.B14.3B51.3B12.3B13.3B174.3B7.B14.3B$57.3B7.B168.B 14.3B78.3B13.3B7.B53.B14.3B12.3B174.3B13.3B7.B$56.3B13.3B166.3B7.B80.
3B12.3B67.3B7.B14.3B174.3B12.3B13.3B$56.3B12.3B166.3B13.3B73.B14.3B 66.3B13.3B7.B176.B14.3B12.3B$56.B14.3B166.3B12.3B79.3B7.B68.3B12.3B
190.3B7.B14.3B$61.3B7.B168.B14.3B78.3B13.3B61.B14.3B189.3B13.3B7.B$
60.3B13.3B166.3B7.B80.3B12.3B67.3B7.B191.3B12.3B$60.3B12.3B166.3B13. 3B73.B14.3B66.3B13.3B184.B14.3B$60.B14.3B166.3B12.3B79.3B7.B68.3B12.
3B190.3B7.B$65.3B7.B168.B14.3B78.3B13.3B61.B14.3B189.3B13.3B$64.3B
182.3B7.B80.3B12.3B67.3B7.B191.3B12.3B$64.3B181.3B89.B14.3B66.3B13.3B 184.B14.3B$64.B183.3B94.3B7.B68.3B12.3B190.3B7.B$248.B95.3B77.B14.3B 189.3B13.3B$344.3B82.3B7.B191.3B12.3B$344.B83.3B200.B14.3B$428.3B205.
3B7.B$428.B206.3B$635.3B$635.B!  bprentice Posts: 686 Joined: September 10th, 2009, 6:20 pm Location: Coos Bay, Oregon ### Re: Universality proof question c0b0p0 wrote:Here is a way to create puffers of arbitrarily high periods. c0b0p0 wrote:I found a smoking ship that generates a fairly large spark. Well done! By combining variations of your puffer and ship a large assortment of additional puffers and ships can easily be constructed. Some examples: Code: Select all x = 271, y = 261, rule = BTCA1 5.A$4.4A$5.5A$2.A.ABABABA$.3AB.2AB3A$.4A.5A$3ABA4.A$.ABA$.3A28$35.3A
$36.A$36.A4$52.A$50.3A$50.2ABA$49.2ABA$51.A12$110.2B$111.3A$107.A.5A
$107.9A$108.2A.5A$105.A3.5AB2A$106.A2.2BA.BABA$102.2A5.2BA2.ABA$103.
2A3.BABA3.A$102.4A2B3A12.A$67.3A30.B.2A.A3B2A11.4A$68.A31.B8A12.2AB AB2A$68.A32.5A15.8A$101.5AB13.3A2.4A$103.2AB2A13.2AB2.4A$103.3A2BA12. 3A3.A6.A6.A$105.3A13.3A3.A13.A$120.2AB2A8.3A$118.2A2BA6.2A4.A$118.2A BA6.3A2.A$113.A3.A2B2A5.4A.BA$111.7AB2A7.AB2.B$110.7ABA4.B2A3.A$109. 2ABA.B3A4.B3A8.A$110.3A4.A4.4A4.5A$110.AB2A8.3A7.A$111.4A17.A$96.A14. 6A3.A$96.A15.3A4.3A$95.3A16.A3.3ABA$118.3A3.A$100.A23.A$98.3A19.2B2.
3A$99.3A15.A.2A3.A$99.A15.A.A5.2A$117.2A2$153.2A$151.6A$149.8A$148. 3AB2A3BA$115.2A30.4A4.5A$99.3A44.2ABAB.6A.2A$100.A44.2A2BA2.AB3A3B2A
$100.A43.A2B2A2.B4AB4A$143.AB3A.A4.2A3BAB$142.2AB2A2.2AB.B2A2BA$141.
2AB2A7.6A.B3A$140.2A2BA6.10AB2A$139.5A.2A6.2A5.AB2A$139.3AB3.A14.A4. 2A$138.2AB3.B.B.A14.6A$138.3A.3A3.A12.A2.2A2B2A$137.4A.ABA.B3A12.3A
.B2A$137.3AB.8A14.A2.2A$138.2AB8A14.2A.A$138.2AB3A3B2A14.A$140.3ABA
2B2A14.2AB$141.A.BAB3A14.A2.B$128.A12.2AB2A2.A16.BA$128.A13.3AB.B2A$127.3A13.A3.A2BA.A23.AB$147.3A3.A21.A3B$132.A14.3A.A.A.4A16.3B$130. 3A18.5A.A7.AB9.B$131.3A17.2A4.B.B4.A3B12.AB$131.A18.2A2B.A2.BA4.3B12. A3B$150.2AB2A10.B13.3B$151.4A14.AB9.B$152.A15.A3B12.AB$168.3B12.A3B$169.B13.3B$173.AB9.B$172.A3B12.AB$131.3A38.3B12.A3B$132.A29.AB9.B13.
3B$132.A28.A3B12.AB9.B$161.3B12.A3B12.AB$162.B13.3B12.A3B$166.AB9.B
13.3B$165.A3B12.AB9.B$165.3B12.A3B12.AB$166.B13.3B12.A3B$170.AB9.B13.
3B$169.A3B12.AB9.B$169.3B12.A3B12.AB$170.B13.3B12.A3B$174.AB9.B13.3B
$173.A3B12.AB9.B$173.3B12.A3B12.AB$174.B13.3B12.A3B$160.A17.AB9.B13.
3B$160.A16.A3B12.AB9.B$159.3A15.3B12.A3B5.A6.AB$178.B13.3B5.4A3.A3B$164.A17.AB9.B7.A5.3B$162.3A16.A3B12.AB2.A3.A2.B$163.3A15.3B12.A3B5.
A$163.A18.B13.3B4.4A$186.AB9.B7.A$185.A3B5.A6.AB$185.3B5.4A3.A3B$186. B7.A5.3B$190.AB2.A3.A2.B$189.A3B5.A$189.3B4.4A$163.3A24.B7.A$164.A29.
AB$164.A28.A3B$193.3B$194.B8$209.A$207.5A$207.3ABA$206.4ABA$206.2A2B
2A$192.A13.7A$192.A12.5A.3A$191.3A11.4AB.B3A$204.5A.A.B$196.A13.3A$
194.3A13.3A$195.3A13.A$195.A2$204.3A$205.3A$205.A4$195.3A$196.A$196.
A9$223.A$220.5A$219.2A.2ABA$218.A2B3.3A$218.3A.B2.A$218.3A.B2.3A$218. 2AB4.A.A$225.B.B.A$223.3A.A$224.4A$224.3A9.B$230.A4.BA2B$229.3A2.BA 2B$225.B2.3ABA2.3B$224.BA2B.A3B2.B4.B$223.BA2B3.AB3A4.BA2B$224.3B5. 2A4.BA2B$224.B4.B2.A6.3B$228.BA2B7.B4.B$227.BA2B12.BA2B$228.3B11.BA 2B$228.B4.B9.3B$232.BA2B7.B4.B$231.BA2B12.BA2B$232.3B11.BA2B$232.B4.
B9.3B$236.BA2B7.B4.B$235.BA2B12.BA2B$236.3B11.BA2B$236.B4.B9.3B$240. BA2B7.B4.B$239.BA2B7.A4.BA2B$240.3B6.3A2.BA2B$240.B4.B2.3ABA2.3B$244. BA2B.A3B2.B4.B$243.BA2B3.ABABA4.BA2B$244.3B5.3BA2.BA2B$244.B4.B2.AB
ABA2.3B$248.BA2B.A3B2.B4.B$247.BA2B3.ABABA4.BA2B$248.3B5.3BA2.BA2B$
248.B4.B2.ABABA2.3B$252.BA2B.A3B2.B4.B$251.BA2B3.ABABA4.BA2B$252.3B 5.3BA2.BA2B$252.B4.B2.ABABA2.3B$256.BA2B.A3B2.B$255.BA2B3.AB3A$256. 3B5.2A$256.B4.B2.A$260.BA2B$259.BA2B$260.3B$260.B!


Code: Select all

x = 26, y = 26, rule = BTCA1
17.B$15.3B$15.3B$6.B7.3B$4.3B14.B$4.3B12.3B$3.3B13.3B$10.B7.3B$8.3B
14.B$8.3B12.3B$7.3B13.3B$14.B7.3B$12.3B4.3A$12.3B4.3A$3.B7.3B5.3A$. 3B14.B$.3B12.3B$3B13.3B$7.B7.3B$5.3B4.3A$5.3B4.3A$4.3B5.3A$11.B$9.3B$9.3B$8.3B!  Code: Select all x = 35, y = 35, rule = BTCA1 16.B$15.BA2B$14.BA2B$15.3B$15.B4.B$19.BA2B$18.BA2B$19.3B$19.B4.B$23.
BA2B$22.BA2B$13.B6.A2.3B$12.BA2B3.2A2.B4.B$11.BA2B3.5A4.BA2B$2.B9.3B 5.2A4.BA2B$.BA2B7.B4.B2.A6.3B$BA2B12.BA2B7.B4.B$.3B11.BA2B12.BA2B$. B4.B6.A2.3B11.BA2B$5.BA2B3.2A2.B4.B6.A2.3B$4.BA2B3.5A4.BA2B3.2A2.B$
5.3B5.2A4.BA2B3.5A$5.B4.B2.A6.3B5.2A$9.BA2B7.B4.B2.A$8.BA2B12.BA2B$
9.3B11.BA2B$9.B4.B6.A2.3B$13.BA2B3.2A2.B$12.BA2B3.5A$13.3B5.2A$13.B 4.B2.A$17.BA2B$16.BA2B$17.3B$17.B!  Brian Prentice c0b0p0 Posts: 645 Joined: February 26th, 2014, 4:48 pm ### Re: Universality proof question bprentice wrote: Code: Select all x = 26, y = 26, rule = BTCA1 17.B$15.3B$15.3B$6.B7.3B$4.3B14.B$4.3B12.3B$3.3B13.3B$10.B7.3B$8.3B 14.B$8.3B12.3B$7.3B13.3B$14.B7.3B$12.3B4.3A$12.3B4.3A$3.B7.3B5.3A$.
3B14.B$.3B12.3B$3B13.3B$7.B7.3B$5.3B4.3A$5.3B4.3A$4.3B5.3A$11.B$9.3B
$9.3B$8.3B!

It's unfortunate that nothing which can clean it is known! (Even the shortest-period known smoking ship, shown at the end of this post, is p96.)

Code: Select all

x = 46, y = 43, rule = BTCA1
37.A$35.3A$35.3A$26.A7.3A$24.3A14.A$24.3A12.3A$23.3A13.3A$30.A7.3A$
28.3A14.A$28.3A12.3A$27.3A13.3A$34.A7.3A$32.3A4.3B$32.3A4.3B$23.A7.3A
5.3B$21.3A14.A$21.3A12.3A$17.B2.3A13.3A$15.3B9.A7.3A$15.3B7.3A4.3B$6.
B7.3B8.3A4.3B$4.3B14.B2.3A5.3B$4.3B12.3B9.A$3.3B13.3B7.3A$10.B7.3B8.
3A$8.3B14.B2.3A$8.3B12.3B$7.3B13.3B$14.B7.3B$12.3B4.3A$12.3B4.3A$3.B 7.3B5.3A$.3B14.B$.3B12.3B$3B13.3B$7.B7.3B$5.3B4.3A$5.3B4.3A$4.3B5.3A$11.B$9.3B$9.3B$8.3B!

Here is an infinitely extensible puffer. Its debris is the same no matter how much one extends it.

Code: Select all

x = 315, y = 213, rule = BTCA1
14.A$12.3A$12.3A$3.A7.3A$.3A14.A$.3A12.3A$3A13.3A$7.A7.3A$5.3A14.A$5. 3A12.3A$4.3A13.3A$11.A7.3A$9.3A14.A$9.3A12.3A$8.3A13.3A$15.A7.3A$13.
3A14.A$13.3A12.3A$12.3A13.3A$19.A7.3A$17.3A4.3B$17.3A4.3B$16.3A5.3B$23.A$21.3A$21.3A$12.A7.3A67$156.A$154.3A$154.3A$145.A7.3A$143.3A14.A$
143.3A12.3A$142.3A13.3A$149.A7.3A$147.3A14.A$147.3A12.3A$146.3A13.3A$
153.A7.3A$151.3A14.A$151.3A12.3A$150.3A13.3A$157.A7.3A$155.3A14.A$
155.3A12.3A$154.3A13.3A$161.A7.3A$159.3A4.3B$159.3A4.3B$158.3A5.3B$
165.A$163.3A$163.3A$154.A7.3A67$298.A$296.3A$296.3A$287.A7.3A$285.3A
14.A$285.3A12.3A$284.3A13.3A$291.A7.3A$289.3A14.A$289.3A12.3A$288.3A
13.3A$295.A7.3A$293.3A14.A$293.3A12.3A$292.3A13.3A$299.A7.3A$297.3A
14.A$297.3A12.3A$296.3A13.3A$303.A7.3A$301.3A4.3B$301.3A4.3B$300.3A5.
3B$307.A$305.3A$305.3A$296.A7.3A!

Here is the p96 smoking ship.

Code: Select all

x = 200, y = 204, rule = BTCA1
14.A$12.3A$12.3A$3.A7.3A$.3A14.A$.3A12.3A$3A13.3A$7.A7.3A$5.3A14.A$5. 3A12.3A$4.3A13.3A$11.A7.3A$9.3A14.A$9.3A12.3A$8.3A13.3A$15.A7.3A$13.
3A14.A$13.3A12.3A$12.3A13.3A$19.A7.3A$17.3A14.A$17.3A12.3A$16.3A13.3A
$23.A7.3A$21.3A14.A$21.3A12.3A$20.3A13.3A$27.A7.3A$25.3A14.A$25.3A12. 3A$24.3A13.3A$31.A7.3A$29.3A4.3B$29.3A4.3B$28.3A5.3B$35.A$33.3A$33.3A$24.A7.3A127$171.A$169.3A$169.3A$160.A7.3A$158.3A14.A$158.3A12.3A$157.3A13.3A$164.A7.3A$162.3A14.A$162.3A12.3A$161.3A13.3A$168.A7.3A$166.3A14.A$166.3A12.3A$165.3A13.3A$172.A7.3A$170.3A14.A$170.3A12.3A$169.3A13.3A$176.A7.3A$174.3A14.A$174.3A12.3A$173.3A13.3A$180.A7.3A$178.3A14.A$178.3A12.3A$177.3A13.3A$184.A7.3A$182.3A14.A$182.3A12.3A$181.3A13.3A$188.A7.3A$186.3A4.3B$186.3A4.3B$185.3A5.3B$192.A$190.3A$
190.3A$181.A7.3A!  A for awesome Posts: 2076 Joined: September 13th, 2014, 5:36 pm Location: Pembina University, Home of the Gliders Contact: ### Re: Universality proof question A reaction that doubles the period of a gun: Code: Select all x = 85, y = 33, rule = BTCA1 27.A30.A$25.3A28.3A$26.3A28.3A$26.A30.A5$21.A40.A$21.3A38.3A$20.3A38. 3A$22.A6.B33.A$27.B.B.B22.3B$28.BAB22.2BA2B$26.2BA.A2B20.BABAB$28.BAB
22.2BA2B$27.B.B.B22.3B$29.B3$10.A64.A$8.3A62.3A$9.3A62.3A$9.A64.A2$.A 25.B28.B25.A$.3A10.B11.BA2B26.3B23.3A$3A9.B.B.B8.BA2B27.3B10.3B9.3A$
2.A10.BAB10.3B28.3B8.2BA2B10.A$11.2BA.A2B8.B41.BABAB$13.BAB52.2BA2B$12.B.B.B52.3B$14.B!

praosylen#5847 (Discord)

x₁=ηx
V*_η=c²√(Λη)
K=(Λu²)/2
Pₐ=1−1/(∫^∞_t₀(p(t)ˡ⁽ᵗ⁾)dt)

$$x_1=\eta x$$
$$V^*_\eta=c^2\sqrt{\Lambda\eta}$$
$$K=\frac{\Lambda u^2}2$$
$$P_a=1-\frac1{\int^\infty_{t_0}p(t)^{l(t)}dt}$$

period54
Posts: 20
Joined: November 12th, 2014, 12:08 pm

### Re: Universality proof question

Some glider-producing puffers (periods: 1024, 512, 256 and 128):

Code: Select all

x = 1471, y = 2376, rule = BTCA1
1352.3B$1351.3B$1351.3B$1351.B$1348.3A5.3B$1348.3A4.3B$1348.3A4.3B$1345.3B7.B$1344.3B13.3B$1344.3B12.3B$1344.B14.3B$1341.3A5.3B7.B$1341.
3A4.3B13.3B$1341.3A4.3B12.3B$1338.3B7.B14.3B$1337.3B13.3B7.B$1337.3B
12.3B13.3B$1337.B14.3B12.3B$1342.3B7.B14.3B$1341.3B13.3B7.B$1341.3B
12.3B13.3B$1341.B14.3B12.3B$1346.3B7.B14.3B$1345.3B13.3B7.B$1345.3B
12.3B13.3B$1345.B14.3B12.3B$1350.3B7.B14.3B$1349.3B13.3B7.B$1349.3B
12.3B13.3B$1349.B14.3B12.3B$1354.3B7.B14.3B$1353.3B13.3B7.B$1353.3B
12.3B13.3B$1353.B14.3B12.3B$1358.3B7.B14.3B$1357.3B13.3B7.B$1357.3B
12.3B13.3B$1357.B14.3B12.3B$1362.3B7.B14.3B$1361.3B13.3B7.B$1361.3B
12.3B13.3B$1361.B14.3B12.3B$1366.3B7.B14.3B$1365.3B13.3B7.B$1365.3B
12.3B13.3B$1365.B14.3B12.3B$1370.3B7.B14.3B$1369.3B13.3B7.B$1369.3B
12.3B13.3B$1369.B14.3B12.3B$1374.3B7.B14.3B$1373.3B13.3B7.B$1373.3B
12.3B13.3B$1373.B14.3B12.3B$1378.3B7.B14.3B$1377.3B13.3B7.B$1377.3B
12.3B13.3B$1377.B14.3B12.3B$1382.3B7.B14.3B$1381.3B13.3B7.B$1381.3B
12.3B13.3B$1381.B14.3B12.3B$1386.3B7.B14.3B$1385.3B13.3B7.B$1385.3B
12.3B13.3B$1385.B14.3B12.3B$1390.3B7.B14.3B$1389.3B13.3B7.B$1389.3B
12.3B13.3B$1389.B14.3B12.3B$1394.3B7.B14.3B$1393.3B13.3B7.B$1393.3B
12.3B13.3B$1393.B14.3B12.3B$1398.3B7.B14.3B$1397.3B13.3B7.B$1397.3B
12.3B13.3B$1397.B14.3B12.3B$1402.3B7.B14.3B$1401.3B13.3B7.B$1401.3B
12.3B13.3B$1401.B14.3B12.3B$1406.3B7.B14.3B$1405.3B13.3B7.B$1405.3B
12.3B13.3B$1405.B14.3B12.3B$1410.3B7.B14.3B$1409.3B13.3B7.B$1409.3B
12.3B13.3B$1409.B14.3B12.3B$1414.3B7.B14.3B$1413.3B13.3B7.B$1413.3B
12.3B13.3B$1413.B14.3B12.3B$1418.3B7.B14.3B$1417.3B13.3B7.B$1417.3B
12.3B13.3B$1417.B14.3B12.3B$1422.3B7.B14.3B$1421.3B13.3B7.B$1421.3B
12.3B13.3B$1421.B14.3B12.3B$1426.3B7.B14.3B$1425.3B13.3B7.B$1425.3B
12.3B13.3B$1425.B14.3B12.3B$1430.3B7.B14.3B$1429.3B13.3B7.B$1429.3B
12.3B13.3B$1429.B14.3B12.3B$1434.3B7.B14.3B$1433.3B13.3B7.B$1433.3B
12.3B13.3B$1433.B14.3B12.3B$1438.3B7.B14.3B$1437.3B13.3B7.B$1437.3B
12.3B13.3B$1437.B14.3B12.3B$1442.3B7.B14.3B$1441.3B13.3B7.B$1441.3B
12.3B$1441.B14.3B$1446.3B7.B$1445.3B$1445.3B$1445.B$1450.3B$1449.3B$
1449.3B$1449.B$1454.3B$1453.3B$1453.3B$1453.B141$1044.3B$1043.3B$
1043.3B$1043.B$1040.3A5.3B$1040.3A4.3B$1040.3A4.3B$1037.3B7.B$1036.3B
13.3B$1036.3B12.3B$1036.B14.3B$1033.3A5.3B7.B$1033.3A4.3B13.3B$1033. 3A4.3B12.3B$1030.3B7.B14.3B$1029.3B13.3B7.B$1029.3B12.3B13.3B$1029.B 14.3B12.3B$1034.3B7.B14.3B$1033.3B13.3B7.B$1033.3B12.3B13.3B$1033.B 14.3B12.3B$1038.3B7.B14.3B$1037.3B13.3B7.B$1037.3B12.3B13.3B$1037.B 14.3B12.3B$1042.3B7.B14.3B$1041.3B13.3B7.B$1041.3B12.3B13.3B$1041.B 14.3B12.3B$1046.3B7.B14.3B$1045.3B13.3B7.B$1045.3B12.3B13.3B$1045.B 14.3B12.3B$1050.3B7.B14.3B$1049.3B13.3B7.B$1049.3B12.3B13.3B$1049.B 14.3B12.3B$1054.3B7.B14.3B$1053.3B13.3B7.B$1053.3B12.3B13.3B$1053.B 14.3B12.3B$1058.3B7.B14.3B$1057.3B13.3B7.B$1057.3B12.3B13.3B$1057.B 14.3B12.3B$1062.3B7.B14.3B$1061.3B13.3B7.B$1061.3B12.3B13.3B$1061.B 14.3B12.3B$1066.3B7.B14.3B$1065.3B13.3B7.B$1065.3B12.3B13.3B$1065.B 14.3B12.3B$1070.3B7.B14.3B$1069.3B13.3B7.B$1069.3B12.3B13.3B$1069.B 14.3B12.3B$1074.3B7.B14.3B$1073.3B13.3B7.B$1073.3B12.3B13.3B$1073.B 14.3B12.3B$1078.3B7.B14.3B$1077.3B13.3B7.B$1077.3B12.3B13.3B$1077.B 14.3B12.3B$1082.3B7.B14.3B$1081.3B13.3B7.B$1081.3B12.3B13.3B$1081.B 14.3B12.3B$1086.3B7.B14.3B$1085.3B13.3B7.B$1085.3B12.3B13.3B$1085.B 14.3B12.3B$1090.3B7.B14.3B$1089.3B13.3B7.B$1089.3B12.3B13.3B$1089.B 14.3B12.3B$1094.3B7.B14.3B$1093.3B13.3B7.B$1093.3B12.3B13.3B$1093.B 14.3B12.3B$1098.3B7.B14.3B$1097.3B13.3B7.B$1097.3B12.3B13.3B$1097.B 14.3B12.3B$1102.3B7.B14.3B$1101.3B13.3B7.B$1101.3B12.3B13.3B$1101.B 14.3B12.3B$1106.3B7.B14.3B$1105.3B13.3B7.B$1105.3B12.3B13.3B$1105.B 14.3B12.3B$1110.3B7.B14.3B$1109.3B13.3B7.B$1109.3B12.3B13.3B$1109.B 14.3B12.3B$1114.3B7.B14.3B$1113.3B13.3B7.B$1113.3B12.3B13.3B$1113.B 14.3B12.3B$1118.3B7.B14.3B$1117.3B13.3B7.B$1117.3B12.3B13.3B$1117.B 14.3B12.3B$1122.3B7.B14.3B$1121.3B13.3B7.B$1121.3B12.3B13.3B$1121.B 14.3B12.3B$1126.3B7.B14.3B$1125.3B13.3B7.B$1125.3B12.3B13.3B$1125.B 14.3B12.3B$1130.3B7.B14.3B$1129.3B13.3B7.B$1129.3B12.3B13.3B$1129.B 14.3B12.3B$1134.3B7.B14.3B$1133.3B13.3B7.B$1133.3B12.3B13.3B$1133.B 14.3B12.3B$1138.3B7.B14.3B$1137.3B13.3B7.B$1137.3B12.3B13.3B$1137.B 14.3B12.3B$1142.3B7.B14.3B$1141.3B13.3B7.B$1141.3B12.3B13.3B$1141.B 14.3B12.3B$1146.3B7.B14.3B$1145.3B13.3B7.B$1145.3B12.3B13.3B$1145.B 14.3B12.3B$1150.3B7.B14.3B$1149.3B13.3B7.B$1149.3B12.3B13.3B$1149.B 14.3B12.3B$1154.3B7.B14.3B$1153.3B13.3B7.B$1153.3B12.3B13.3B$1153.B 14.3B12.3B$1158.3B7.B14.3B$1157.3B13.3B7.B$1157.3B12.3B13.3B$1157.B 14.3B12.3B$1162.3B7.B14.3B$1161.3B13.3B7.B$1161.3B12.3B13.3B$1161.B 14.3B12.3B$1166.3B7.B14.3B$1165.3B13.3B7.B$1165.3B12.3B13.3B$1165.B 14.3B12.3B$1170.3B7.B14.3B$1169.3B13.3B7.B$1169.3B12.3B13.3B$1169.B 14.3B12.3B$1174.3B7.B14.3B$1173.3B13.3B7.B$1173.3B12.3B13.3B$1173.B 14.3B12.3B$1178.3B7.B14.3B$1177.3B13.3B7.B$1177.3B12.3B13.3B$1177.B 14.3B12.3B$1182.3B7.B14.3B$1181.3B13.3B7.B$1181.3B12.3B13.3B$1181.B 14.3B12.3B$1186.3B7.B14.3B$1185.3B13.3B7.B$1185.3B12.3B13.3B$1185.B 14.3B12.3B$1190.3B7.B14.3B$1189.3B13.3B7.B$1189.3B12.3B13.3B$1189.B 14.3B12.3B$1194.3B7.B14.3B$1193.3B13.3B7.B$1193.3B12.3B13.3B$1193.B 14.3B12.3B$1198.3B7.B14.3B$1197.3B13.3B7.B$1197.3B12.3B13.3B$1197.B 14.3B12.3B$1202.3B7.B14.3B$1201.3B13.3B7.B$1201.3B12.3B13.3B$1201.B 14.3B12.3B$1206.3B7.B14.3B$1205.3B13.3B7.B$1205.3B12.3B13.3B$1205.B 14.3B12.3B$1210.3B7.B14.3B$1209.3B13.3B7.B$1209.3B12.3B13.3B$1209.B 14.3B12.3B$1214.3B7.B14.3B$1213.3B13.3B7.B$1213.3B12.3B13.3B$1213.B 14.3B12.3B$1218.3B7.B14.3B$1217.3B13.3B7.B$1217.3B12.3B13.3B$1217.B 14.3B12.3B$1222.3B7.B14.3B$1221.3B13.3B7.B$1221.3B12.3B13.3B$1221.B 14.3B12.3B$1226.3B7.B14.3B$1225.3B13.3B7.B$1225.3B12.3B13.3B$1225.B 14.3B12.3B$1230.3B7.B14.3B$1229.3B13.3B7.B$1229.3B12.3B13.3B$1229.B 14.3B12.3B$1234.3B7.B14.3B$1233.3B13.3B7.B$1233.3B12.3B13.3B$1233.B 14.3B12.3B$1238.3B7.B14.3B$1237.3B13.3B7.B$1237.3B12.3B13.3B$1237.B 14.3B12.3B$1242.3B7.B14.3B$1241.3B13.3B7.B$1241.3B12.3B13.3B$1241.B 14.3B12.3B$1246.3B7.B14.3B$1245.3B13.3B7.B$1245.3B12.3B13.3B$1245.B 14.3B12.3B$1250.3B7.B14.3B$1249.3B13.3B7.B$1249.3B12.3B13.3B$1249.B 14.3B12.3B$1254.3B7.B14.3B$1253.3B13.3B7.B$1253.3B12.3B13.3B$1253.B 14.3B12.3B$1258.3B7.B14.3B$1257.3B13.3B7.B$1257.3B12.3B13.3B$1257.B 14.3B12.3B$1262.3B7.B14.3B$1261.3B13.3B7.B$1261.3B12.3B$1261.B14.3B$
1266.3B7.B$1265.3B$1265.3B$1265.B$1270.3B$1269.3B$1269.3B$1269.B$
1274.3B$1273.3B$1273.3B$1273.B162$625.3B$624.3B$624.3B$624.B$621.3A5.
3B$621.3A4.3B$621.3A4.3B$618.3B7.B$617.3B13.3B$617.3B12.3B$617.B14.3B
$614.3A5.3B7.B$614.3A4.3B13.3B$614.3A4.3B12.3B$611.3B7.B14.3B$610.3B 13.3B7.B$610.3B12.3B13.3B$610.B14.3B12.3B$615.3B7.B14.3B$614.3B13.3B 7.B$614.3B12.3B13.3B$614.B14.3B12.3B$619.3B7.B14.3B$618.3B13.3B7.B$
618.3B12.3B13.3B$618.B14.3B12.3B$623.3B7.B14.3B$622.3B13.3B7.B$622.3B
12.3B13.3B$622.B14.3B12.3B$627.3B7.B14.3B$626.3B13.3B7.B$626.3B12.3B
13.3B$626.B14.3B12.3B$631.3B7.B14.3B$630.3B13.3B7.B$630.3B12.3B13.3B$630.B14.3B12.3B$635.3B7.B14.3B$634.3B13.3B7.B$634.3B12.3B13.3B$634.B 14.3B12.3B$639.3B7.B14.3B$638.3B13.3B7.B$638.3B12.3B13.3B$638.B14.3B 12.3B$643.3B7.B14.3B$642.3B13.3B7.B$642.3B12.3B13.3B$642.B14.3B12.3B$
647.3B7.B14.3B$646.3B13.3B7.B$646.3B12.3B13.3B$646.B14.3B12.3B$651.3B
7.B14.3B$650.3B13.3B7.B$650.3B12.3B13.3B$650.B14.3B12.3B$655.3B7.B14.
3B$654.3B13.3B7.B$654.3B12.3B13.3B$654.B14.3B12.3B$659.3B7.B14.3B$658.3B13.3B7.B$658.3B12.3B13.3B$658.B14.3B12.3B$663.3B7.B14.3B$662.3B 13.3B7.B$662.3B12.3B13.3B$662.B14.3B12.3B$667.3B7.B14.3B$666.3B13.3B 7.B$666.3B12.3B13.3B$666.B14.3B12.3B$671.3B7.B14.3B$670.3B13.3B7.B$
670.3B12.3B13.3B$670.B14.3B12.3B$675.3B7.B14.3B$674.3B13.3B7.B$674.3B
12.3B13.3B$674.B14.3B12.3B$679.3B7.B14.3B$678.3B13.3B7.B$678.3B12.3B
13.3B$678.B14.3B12.3B$683.3B7.B14.3B$682.3B13.3B7.B$682.3B12.3B13.3B$682.B14.3B12.3B$687.3B7.B14.3B$686.3B13.3B7.B$686.3B12.3B13.3B$686.B 14.3B12.3B$691.3B7.B14.3B$690.3B13.3B7.B$690.3B12.3B13.3B$690.B14.3B 12.3B$695.3B7.B14.3B$694.3B13.3B7.B$694.3B12.3B13.3B$694.B14.3B12.3B$
699.3B7.B14.3B$698.3B13.3B7.B$698.3B12.3B13.3B$698.B14.3B12.3B$703.3B
7.B14.3B$702.3B13.3B7.B$702.3B12.3B13.3B$702.B14.3B12.3B$707.3B7.B14.
3B$706.3B13.3B7.B$706.3B12.3B13.3B$706.B14.3B12.3B$711.3B7.B14.3B$710.3B13.3B7.B$710.3B12.3B13.3B$710.B14.3B12.3B$715.3B7.B14.3B$714.3B 13.3B7.B$714.3B12.3B13.3B$714.B14.3B12.3B$719.3B7.B14.3B$718.3B13.3B 7.B$718.3B12.3B13.3B$718.B14.3B12.3B$723.3B7.B14.3B$722.3B13.3B7.B$
722.3B12.3B13.3B$722.B14.3B12.3B$727.3B7.B14.3B$726.3B13.3B7.B$726.3B
12.3B13.3B$726.B14.3B12.3B$731.3B7.B14.3B$730.3B13.3B7.B$730.3B12.3B
13.3B$730.B14.3B12.3B$735.3B7.B14.3B$734.3B13.3B7.B$734.3B12.3B13.3B$734.B14.3B12.3B$739.3B7.B14.3B$738.3B13.3B7.B$738.3B12.3B13.3B$738.B 14.3B12.3B$743.3B7.B14.3B$742.3B13.3B7.B$742.3B12.3B13.3B$742.B14.3B 12.3B$747.3B7.B14.3B$746.3B13.3B7.B$746.3B12.3B13.3B$746.B14.3B12.3B$
751.3B7.B14.3B$750.3B13.3B7.B$750.3B12.3B13.3B$750.B14.3B12.3B$755.3B
7.B14.3B$754.3B13.3B7.B$754.3B12.3B13.3B$754.B14.3B12.3B$759.3B7.B14.
3B$758.3B13.3B7.B$758.3B12.3B13.3B$758.B14.3B12.3B$763.3B7.B14.3B$762.3B13.3B7.B$762.3B12.3B13.3B$762.B14.3B12.3B$767.3B7.B14.3B$766.3B 13.3B7.B$766.3B12.3B13.3B$766.B14.3B12.3B$771.3B7.B14.3B$770.3B13.3B 7.B$770.3B12.3B13.3B$770.B14.3B12.3B$775.3B7.B14.3B$774.3B13.3B7.B$
774.3B12.3B13.3B$774.B14.3B12.3B$779.3B7.B14.3B$778.3B13.3B7.B$778.3B
12.3B13.3B$778.B14.3B12.3B$783.3B7.B14.3B$782.3B13.3B7.B$782.3B12.3B
13.3B$782.B14.3B12.3B$787.3B7.B14.3B$786.3B13.3B7.B$786.3B12.3B13.3B$786.B14.3B12.3B$791.3B7.B14.3B$790.3B13.3B7.B$790.3B12.3B13.3B$790.B 14.3B12.3B$795.3B7.B14.3B$794.3B13.3B7.B$794.3B12.3B13.3B$794.B14.3B 12.3B$799.3B7.B14.3B$798.3B13.3B7.B$798.3B12.3B13.3B$798.B14.3B12.3B$
803.3B7.B14.3B$802.3B13.3B7.B$802.3B12.3B13.3B$802.B14.3B12.3B$807.3B
7.B14.3B$806.3B13.3B7.B$806.3B12.3B13.3B$806.B14.3B12.3B$811.3B7.B14.
3B$810.3B13.3B7.B$810.3B12.3B13.3B$810.B14.3B12.3B$815.3B7.B14.3B$814.3B13.3B7.B$814.3B12.3B13.3B$814.B14.3B12.3B$819.3B7.B14.3B$818.3B 13.3B7.B$818.3B12.3B13.3B$818.B14.3B12.3B$823.3B7.B14.3B$822.3B13.3B 7.B$822.3B12.3B13.3B$822.B14.3B12.3B$827.3B7.B14.3B$826.3B13.3B7.B$
826.3B12.3B13.3B$826.B14.3B12.3B$831.3B7.B14.3B$830.3B13.3B7.B$830.3B
12.3B13.3B$830.B14.3B12.3B$835.3B7.B14.3B$834.3B13.3B7.B$834.3B12.3B
13.3B$834.B14.3B12.3B$839.3B7.B14.3B$838.3B13.3B7.B$838.3B12.3B13.3B$838.B14.3B12.3B$843.3B7.B14.3B$842.3B13.3B7.B$842.3B12.3B13.3B$842.B 14.3B12.3B$847.3B7.B14.3B$846.3B13.3B7.B$846.3B12.3B13.3B$846.B14.3B 12.3B$851.3B7.B14.3B$850.3B13.3B7.B$850.3B12.3B13.3B$850.B14.3B12.3B$
855.3B7.B14.3B$854.3B13.3B7.B$854.3B12.3B13.3B$854.B14.3B12.3B$859.3B
7.B14.3B$858.3B13.3B7.B$858.3B12.3B13.3B$858.B14.3B12.3B$863.3B7.B14.
3B$862.3B13.3B7.B$862.3B12.3B13.3B$862.B14.3B12.3B$867.3B7.B14.3B$866.3B13.3B7.B$866.3B12.3B13.3B$866.B14.3B12.3B$871.3B7.B14.3B$870.3B 13.3B7.B$870.3B12.3B13.3B$870.B14.3B12.3B$875.3B7.B14.3B$874.3B13.3B 7.B$874.3B12.3B13.3B$874.B14.3B12.3B$879.3B7.B14.3B$878.3B13.3B7.B$
878.3B12.3B13.3B$878.B14.3B12.3B$883.3B7.B14.3B$882.3B13.3B7.B$882.3B
12.3B13.3B$882.B14.3B12.3B$887.3B7.B14.3B$886.3B13.3B7.B$886.3B12.3B
13.3B$886.B14.3B12.3B$891.3B7.B14.3B$890.3B13.3B7.B$890.3B12.3B13.3B$890.B14.3B12.3B$895.3B7.B14.3B$894.3B13.3B7.B$894.3B12.3B13.3B$894.B 14.3B12.3B$899.3B7.B14.3B$898.3B13.3B7.B$898.3B12.3B13.3B$898.B14.3B 12.3B$903.3B7.B14.3B$902.3B13.3B7.B$902.3B12.3B13.3B$902.B14.3B12.3B$
907.3B7.B14.3B$906.3B13.3B7.B$906.3B12.3B13.3B$906.B14.3B12.3B$911.3B
7.B14.3B$910.3B13.3B7.B$910.3B12.3B13.3B$910.B14.3B12.3B$915.3B7.B14.
3B$914.3B13.3B7.B$914.3B12.3B13.3B$914.B14.3B12.3B$919.3B7.B14.3B$918.3B13.3B7.B$918.3B12.3B13.3B$918.B14.3B12.3B$923.3B7.B14.3B$922.3B 13.3B7.B$922.3B12.3B13.3B$922.B14.3B12.3B$927.3B7.B14.3B$926.3B13.3B 7.B$926.3B12.3B13.3B$926.B14.3B12.3B$931.3B7.B14.3B$930.3B13.3B7.B$
930.3B12.3B13.3B$930.B14.3B12.3B$935.3B7.B14.3B$934.3B13.3B7.B$934.3B
12.3B13.3B$934.B14.3B12.3B$939.3B7.B14.3B$938.3B13.3B7.B$938.3B12.3B
13.3B$938.B14.3B12.3B$943.3B7.B14.3B$942.3B13.3B7.B$942.3B12.3B13.3B$942.B14.3B12.3B$947.3B7.B14.3B$946.3B13.3B7.B$946.3B12.3B13.3B$946.B 14.3B12.3B$951.3B7.B14.3B$950.3B13.3B7.B$950.3B12.3B13.3B$950.B14.3B 12.3B$955.3B7.B14.3B$954.3B13.3B7.B$954.3B12.3B13.3B$954.B14.3B12.3B$
959.3B7.B14.3B$958.3B13.3B7.B$958.3B12.3B13.3B$958.B14.3B12.3B$963.3B
7.B14.3B$962.3B13.3B7.B$962.3B12.3B13.3B$962.B14.3B12.3B$967.3B7.B14.
3B$966.3B13.3B7.B$966.3B12.3B13.3B$966.B14.3B12.3B$971.3B7.B14.3B$970.3B13.3B7.B$970.3B12.3B13.3B$970.B14.3B12.3B$975.3B7.B14.3B$974.3B 13.3B7.B$974.3B12.3B13.3B$974.B14.3B12.3B$979.3B7.B14.3B$978.3B13.3B 7.B$978.3B12.3B13.3B$978.B14.3B12.3B$983.3B7.B14.3B$982.3B13.3B7.B$
982.3B12.3B13.3B$982.B14.3B12.3B$987.3B7.B14.3B$986.3B13.3B7.B$986.3B
12.3B13.3B$986.B14.3B12.3B$991.3B7.B14.3B$990.3B13.3B7.B$990.3B12.3B
13.3B$990.B14.3B12.3B$995.3B7.B14.3B$994.3B13.3B7.B$994.3B12.3B13.3B$994.B14.3B12.3B$999.3B7.B14.3B$998.3B13.3B7.B$998.3B12.3B13.3B$998.B 14.3B12.3B$1003.3B7.B14.3B$1002.3B13.3B7.B$1002.3B12.3B13.3B$1002.B 14.3B12.3B$1007.3B7.B14.3B$1006.3B13.3B7.B$1006.3B12.3B13.3B$1006.B 14.3B12.3B$1011.3B7.B14.3B$1010.3B13.3B7.B$1010.3B12.3B13.3B$1010.B 14.3B12.3B$1015.3B7.B14.3B$1014.3B13.3B7.B$1014.3B12.3B13.3B$1014.B 14.3B12.3B$1019.3B7.B14.3B$1018.3B13.3B7.B$1018.3B12.3B13.3B$1018.B 14.3B12.3B$1023.3B7.B14.3B$1022.3B13.3B7.B$1022.3B12.3B13.3B$1022.B 14.3B12.3B$1027.3B7.B14.3B$1026.3B13.3B7.B$1026.3B12.3B13.3B$1026.B 14.3B12.3B$1031.3B7.B14.3B$1030.3B13.3B7.B$1030.3B12.3B13.3B$1030.B 14.3B12.3B$1035.3B7.B14.3B$1034.3B13.3B7.B$1034.3B12.3B13.3B$1034.B 14.3B12.3B$1039.3B7.B14.3B$1038.3B13.3B7.B$1038.3B12.3B13.3B$1038.B 14.3B12.3B$1043.3B7.B14.3B$1042.3B13.3B7.B$1042.3B12.3B13.3B$1042.B 14.3B12.3B$1047.3B7.B14.3B$1046.3B13.3B7.B$1046.3B12.3B13.3B$1046.B 14.3B12.3B$1051.3B7.B14.3B$1050.3B13.3B7.B$1050.3B12.3B13.3B$1050.B 14.3B12.3B$1055.3B7.B14.3B$1054.3B13.3B7.B$1054.3B12.3B13.3B$1054.B 14.3B12.3B$1059.3B7.B14.3B$1058.3B13.3B7.B$1058.3B12.3B13.3B$1058.B 14.3B12.3B$1063.3B7.B14.3B$1062.3B13.3B7.B$1062.3B12.3B13.3B$1062.B 14.3B12.3B$1067.3B7.B14.3B$1066.3B13.3B7.B$1066.3B12.3B13.3B$1066.B 14.3B12.3B$1071.3B7.B14.3B$1070.3B13.3B7.B$1070.3B12.3B13.3B$1070.B 14.3B12.3B$1075.3B7.B14.3B$1074.3B13.3B7.B$1074.3B12.3B13.3B$1074.B 14.3B12.3B$1079.3B7.B14.3B$1078.3B13.3B7.B$1078.3B12.3B13.3B$1078.B 14.3B12.3B$1083.3B7.B14.3B$1082.3B13.3B7.B$1082.3B12.3B13.3B$1082.B 14.3B12.3B$1087.3B7.B14.3B$1086.3B13.3B7.B$1086.3B12.3B13.3B$1086.B 14.3B12.3B$1091.3B7.B14.3B$1090.3B13.3B7.B$1090.3B12.3B13.3B$1090.B 14.3B12.3B$1095.3B7.B14.3B$1094.3B13.3B7.B$1094.3B12.3B13.3B$1094.B 14.3B12.3B$1099.3B7.B14.3B$1098.3B13.3B7.B$1098.3B12.3B$1098.B14.3B$
1103.3B7.B$1102.3B$1102.3B$1102.B$1107.3B$1106.3B$1106.3B$1106.B$
1111.3B$1110.3B$1110.3B$1110.B132$15.3B$14.3B$14.3B$14.B$11.3A5.3B$11.3A4.3B$11.3A4.3B$8.3B7.B$7.3B13.3B$7.3B12.3B$7.B14.3B$4.3A5.3B7.B$
4.3A4.3B13.3B$4.3A4.3B12.3B$.3B7.B14.3B$3B13.3B7.B$3B12.3B13.3B$B14. 3B12.3B$5.3B7.B14.3B$4.3B13.3B7.B$4.3B12.3B13.3B$4.B14.3B12.3B$9.3B7.
B14.3B$8.3B13.3B7.B$8.3B12.3B13.3B$8.B14.3B12.3B$13.3B7.B14.3B$12.3B 13.3B7.B$12.3B12.3B13.3B$12.B14.3B12.3B$17.3B7.B14.3B$16.3B13.3B7.B$
16.3B12.3B13.3B$16.B14.3B12.3B$21.3B7.B14.3B$20.3B13.3B7.B$20.3B12.3B
13.3B$20.B14.3B12.3B$25.3B7.B14.3B$24.3B13.3B7.B$24.3B12.3B13.3B$24.B 14.3B12.3B$29.3B7.B14.3B$28.3B13.3B7.B$28.3B12.3B13.3B$28.B14.3B12.3B$33.3B7.B14.3B$32.3B13.3B7.B$32.3B12.3B13.3B$32.B14.3B12.3B$37.3B7.B
14.3B$36.3B13.3B7.B$36.3B12.3B13.3B$36.B14.3B12.3B$41.3B7.B14.3B$40. 3B13.3B7.B$40.3B12.3B13.3B$40.B14.3B12.3B$45.3B7.B14.3B$44.3B13.3B7.B$44.3B12.3B13.3B$44.B14.3B12.3B$49.3B7.B14.3B$48.3B13.3B7.B$48.3B12.
3B13.3B$48.B14.3B12.3B$53.3B7.B14.3B$52.3B13.3B7.B$52.3B12.3B13.3B$52.B14.3B12.3B$57.3B7.B14.3B$56.3B13.3B7.B$56.3B12.3B13.3B$56.B14.3B 12.3B$61.3B7.B14.3B$60.3B13.3B7.B$60.3B12.3B13.3B$60.B14.3B12.3B$65.
3B7.B14.3B$64.3B13.3B7.B$64.3B12.3B13.3B$64.B14.3B12.3B$69.3B7.B14.3B
$68.3B13.3B7.B$68.3B12.3B13.3B$68.B14.3B12.3B$73.3B7.B14.3B$72.3B13. 3B7.B$72.3B12.3B13.3B$72.B14.3B12.3B$77.3B7.B14.3B$76.3B13.3B7.B$76.
3B12.3B13.3B$76.B14.3B12.3B$81.3B7.B14.3B$80.3B13.3B7.B$80.3B12.3B13.
3B$80.B14.3B12.3B$85.3B7.B14.3B$84.3B13.3B7.B$84.3B12.3B13.3B$84.B14. 3B12.3B$89.3B7.B14.3B$88.3B13.3B7.B$88.3B12.3B13.3B$88.B14.3B12.3B$
93.3B7.B14.3B$92.3B13.3B7.B$92.3B12.3B13.3B$92.B14.3B12.3B$97.3B7.B
14.3B$96.3B13.3B7.B$96.3B12.3B13.3B$96.B14.3B12.3B$101.3B7.B14.3B$100.3B13.3B7.B$100.3B12.3B13.3B$100.B14.3B12.3B$105.3B7.B14.3B$104.3B 13.3B7.B$104.3B12.3B13.3B$104.B14.3B12.3B$109.3B7.B14.3B$108.3B13.3B 7.B$108.3B12.3B13.3B$108.B14.3B12.3B$113.3B7.B14.3B$112.3B13.3B7.B$
112.3B12.3B13.3B$112.B14.3B12.3B$117.3B7.B14.3B$116.3B13.3B7.B$116.3B
12.3B13.3B$116.B14.3B12.3B$121.3B7.B14.3B$120.3B13.3B7.B$120.3B12.3B
13.3B$120.B14.3B12.3B$125.3B7.B14.3B$124.3B13.3B7.B$124.3B12.3B13.3B$124.B14.3B12.3B$129.3B7.B14.3B$128.3B13.3B7.B$128.3B12.3B13.3B$128.B 14.3B12.3B$133.3B7.B14.3B$132.3B13.3B7.B$132.3B12.3B13.3B$132.B14.3B 12.3B$137.3B7.B14.3B$136.3B13.3B7.B$136.3B12.3B13.3B$136.B14.3B12.3B$
141.3B7.B14.3B$140.3B13.3B7.B$140.3B12.3B13.3B$140.B14.3B12.3B$145.3B
7.B14.3B$144.3B13.3B7.B$144.3B12.3B13.3B$144.B14.3B12.3B$149.3B7.B14.
3B$148.3B13.3B7.B$148.3B12.3B13.3B$148.B14.3B12.3B$153.3B7.B14.3B$152.3B13.3B7.B$152.3B12.3B13.3B$152.B14.3B12.3B$157.3B7.B14.3B$156.3B 13.3B7.B$156.3B12.3B13.3B$156.B14.3B12.3B$161.3B7.B14.3B$160.3B13.3B 7.B$160.3B12.3B13.3B$160.B14.3B12.3B$165.3B7.B14.3B$164.3B13.3B7.B$
164.3B12.3B13.3B$164.B14.3B12.3B$169.3B7.B14.3B$168.3B13.3B7.B$168.3B
12.3B13.3B$168.B14.3B12.3B$173.3B7.B14.3B$172.3B13.3B7.B$172.3B12.3B
13.3B$172.B14.3B12.3B$177.3B7.B14.3B$176.3B13.3B7.B$176.3B12.3B13.3B$176.B14.3B12.3B$181.3B7.B14.3B$180.3B13.3B7.B$180.3B12.3B13.3B$180.B 14.3B12.3B$185.3B7.B14.3B$184.3B13.3B7.B$184.3B12.3B13.3B$184.B14.3B 12.3B$189.3B7.B14.3B$188.3B13.3B7.B$188.3B12.3B13.3B$188.B14.3B12.3B$
193.3B7.B14.3B$192.3B13.3B7.B$192.3B12.3B13.3B$192.B14.3B12.3B$197.3B
7.B14.3B$196.3B13.3B7.B$196.3B12.3B13.3B$196.B14.3B12.3B$201.3B7.B14.
3B$200.3B13.3B7.B$200.3B12.3B13.3B$200.B14.3B12.3B$205.3B7.B14.3B$204.3B13.3B7.B$204.3B12.3B13.3B$204.B14.3B12.3B$209.3B7.B14.3B$208.3B 13.3B7.B$208.3B12.3B13.3B$208.B14.3B12.3B$213.3B7.B14.3B$212.3B13.3B 7.B$212.3B12.3B13.3B$212.B14.3B12.3B$217.3B7.B14.3B$216.3B13.3B7.B$
216.3B12.3B13.3B$216.B14.3B12.3B$221.3B7.B14.3B$220.3B13.3B7.B$220.3B
12.3B13.3B$220.B14.3B12.3B$225.3B7.B14.3B$224.3B13.3B7.B$224.3B12.3B
13.3B$224.B14.3B12.3B$229.3B7.B14.3B$228.3B13.3B7.B$228.3B12.3B13.3B$228.B14.3B12.3B$233.3B7.B14.3B$232.3B13.3B7.B$232.3B12.3B13.3B$232.B 14.3B12.3B$237.3B7.B14.3B$236.3B13.3B7.B$236.3B12.3B13.3B$236.B14.3B 12.3B$241.3B7.B14.3B$240.3B13.3B7.B$240.3B12.3B13.3B$240.B14.3B12.3B$
245.3B7.B14.3B$244.3B13.3B7.B$244.3B12.3B13.3B$244.B14.3B12.3B$249.3B
7.B14.3B$248.3B13.3B7.B$248.3B12.3B13.3B$248.B14.3B12.3B$253.3B7.B14.
3B$252.3B13.3B7.B$252.3B12.3B13.3B$252.B14.3B12.3B$257.3B7.B14.3B$256.3B13.3B7.B$256.3B12.3B13.3B$256.B14.3B12.3B$261.3B7.B14.3B$260.3B 13.3B7.B$260.3B12.3B13.3B$260.B14.3B12.3B$265.3B7.B14.3B$264.3B13.3B 7.B$264.3B12.3B13.3B$264.B14.3B12.3B$269.3B7.B14.3B$268.3B13.3B7.B$
268.3B12.3B13.3B$268.B14.3B12.3B$273.3B7.B14.3B$272.3B13.3B7.B$272.3B
12.3B13.3B$272.B14.3B12.3B$277.3B7.B14.3B$276.3B13.3B7.B$276.3B12.3B
13.3B$276.B14.3B12.3B$281.3B7.B14.3B$280.3B13.3B7.B$280.3B12.3B13.3B$280.B14.3B12.3B$285.3B7.B14.3B$284.3B13.3B7.B$284.3B12.3B13.3B$284.B 14.3B12.3B$289.3B7.B14.3B$288.3B13.3B7.B$288.3B12.3B13.3B$288.B14.3B 12.3B$293.3B7.B14.3B$292.3B13.3B7.B$292.3B12.3B13.3B$292.B14.3B12.3B$
297.3B7.B14.3B$296.3B13.3B7.B$296.3B12.3B13.3B$296.B14.3B12.3B$301.3B
7.B14.3B$300.3B13.3B7.B$300.3B12.3B13.3B$300.B14.3B12.3B$305.3B7.B14.
3B$304.3B13.3B7.B$304.3B12.3B13.3B$304.B14.3B12.3B$309.3B7.B14.3B$308.3B13.3B7.B$308.3B12.3B13.3B$308.B14.3B12.3B$313.3B7.B14.3B$312.3B 13.3B7.B$312.3B12.3B13.3B$312.B14.3B12.3B$317.3B7.B14.3B$316.3B13.3B 7.B$316.3B12.3B13.3B$316.B14.3B12.3B$321.3B7.B14.3B$320.3B13.3B7.B$
320.3B12.3B13.3B$320.B14.3B12.3B$325.3B7.B14.3B$324.3B13.3B7.B$324.3B
12.3B13.3B$324.B14.3B12.3B$329.3B7.B14.3B$328.3B13.3B7.B$328.3B12.3B
13.3B$328.B14.3B12.3B$333.3B7.B14.3B$332.3B13.3B7.B$332.3B12.3B13.3B$332.B14.3B12.3B$337.3B7.B14.3B$336.3B13.3B7.B$336.3B12.3B13.3B$336.B 14.3B12.3B$341.3B7.B14.3B$340.3B13.3B7.B$340.3B12.3B13.3B$340.B14.3B 12.3B$345.3B7.B14.3B$344.3B13.3B7.B$344.3B12.3B13.3B$344.B14.3B12.3B$
349.3B7.B14.3B$348.3B13.3B7.B$348.3B12.3B13.3B$348.B14.3B12.3B$353.3B
7.B14.3B$352.3B13.3B7.B$352.3B12.3B13.3B$352.B14.3B12.3B$357.3B7.B14.
3B$356.3B13.3B7.B$356.3B12.3B13.3B$356.B14.3B12.3B$361.3B7.B14.3B$360.3B13.3B7.B$360.3B12.3B13.3B$360.B14.3B12.3B$365.3B7.B14.3B$364.3B 13.3B7.B$364.3B12.3B13.3B$364.B14.3B12.3B$369.3B7.B14.3B$368.3B13.3B 7.B$368.3B12.3B13.3B$368.B14.3B12.3B$373.3B7.B14.3B$372.3B13.3B7.B$
372.3B12.3B13.3B$372.B14.3B12.3B$377.3B7.B14.3B$376.3B13.3B7.B$376.3B
12.3B13.3B$376.B14.3B12.3B$381.3B7.B14.3B$380.3B13.3B7.B$380.3B12.3B
13.3B$380.B14.3B12.3B$385.3B7.B14.3B$384.3B13.3B7.B$384.3B12.3B13.3B$384.B14.3B12.3B$389.3B7.B14.3B$388.3B13.3B7.B$388.3B12.3B13.3B$388.B 14.3B12.3B$393.3B7.B14.3B$392.3B13.3B7.B$392.3B12.3B13.3B$392.B14.3B 12.3B$397.3B7.B14.3B$396.3B13.3B7.B$396.3B12.3B13.3B$396.B14.3B12.3B$
401.3B7.B14.3B$400.3B13.3B7.B$400.3B12.3B13.3B$400.B14.3B12.3B$405.3B
7.B14.3B$404.3B13.3B7.B$404.3B12.3B13.3B$404.B14.3B12.3B$409.3B7.B14.
3B$408.3B13.3B7.B$408.3B12.3B13.3B$408.B14.3B12.3B$413.3B7.B14.3B$412.3B13.3B7.B$412.3B12.3B13.3B$412.B14.3B12.3B$417.3B7.B14.3B$416.3B 13.3B7.B$416.3B12.3B13.3B$416.B14.3B12.3B$421.3B7.B14.3B$420.3B13.3B 7.B$420.3B12.3B13.3B$420.B14.3B12.3B$425.3B7.B14.3B$424.3B13.3B7.B$
424.3B12.3B13.3B$424.B14.3B12.3B$429.3B7.B14.3B$428.3B13.3B7.B$428.3B
12.3B13.3B$428.B14.3B12.3B$433.3B7.B14.3B$432.3B13.3B7.B$432.3B12.3B
13.3B$432.B14.3B12.3B$437.3B7.B14.3B$436.3B13.3B7.B$436.3B12.3B13.3B$436.B14.3B12.3B$441.3B7.B14.3B$440.3B13.3B7.B$440.3B12.3B13.3B$440.B 14.3B12.3B$445.3B7.B14.3B$444.3B13.3B7.B$444.3B12.3B13.3B$444.B14.3B 12.3B$449.3B7.B14.3B$448.3B13.3B7.B$448.3B12.3B13.3B$448.B14.3B12.3B$
453.3B7.B14.3B$452.3B13.3B7.B$452.3B12.3B13.3B$452.B14.3B12.3B$457.3B
7.B14.3B$456.3B13.3B7.B$456.3B12.3B13.3B$456.B14.3B12.3B$461.3B7.B14.
3B$460.3B13.3B7.B$460.3B12.3B13.3B$460.B14.3B12.3B$465.3B7.B14.3B$464.3B13.3B7.B$464.3B12.3B13.3B$464.B14.3B12.3B$469.3B7.B14.3B$468.3B 13.3B7.B$468.3B12.3B13.3B$468.B14.3B12.3B$473.3B7.B14.3B$472.3B13.3B 7.B$472.3B12.3B13.3B$472.B14.3B12.3B$477.3B7.B14.3B$476.3B13.3B7.B$
476.3B12.3B13.3B$476.B14.3B12.3B$481.3B7.B14.3B$480.3B13.3B7.B$480.3B
12.3B13.3B$480.B14.3B12.3B$485.3B7.B14.3B$484.3B13.3B7.B$484.3B12.3B
13.3B$484.B14.3B12.3B$489.3B7.B14.3B$488.3B13.3B7.B$488.3B12.3B13.3B$488.B14.3B12.3B$493.3B7.B14.3B$492.3B13.3B7.B$492.3B12.3B13.3B$492.B 14.3B12.3B$497.3B7.B14.3B$496.3B13.3B7.B$496.3B12.3B13.3B$496.B14.3B 12.3B$501.3B7.B14.3B$500.3B13.3B7.B$500.3B12.3B13.3B$500.B14.3B12.3B$
505.3B7.B14.3B$504.3B13.3B7.B$504.3B12.3B13.3B$504.B14.3B12.3B$509.3B
7.B14.3B$508.3B13.3B7.B$508.3B12.3B13.3B$508.B14.3B12.3B$513.3B7.B14.
3B$512.3B13.3B7.B$512.3B12.3B13.3B$512.B14.3B12.3B$517.3B7.B14.3B$516.3B13.3B7.B$516.3B12.3B13.3B$516.B14.3B12.3B$521.3B7.B14.3B$520.3B 13.3B7.B$520.3B12.3B13.3B$520.B14.3B12.3B$525.3B7.B14.3B$524.3B13.3B 7.B$524.3B12.3B13.3B$524.B14.3B12.3B$529.3B7.B14.3B$528.3B13.3B7.B$
528.3B12.3B13.3B$528.B14.3B12.3B$533.3B7.B14.3B$532.3B13.3B7.B$532.3B
12.3B13.3B$532.B14.3B12.3B$537.3B7.B14.3B$536.3B13.3B7.B$536.3B12.3B
13.3B$536.B14.3B12.3B$541.3B7.B14.3B$540.3B13.3B7.B$540.3B12.3B13.3B$540.B14.3B12.3B$545.3B7.B14.3B$544.3B13.3B7.B$544.3B12.3B13.3B$544.B 14.3B12.3B$549.3B7.B14.3B$548.3B13.3B7.B$548.3B12.3B13.3B$548.B14.3B 12.3B$553.3B7.B14.3B$552.3B13.3B7.B$552.3B12.3B13.3B$552.B14.3B12.3B$
557.3B7.B14.3B$556.3B13.3B7.B$556.3B12.3B13.3B$556.B14.3B12.3B$561.3B
7.B14.3B$560.3B13.3B7.B$560.3B12.3B13.3B$560.B14.3B12.3B$565.3B7.B14.
3B$564.3B13.3B7.B$564.3B12.3B13.3B$564.B14.3B12.3B$569.3B7.B14.3B$568.3B13.3B7.B$568.3B12.3B13.3B$568.B14.3B12.3B$573.3B7.B14.3B$572.3B 13.3B7.B$572.3B12.3B13.3B$572.B14.3B12.3B$577.3B7.B14.3B$576.3B13.3B 7.B$576.3B12.3B13.3B$576.B14.3B12.3B$581.3B7.B14.3B$580.3B13.3B7.B$
580.3B12.3B13.3B$580.B14.3B12.3B$585.3B7.B14.3B$584.3B13.3B7.B$584.3B
12.3B13.3B$584.B14.3B12.3B$589.3B7.B14.3B$588.3B13.3B7.B$588.3B12.3B
13.3B$588.B14.3B12.3B$593.3B7.B14.3B$592.3B13.3B7.B$592.3B12.3B13.3B$592.B14.3B12.3B$597.3B7.B14.3B$596.3B13.3B7.B$596.3B12.3B13.3B$596.B 14.3B12.3B$601.3B7.B14.3B$600.3B13.3B7.B$600.3B12.3B13.3B$600.B14.3B 12.3B$605.3B7.B14.3B$604.3B13.3B7.B$604.3B12.3B13.3B$604.B14.3B12.3B$
609.3B7.B14.3B$608.3B13.3B7.B$608.3B12.3B13.3B$608.B14.3B12.3B$613.3B
7.B14.3B$612.3B13.3B7.B$612.3B12.3B13.3B$612.B14.3B12.3B$617.3B7.B14.
3B$616.3B13.3B7.B$616.3B12.3B13.3B$616.B14.3B12.3B$621.3B7.B14.3B$620.3B13.3B7.B$620.3B12.3B13.3B$620.B14.3B12.3B$625.3B7.B14.3B$624.3B 13.3B7.B$624.3B12.3B13.3B$624.B14.3B12.3B$629.3B7.B14.3B$628.3B13.3B 7.B$628.3B12.3B13.3B$628.B14.3B12.3B$633.3B7.B14.3B$632.3B13.3B7.B$
632.3B12.3B13.3B$632.B14.3B12.3B$637.3B7.B14.3B$636.3B13.3B7.B$636.3B
12.3B13.3B$636.B14.3B12.3B$641.3B7.B14.3B$640.3B13.3B7.B$640.3B12.3B
13.3B$640.B14.3B12.3B$645.3B7.B14.3B$644.3B13.3B7.B$644.3B12.3B13.3B$644.B14.3B12.3B$649.3B7.B14.3B$648.3B13.3B7.B$648.3B12.3B13.3B$648.B 14.3B12.3B$653.3B7.B14.3B$652.3B13.3B7.B$652.3B12.3B13.3B$652.B14.3B 12.3B$657.3B7.B14.3B$656.3B13.3B7.B$656.3B12.3B13.3B$656.B14.3B12.3B$
661.3B7.B14.3B$660.3B13.3B7.B$660.3B12.3B13.3B$660.B14.3B12.3B$665.3B
7.B14.3B$664.3B13.3B7.B$664.3B12.3B13.3B$664.B14.3B12.3B$669.3B7.B14.
3B$668.3B13.3B7.B$668.3B12.3B13.3B$668.B14.3B12.3B$673.3B7.B14.3B$672.3B13.3B7.B$672.3B12.3B13.3B$672.B14.3B12.3B$677.3B7.B14.3B$676.3B 13.3B7.B$676.3B12.3B13.3B$676.B14.3B12.3B$681.3B7.B14.3B$680.3B13.3B 7.B$680.3B12.3B13.3B$680.B14.3B12.3B$685.3B7.B14.3B$684.3B13.3B7.B$
684.3B12.3B13.3B$684.B14.3B12.3B$689.3B7.B14.3B$688.3B13.3B7.B$688.3B
12.3B13.3B$688.B14.3B12.3B$693.3B7.B14.3B$692.3B13.3B7.B$692.3B12.3B
13.3B$692.B14.3B12.3B$697.3B7.B14.3B$696.3B13.3B7.B$696.3B12.3B13.3B$696.B14.3B12.3B$701.3B7.B14.3B$700.3B13.3B7.B$700.3B12.3B13.3B$700.B 14.3B12.3B$705.3B7.B14.3B$704.3B13.3B7.B$704.3B12.3B13.3B$704.B14.3B 12.3B$709.3B7.B14.3B$708.3B13.3B7.B$708.3B12.3B13.3B$708.B14.3B12.3B$
713.3B7.B14.3B$712.3B13.3B7.B$712.3B12.3B13.3B$712.B14.3B12.3B$717.3B
7.B14.3B$716.3B13.3B7.B$716.3B12.3B13.3B$716.B14.3B12.3B$721.3B7.B14.
3B$720.3B13.3B7.B$720.3B12.3B13.3B$720.B14.3B12.3B$725.3B7.B14.3B$724.3B13.3B7.B$724.3B12.3B13.3B$724.B14.3B12.3B$729.3B7.B14.3B$728.3B 13.3B7.B$728.3B12.3B13.3B$728.B14.3B12.3B$733.3B7.B14.3B$732.3B13.3B 7.B$732.3B12.3B13.3B$732.B14.3B12.3B$737.3B7.B14.3B$736.3B13.3B7.B$
736.3B12.3B13.3B$736.B14.3B12.3B$741.3B7.B14.3B$740.3B13.3B7.B$740.3B
12.3B13.3B$740.B14.3B12.3B$745.3B7.B14.3B$744.3B13.3B7.B$744.3B12.3B
13.3B$744.B14.3B12.3B$749.3B7.B14.3B$748.3B13.3B7.B$748.3B12.3B13.3B$748.B14.3B12.3B$753.3B7.B14.3B$752.3B13.3B7.B$752.3B12.3B13.3B$752.B 14.3B12.3B$757.3B7.B14.3B$756.3B13.3B7.B$756.3B12.3B13.3B$756.B14.3B 12.3B$761.3B7.B14.3B$760.3B13.3B7.B$760.3B12.3B13.3B$760.B14.3B12.3B$
765.3B7.B14.3B$764.3B13.3B7.B$764.3B12.3B13.3B$764.B14.3B12.3B$769.3B
7.B14.3B$768.3B13.3B7.B$768.3B12.3B13.3B$768.B14.3B12.3B$773.3B7.B14.
3B$772.3B13.3B7.B$772.3B12.3B13.3B$772.B14.3B12.3B$777.3B7.B14.3B$776.3B13.3B7.B$776.3B12.3B13.3B$776.B14.3B12.3B$781.3B7.B14.3B$780.3B 13.3B7.B$780.3B12.3B13.3B$780.B14.3B12.3B$785.3B7.B14.3B$784.3B13.3B 7.B$784.3B12.3B13.3B$784.B14.3B12.3B$789.3B7.B14.3B$788.3B13.3B7.B$
788.3B12.3B13.3B$788.B14.3B12.3B$793.3B7.B14.3B$792.3B13.3B7.B$792.3B
12.3B13.3B$792.B14.3B12.3B$797.3B7.B14.3B$796.3B13.3B7.B$796.3B12.3B
13.3B$796.B14.3B12.3B$801.3B7.B14.3B$800.3B13.3B7.B$800.3B12.3B13.3B$800.B14.3B12.3B$805.3B7.B14.3B$804.3B13.3B7.B$804.3B12.3B13.3B$804.B 14.3B12.3B$809.3B7.B14.3B$808.3B13.3B7.B$808.3B12.3B13.3B$808.B14.3B 12.3B$813.3B7.B14.3B$812.3B13.3B7.B$812.3B12.3B13.3B$812.B14.3B12.3B$
817.3B7.B14.3B$816.3B13.3B7.B$816.3B12.3B13.3B$816.B14.3B12.3B$821.3B
7.B14.3B$820.3B13.3B7.B$820.3B12.3B13.3B$820.B14.3B12.3B$825.3B7.B14.
3B$824.3B13.3B7.B$824.3B12.3B13.3B$824.B14.3B12.3B$829.3B7.B14.3B$828.3B13.3B7.B$828.3B12.3B13.3B$828.B14.3B12.3B$833.3B7.B14.3B$832.3B 13.3B7.B$832.3B12.3B13.3B$832.B14.3B12.3B$837.3B7.B14.3B$836.3B13.3B 7.B$836.3B12.3B13.3B$836.B14.3B12.3B$841.3B7.B14.3B$840.3B13.3B7.B$
840.3B12.3B13.3B$840.B14.3B12.3B$845.3B7.B14.3B$844.3B13.3B7.B$844.3B
12.3B13.3B$844.B14.3B12.3B$849.3B7.B14.3B$848.3B13.3B7.B$848.3B12.3B
13.3B$848.B14.3B12.3B$853.3B7.B14.3B$852.3B13.3B7.B$852.3B12.3B13.3B$852.B14.3B12.3B$857.3B7.B14.3B$856.3B13.3B7.B$856.3B12.3B13.3B$856.B 14.3B12.3B$861.3B7.B14.3B$860.3B13.3B7.B$860.3B12.3B13.3B$860.B14.3B 12.3B$865.3B7.B14.3B$864.3B13.3B7.B$864.3B12.3B13.3B$864.B14.3B12.3B$
869.3B7.B14.3B$868.3B13.3B7.B$868.3B12.3B13.3B$868.B14.3B12.3B$873.3B
7.B14.3B$872.3B13.3B7.B$872.3B12.3B13.3B$872.B14.3B12.3B$877.3B7.B14.
3B$876.3B13.3B7.B$876.3B12.3B13.3B$876.B14.3B12.3B$881.3B7.B14.3B$880.3B13.3B7.B$880.3B12.3B13.3B$880.B14.3B12.3B$885.3B7.B14.3B$884.3B 13.3B7.B$884.3B12.3B13.3B$884.B14.3B12.3B$889.3B7.B14.3B$888.3B13.3B 7.B$888.3B12.3B13.3B$888.B14.3B12.3B$893.3B7.B14.3B$892.3B13.3B7.B$
892.3B12.3B13.3B$892.B14.3B12.3B$897.3B7.B14.3B$896.3B13.3B7.B$896.3B
12.3B13.3B$896.B14.3B12.3B$901.3B7.B14.3B$900.3B13.3B7.B$900.3B12.3B
13.3B$900.B14.3B12.3B$905.3B7.B14.3B$904.3B13.3B7.B$904.3B12.3B13.3B$904.B14.3B12.3B$909.3B7.B14.3B$908.3B13.3B7.B$908.3B12.3B13.3B$908.B 14.3B12.3B$913.3B7.B14.3B$912.3B13.3B7.B$912.3B12.3B13.3B$912.B14.3B 12.3B$917.3B7.B14.3B$916.3B13.3B7.B$916.3B12.3B13.3B$916.B14.3B12.3B$
921.3B7.B14.3B$920.3B13.3B7.B$920.3B12.3B13.3B$920.B14.3B12.3B$925.3B
7.B14.3B$924.3B13.3B7.B$924.3B12.3B13.3B$924.B14.3B12.3B$929.3B7.B14.
3B$928.3B13.3B7.B$928.3B12.3B13.3B$928.B14.3B12.3B$933.3B7.B14.3B$932.3B13.3B7.B$932.3B12.3B13.3B$932.B14.3B12.3B$937.3B7.B14.3B$936.3B 13.3B7.B$936.3B12.3B13.3B$936.B14.3B12.3B$941.3B7.B14.3B$940.3B13.3B 7.B$940.3B12.3B13.3B$940.B14.3B12.3B$945.3B7.B14.3B$944.3B13.3B7.B$
944.3B12.3B13.3B$944.B14.3B12.3B$949.3B7.B14.3B$948.3B13.3B7.B$948.3B
12.3B13.3B$948.B14.3B12.3B$953.3B7.B14.3B$952.3B13.3B7.B$952.3B12.3B
13.3B$952.B14.3B12.3B$957.3B7.B14.3B$956.3B13.3B7.B$956.3B12.3B13.3B$956.B14.3B12.3B$961.3B7.B14.3B$960.3B13.3B7.B$960.3B12.3B13.3B$960.B 14.3B12.3B$965.3B7.B14.3B$964.3B13.3B7.B$964.3B12.3B13.3B$964.B14.3B 12.3B$969.3B7.B14.3B$968.3B13.3B7.B$968.3B12.3B13.3B$968.B14.3B12.3B$
973.3B7.B14.3B$972.3B13.3B7.B$972.3B12.3B13.3B$972.B14.3B12.3B$977.3B
7.B14.3B$976.3B13.3B7.B$976.3B12.3B13.3B$976.B14.3B12.3B$981.3B7.B14.
3B$980.3B13.3B7.B$980.3B12.3B13.3B$980.B14.3B12.3B$985.3B7.B14.3B$984.3B13.3B7.B$984.3B12.3B13.3B$984.B14.3B12.3B$989.3B7.B14.3B$988.3B 13.3B7.B$988.3B12.3B13.3B$988.B14.3B12.3B$993.3B7.B14.3B$992.3B13.3B 7.B$992.3B12.3B13.3B$992.B14.3B12.3B$997.3B7.B14.3B$996.3B13.3B7.B$
996.3B12.3B13.3B$996.B14.3B12.3B$1001.3B7.B14.3B$1000.3B13.3B7.B$
1000.3B12.3B$1000.B14.3B$1005.3B7.B$1004.3B$1004.3B$1004.B$1009.3B$1008.3B$1008.3B$1008.B$1013.3B$1012.3B$1012.3B$1012.B!  period54 Posts: 20 Joined: November 12th, 2014, 12:08 pm ### Re: Universality proof question So, I made something similar to sliding block memory. It is extremely ugly and practically useless, but at least it works, and probably proves that this rule is Turing complete. Construction: Code: Select all x = 2815, y = 2943, rule = BTCA1 15.3B$14.3B$14.3B$14.B$11.3A5.3B$11.3A4.3B$11.3A4.3B$8.3B7.B$7.3B13. 3B$7.3B12.3B$7.B14.3B$4.3A5.3B7.B$4.3A4.3B13.3B$4.3A4.3B12.3B$.3B7.B 14.3B$3B13.3B7.B$3B12.3B13.3B$B14.3B12.3B$5.3B7.B14.3B$4.3B13.3B7.B$4.3B12.3B13.3B$4.B14.3B12.3B$9.3B7.B14.3B$8.3B13.3B7.B$8.3B12.3B13.3B$8.B14.3B12.3B$13.3B7.B14.3B$12.3B13.3B7.B$12.3B12.3B13.3B$12.B14.3B
12.3B$17.3B7.B14.3B$16.3B13.3B7.B$16.3B12.3B13.3B$16.B14.3B12.3B$21. 3B7.B14.3B$20.3B13.3B7.B$20.3B12.3B13.3B$20.B14.3B12.3B$25.3B7.B14.3B$24.3B13.3B7.B$24.3B12.3B13.3B$24.B14.3B12.3B$29.3B7.B14.3B$28.3B13.
3B7.B$28.3B12.3B13.3B$28.B14.3B12.3B$33.3B7.B14.3B$32.3B13.3B7.B$32. 3B12.3B13.3B$32.B14.3B12.3B$37.3B7.B14.3B$36.3B13.3B7.B$36.3B12.3B13. 3B$36.B14.3B12.3B$41.3B7.B14.3B$40.3B13.3B7.B$40.3B12.3B13.3B$40.B14.
3B12.3B$45.3B7.B14.3B$44.3B13.3B7.B$44.3B12.3B13.3B$44.B14.3B12.3B$49.3B7.B14.3B$48.3B13.3B7.B$48.3B12.3B13.3B$48.B14.3B12.3B$53.3B7.B 14.3B$52.3B13.3B7.B$52.3B12.3B13.3B$52.B14.3B12.3B$57.3B7.B14.3B$56.
3B13.3B7.B$56.3B12.3B13.3B$56.B14.3B12.3B$61.3B7.B14.3B$60.3B13.3B7.B
$60.3B12.3B13.3B$60.B14.3B12.3B$65.3B7.B14.3B$64.3B13.3B7.B$64.3B12. 3B13.3B$64.B14.3B12.3B$69.3B7.B14.3B$68.3B13.3B7.B$68.3B12.3B13.3B$
68.B14.3B12.3B$73.3B7.B14.3B$72.3B13.3B7.B$72.3B12.3B13.3B$72.B14.3B
12.3B$77.3B7.B14.3B$76.3B13.3B7.B$76.3B12.3B13.3B$76.B14.3B12.3B$81. 3B7.B14.3B$80.3B13.3B7.B$80.3B12.3B13.3B$80.B14.3B12.3B$85.3B7.B14.3B$84.3B13.3B7.B$84.3B12.3B13.3B$84.B14.3B12.3B$89.3B7.B14.3B$88.3B13.
3B7.B$88.3B12.3B13.3B$88.B14.3B12.3B$93.3B7.B14.3B$92.3B13.3B7.B$92. 3B12.3B13.3B$92.B14.3B12.3B$97.3B7.B14.3B$96.3B13.3B7.B$96.3B12.3B13. 3B$96.B14.3B12.3B$101.3B7.B14.3B$100.3B13.3B7.B$100.3B12.3B13.3B$100.
B14.3B12.3B$105.3B7.B14.3B$104.3B13.3B7.B$104.3B12.3B13.3B$104.B14.3B
12.3B$109.3B7.B14.3B$108.3B13.3B7.B$108.3B12.3B13.3B$108.B14.3B12.3B$113.3B7.B14.3B$112.3B13.3B7.B$112.3B12.3B13.3B$112.B14.3B12.3B$117.3B 7.B14.3B$116.3B13.3B7.B$116.3B12.3B13.3B$116.B14.3B12.3B$121.3B7.B14. 3B$120.3B13.3B7.B$120.3B12.3B13.3B$120.B14.3B12.3B$125.3B7.B14.3B$
124.3B13.3B7.B$124.3B12.3B13.3B$124.B14.3B12.3B$129.3B7.B14.3B$128.3B
13.3B7.B$128.3B12.3B13.3B$128.B14.3B12.3B$133.3B7.B14.3B$132.3B13.3B
7.B$132.3B12.3B13.3B$132.B14.3B12.3B$137.3B7.B14.3B$136.3B13.3B7.B$136.3B12.3B13.3B$136.B14.3B12.3B$141.3B7.B14.3B$140.3B13.3B7.B$140.3B 12.3B13.3B$140.B14.3B12.3B$145.3B7.B14.3B$144.3B13.3B7.B$144.3B12.3B 13.3B$144.B14.3B12.3B$149.3B7.B14.3B$148.3B13.3B7.B$148.3B12.3B13.3B$
148.B14.3B12.3B$153.3B7.B14.3B$152.3B13.3B7.B$152.3B12.3B13.3B$152.B
14.3B12.3B$157.3B7.B14.3B$156.3B13.3B7.B$156.3B12.3B13.3B$156.B14.3B
12.3B$161.3B7.B14.3B$160.3B13.3B7.B$160.3B12.3B13.3B$160.B14.3B12.3B$165.3B7.B14.3B$164.3B13.3B7.B$164.3B12.3B13.3B$164.B14.3B12.3B$169.3B 7.B14.3B$168.3B13.3B7.B$168.3B12.3B13.3B$168.B14.3B12.3B$173.3B7.B14. 3B$172.3B13.3B7.B$172.3B12.3B13.3B$172.B14.3B12.3B$177.3B7.B14.3B$
176.3B13.3B7.B$176.3B12.3B13.3B$176.B14.3B12.3B$181.3B7.B14.3B$180.3B
13.3B7.B$180.3B12.3B13.3B$180.B14.3B12.3B$185.3B7.B14.3B$184.3B13.3B
7.B$184.3B12.3B13.3B$184.B14.3B12.3B$189.3B7.B14.3B$188.3B13.3B7.B$188.3B12.3B13.3B$188.B14.3B12.3B$193.3B7.B14.3B$192.3B13.3B7.B$192.3B 12.3B13.3B$192.B14.3B12.3B$197.3B7.B14.3B$196.3B13.3B7.B$196.3B12.3B 13.3B$196.B14.3B12.3B$201.3B7.B14.3B$200.3B13.3B7.B$200.3B12.3B13.3B$
200.B14.3B12.3B$205.3B7.B14.3B$204.3B13.3B7.B$204.3B12.3B13.3B$204.B
14.3B12.3B$209.3B7.B14.3B$208.3B13.3B7.B$208.3B12.3B13.3B$208.B14.3B
12.3B$213.3B7.B14.3B$212.3B13.3B7.B$212.3B12.3B13.3B$212.B14.3B12.3B$217.3B7.B14.3B$216.3B13.3B7.B$216.3B12.3B13.3B$216.B14.3B12.3B$221.3B 7.B14.3B$220.3B13.3B7.B$220.3B12.3B13.3B$220.B14.3B12.3B$225.3B7.B14. 3B$224.3B13.3B7.B$224.3B12.3B13.3B$224.B14.3B12.3B$229.3B7.B14.3B$
228.3B13.3B7.B$228.3B12.3B13.3B$228.B14.3B12.3B$233.3B7.B14.3B$232.3B
13.3B7.B$232.3B12.3B$232.B14.3B$237.3B7.B$236.3B$236.3B$236.B$241.3B$
240.3B$240.3B$240.B$245.3B$244.3B$244.3B$244.B1306$1567.2A$1567.2A
1113$2696.AB$2695.A3B$2695.3B$2696.B$2700.AB$2699.A3B$2699.3B$2700.B$2704.AB$2703.A3B$2703.3B$2693.AB9.B$2692.A3B12.AB$2692.3B12.A3B$2682. AB9.B13.3B$2681.A3B12.AB9.B$2681.3B12.A3B12.AB$2682.B13.3B12.A3B$2686.AB9.B13.3B$2685.A3B12.AB9.B$2685.3B12.A3B12.AB$2686.B13.3B12.A3B
$2690.AB9.B13.3B$2689.A3B12.AB9.B$2689.3B12.A3B12.AB$2690.B13.3B12.A
3B$2694.AB9.B13.3B$2693.A3B12.AB9.B$2693.3B12.A3B12.AB$2694.B13.3B12.
A3B$2698.AB9.B13.3B$2697.A3B12.AB9.B$2697.3B12.A3B12.AB$2698.B13.3B
12.A3B$2702.AB9.B13.3B$2701.A3B12.AB9.B$2701.3B12.A3B12.AB$2702.B13.
3B12.A3B$2706.AB9.B13.3B$2705.A3B12.AB9.B$2705.3B12.A3B12.AB$2706.B
13.3B12.A3B$2710.AB9.B13.3B$2709.A3B12.AB9.B$2709.3B12.A3B12.AB$2710.
B13.3B12.A3B$2714.AB9.B13.3B$2713.A3B12.AB9.B$2713.3B12.A3B12.AB$
2714.B13.3B12.A3B$2718.AB9.B13.3B$2717.A3B12.AB9.B$2717.3B12.A3B12.AB$2718.B13.3B12.A3B$2722.AB9.B13.3B$2721.A3B12.AB9.B$2721.3B12.A3B12.A B$2722.B13.3B12.A3B$2726.AB9.B13.3B$2725.A3B12.AB9.B$2725.3B12.A3B12. AB$2726.B13.3B12.A3B$2730.AB9.B13.3B$2729.A3B12.AB9.B$2729.3B12.A3B 12.AB$2730.B13.3B12.A3B$2734.AB9.B13.3B$2733.A3B12.AB9.B$2733.3B12.A 3B12.AB$2734.B13.3B12.A3B$2738.AB9.B13.3B$2737.A3B12.AB9.B$2737.3B12. A3B12.AB$2738.B13.3B12.A3B$2742.AB9.B13.3B$2741.A3B12.AB9.B$2741.3B 12.A3B12.AB$2742.B13.3B12.A3B$2746.AB9.B13.3B$2745.A3B12.AB9.B$2745. 3B12.A3B12.AB$2746.B13.3B12.A3B$2750.AB9.B13.3B$2749.A3B12.AB9.B$2749.3B12.A3B12.AB$2750.B13.3B12.A3B$2754.AB9.B13.3B$2753.A3B12.AB9.B
$2753.3B12.A3B12.AB$2754.B13.3B12.A3B$2758.AB9.B13.3B$2757.A3B12.AB9.
B$2757.3B12.A3B12.AB$2758.B13.3B12.A3B$2762.AB9.B13.3B$2761.A3B12.AB
9.B$2761.3B12.A3B12.AB$2762.B13.3B12.A3B$2766.AB9.B13.3B$2765.A3B12.A
B9.B$2765.3B12.A3B12.AB$2766.B13.3B12.A3B$2770.AB9.B13.3B$2769.A3B12.
AB9.B$2769.3B12.A3B12.AB$2770.B13.3B12.A3B$2774.AB9.B13.3B$2773.A3B
12.AB9.B$2773.3B12.A3B12.AB$2774.B13.3B12.A3B$2778.AB9.B13.3B$2777.A
3B12.AB9.B$2777.3B12.A3B12.AB$2778.B13.3B12.A3B$2782.AB9.B13.3B$2781.
A3B12.AB9.B$2781.3B12.A3B12.AB$2782.B13.3B12.A3B$2786.AB9.B9.A3.3B$
2785.A3B12.AB3.4A2.B$2785.3B12.A3B3.ABA$2786.B13.3B4.4A$2790.AB9.B7.A$2789.A3B12.AB$2789.3B12.A3B$2790.B9.A3.3B$2794.AB3.4A2.B$2793.A3B3.A
BA$1358.B1434.3B4.4A$1358.3B1433.B7.A$1358.3B1437.AB$1359.3B1435.A3B$1354.B1442.3B$1354.3B1441.B$1354.3B$1355.3B$1350.B$1350.3B$1350.3B$
1351.3B7.B$1346.B14.3B$1346.3B12.3B$1346.3B13.3B7.B$1347.3B7.B14.3B$1342.B14.3B12.3B$1342.3B12.3B13.3B$1342.3B13.3B7.B$1343.3B7.B14.3B$1338.B14.3B12.3B$1338.3B12.3B13.3B$1338.3B13.3B7.B$1339.3B7.B14.3B$1334.B14.3B12.3B$1334.3B12.3B13.3B$1334.3B13.3B7.B$1335.3B7.B14.3B$1330.B14.3B12.3B$1330.3B12.3B13.3B$1330.3B13.3B7.B$1331.3B7.B14.3B$1326.B14.3B12.3B$1326.3B12.3B13.3B$1326.3B13.3B7.B$1327.3B7.B14.3B$1322.B14.3B12.3B$1322.3B12.3B13.3B$1322.3B13.3B7.B$1323.3B7.B14.3B$1318.B14.3B12.3B$1318.3B12.3B13.3B$1318.3B13.3B7.B$1319.3B7.B14.3B$1314.B14.3B12.3B$1314.3B12.3B13.3B$1314.3B13.3B7.B$1315.3B7.B14.3B$1310.B14.3B12.3B$1310.3B12.3B13.3B$1310.3B13.3B7.B$1311.3B7.B14.3B$1306.B14.3B12.3B$1306.3B12.3B13.3B$1306.3B13.3B7.B$1307.3B7.B14.3B$1302.B14.3B12.3B$1302.3B12.3B13.3B$1302.3B13.3B7.B$1303.3B7.B14.3B$1298.B14.3B12.3B$1298.3B12.3B13.3B$1298.3B13.3B7.B$1299.3B7.B14.3B$1294.B14.3B12.3B$1294.3B12.3B13.3B$1294.3B13.3B7.B$1295.3B7.B14.3B$1290.B14.3B12.3B$1290.3B12.3B13.3B$1290.3B13.3B7.B$1291.3B7.B14.3B$1286.B14.3B12.3B$1286.3B12.3B13.3B$1286.3B13.3B7.B$1287.3B7.B14.3B$1282.B14.3B12.3B$1282.3B12.3B13.3B$1282.3B13.3B7.B$1283.3B7.B14.3B$1278.B14.3B12.3B$1278.3B12.3B13.3B$1278.3B13.3B7.B$1279.3B7.B14.3B$1274.B14.3B12.3B$1274.3B12.3B13.3B$1274.3B13.3B7.B$1275.3B7.B14.3B$1270.B14.3B12.3B$1270.3B12.3B13.3B$1270.3B13.3B7.B$1271.3B7.B14.3B$1266.B14.3B12.3B$1266.3B12.3B13.3B$1266.3B13.3B7.B$1267.3B7.B14.3B$1262.B14.3B12.3B$1262.3B12.3B13.3B$1262.3B13.3B7.B$1263.3B7.B14.3B$1258.B14.3B12.3B$1258.3B12.3B13.3B$1258.3B13.3B7.B$1259.3B7.B14.3B$1254.B14.3B12.3B$1254.3B12.3B13.3B$1254.3B13.3B7.B$1255.3B7.B14.3B$1250.B14.3B12.3B$1250.3B12.3B13.3B$1250.3B13.3B7.B$1251.3B7.B14.3B$1246.B14.3B12.3B$1246.3B12.3B13.3B$1246.3B13.3B7.B$1247.3B7.B14.3B$1242.B14.3B12.3B$1242.3B12.3B13.3B$1242.3B3.3A7.3B7.B$1243.3B2.3A2.B
14.3B$1246.5A2.3B12.3B$1246.3A4.3B13.3B$1246.3A5.3B7.B$1249.B14.3B$1249.3B12.3B$1249.3B3.3A7.3B$1250.3B2.3A2.B$1253.5A2.3B$1253.3A4.3B$
1253.3A5.3B$1256.B$1256.3B$1256.3B$1257.3B!

I'll try to explain how it works:
There are three puffers: A, B and C. A has period of 256, B and C - 128.
Glider streams from B and C collide, using reaction A for awesome found, resulting in period 256 stream B/C.
Single block is pulled by both glider stream from A and B/C. Since both streams have same period, it stays in place.

It can be incremented by skipping one glider in B/C stream and decremented by skipping one in C.

You can check if block is in particular position using this reaction:

Code: Select all

x = 24, y = 29, rule = BTCA1
20.B$20.3B$20.3B$21.3B15$2.B$2.3B$2.3B$3.3B6$2A$2A!  knightlife Posts: 566 Joined: May 31st, 2009, 12:08 am ### Re: Universality proof question Spaceship converter is periodic: Code: Select all x = 119, y = 23, rule = BTCA1 101.B$99.3B$100.3B$97.B.AB$97.3B$96.3B$98.B4$107.3A$.B31.B33.B30.2B6. 3A$3B31.2B29.3B29.BA2B5.3A$4B26.B3.2B27.2B2A2B28.4B5.A$4B26.B3.2B27.
2B2A2B28.4B$3B31.2B29.3B29.BA2B$.B31.B33.B30.2B17.B$115.3B$116.3B$113.B.AB$113.3B$112.3B$114.B!

Stream shown is p560 but could be lower period. The p70 osc can be increased by 12n (n=1,2,3...) and possibly decreased. Other combinations of periods may be possible because the reflector on one end of the osc can be a different phase giving +/- 6n possibilities.