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Smallest Oscillators Supporting Specific Periods

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Re: Smallest Oscillators Supporting Specific Periods

Postby Naszvadi » April 2nd, 2018, 2:44 pm

Macbi wrote:All 2 cell populations:

...
p8
x = 3, y = 3, rule = B2en3cejq4cjkt5jkny6i8/S01c2in3-knr4acknqry5qry6aik
2bo2$o!

...



You probably forgot this 2bit p8: ../forums/viewtopic.php?f=12&t=3020#p53012
#C Moore neighbourhood still violates symmetry
x = 2, y = 4, rule = B02/S1V
o3$bo!
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Re: Smallest Oscillators Supporting Specific Periods

Postby wwei23 » April 2nd, 2018, 2:51 pm

Naszvadi wrote:
Macbi wrote:All 2 cell populations:

...
p8
x = 3, y = 3, rule = B2en3cejq4cjkt5jkny6i8/S01c2in3-knr4acknqry5qry6aik
2bo2$o!

...



You probably forgot this 2bit p8: ../forums/viewtopic.php?f=12&t=3020#p53012
#C Moore neighbourhood still violates symmetry
x = 2, y = 4, rule = B02/S1V
o3$bo!

I don’t allow B0 in my tabulations.
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Re: Smallest Oscillators Supporting Specific Periods

Postby Majestas32 » April 3rd, 2018, 12:19 pm

:facepalm:
Please, stop spam searching Snowflakes.
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Re: Smallest Oscillators Supporting Specific Periods

Postby wwei23 » April 3rd, 2018, 12:48 pm

We have a P55, but not a 2 cell one.
On the tabulations. Red means bad, like a 3-cell oscillator, or a period without an oscillator.
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Re: Smallest Oscillators Supporting Specific Periods

Postby 2718281828 » April 20th, 2018, 8:16 am

Some oscillators, from snowflakes, with cells 19, 14, 15, 16:
x = 42, y = 61, rule = B2ci3ai4c8/S02ae3eijkq4iz5ar6i7e
3$11bo$9bo$4bo$22b2o6bo4b2o$7bo2bo11bo12bo$22b2o6bo6bo$4bo$9bo$11bo7$
11bo16bo2$4bo30bo2$5bobo4bo14bo4bobo2$4bo30bo2$11bo16bo7$11bo15bo$29bo
$4bo29bo2$5bobo4bo15bo2bo2$4bo29bo$29bo$11bo15bo5$29bo$11bo15bo$27b2o$
4bo2$5bobo4bo$30bo$4bo$27b2o$11bo15bo$29bo!

I know, the populations are quite high, but they can cover all large integers (esp. all > 100) by shifting one part (either left or right), so we have
for
4N: 19 cells
4N+1: 14 cells
4N+2: 15 cells
4N+3: 16 cells
I think this is not optimal but a start for large period oscillators.

EDIT1:
4N: 7 cells
x = 9, y = 13, rule = B2ek3cei4acjr5c8/S02ack3kn4aen
5bo2$3bo$o3bo$3bo5$8bo3$3bo!
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Re: Smallest Oscillators Supporting Specific Periods

Postby AbhpzTa » April 20th, 2018, 10:12 am

2718281828 wrote:Some oscillators, from snowflakes, with cells 19, 14, 15, 16:
x = 42, y = 61, rule = B2ci3ai4c8/S02ae3eijkq4iz5ar6i7e
3$11bo$9bo$4bo$22b2o6bo4b2o$7bo2bo11bo12bo$22b2o6bo6bo$4bo$9bo$11bo7$
11bo16bo2$4bo30bo2$5bobo4bo14bo4bobo2$4bo30bo2$11bo16bo7$11bo15bo$29bo
$4bo29bo2$5bobo4bo15bo2bo2$4bo29bo$29bo$11bo15bo5$29bo$11bo15bo$27b2o$
4bo2$5bobo4bo$30bo$4bo$27b2o$11bo15bo$29bo!

I know, the populations are quite high, but they can cover all large integers (esp. all > 100) by shifting one part (either left or right), so we have
for
4N: 19 cells
4N+1: 14 cells
4N+2: 15 cells
4N+3: 16 cells
I think this is not optimal but a start for large period oscillators.

EDIT1:
4N: 7 cells
x = 9, y = 13, rule = B2ek3cei4acjr5c8/S02ack3kn4aen
5bo2$3bo$o3bo$3bo5$8bo3$3bo!

4N: 4 cells (N >= 2)
x = 9, y = 24, rule = B2ei3i/S012ce3j
2bo$o4bo$2bo5$2bo$o5bo$2bo5$2bo$o6bo$2bo5$2bo$o7bo$2bo!

4N+2: 4 cells (N >= 2)
x = 9, y = 24, rule = B2ei3i/S012ce3j4t
2bo$o4bo$2bo5$2bo$o5bo$2bo5$2bo$o6bo$2bo5$2bo$o7bo$2bo!
Iteration of sigma(n)+tau(n)-n [sigma(n)+tau(n)-n : OEIS A163163] (e.g. 16,20,28,34,24,44,46,30,50,49,11,3,3, ...) :
965808 is period 336 (max = 207085118608).
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Re: Smallest Oscillators Supporting Specific Periods

Postby jimmyChen2013 » June 1st, 2018, 12:47 am

p448 8 cell
x = 4, y = 4, rule = B2in34ce5q/S2-n34cejnyz
b2o$o2bo$3bo$b3o!
Failed Replicator!
x = 4, y = 4, rule = B34ce5cen67c8/S2-i3-jqry4cent5j67c8
bo$obo$bobo$2bo!

(That I wish was not failed D:)
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Re: Smallest Oscillators Supporting Specific Periods

Postby Redstoneboi » June 1st, 2018, 12:59 am

i guess the rest of the posts should either improve on the 4n + x populations, or find an oscillator whose population is lower than the 4n + x counterpart.

should we also count the bounding box?
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Re: Smallest Oscillators Supporting Specific Periods

Postby 2718281828 » June 1st, 2018, 2:36 am

4 cells for 4N+1 (N>2):
x = 9, y = 22, rule = B2ei3i4r/S012cek3j4t
5bo$2bo$o$2bo3$6bo$2bo$o$2bo3$7bo$2bo$o$2bo3$8bo$2bo$o$2bo!

so only 4N+3 is missing to be optimized to 4 cells.

So far, we have 6 cells for 4N+3:
x = 14, y = 28, rule = B2ei3i4r/S012cek3j4t
10bo$5bo2bo2$5bo2bo$3bo3$11bo$5bo2bo2$5bo2bo$2bo4$12bo$5bo2bo2$5bo2bo$
bo4$13bo$5bo2bo2$5bo2bo$o!
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Re: Smallest Oscillators Supporting Specific Periods

Postby Macbi » June 1st, 2018, 1:07 pm

Does anyone think it might be possible to get any families of oscillators with three cells? I'm imagining that two of the cells are close together and create a spacefiller which evolves until it encounters the third cell. The third cell causes a collapse which sends the spacefiller back to its original state.
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Re: Smallest Oscillators Supporting Specific Periods

Postby 2718281828 » June 1st, 2018, 1:53 pm

Macbi wrote:Does anyone think it might be possible to get any families of oscillators with three cells? I'm imagining that two of the cells are close together and create a spacefiller which evolves until it encounters the third cell. The third cell causes a collapse which sends the spacefiller back to its original state.


Two cells are always symmetric, so a 2 cell spacefiller will always expand in at least in two directions. Thus, I think it is not feasible, as a cell would be required on both ends, which makes again 4. However, this might be an option for a 4N+3, 4 cell variant.

If we would use a 2 cells spacefiller with only 1 cell which gets hit, then a fuse could run faster back and catch the other spacefiller (due to symmetry), react and create a cell reflector plus a spaceship on the right line. So I think theoretically this might be feasible, but the reflector itself will still have at least 4 cells, just the predecessor would have 3 cells.
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Re: Smallest Oscillators Supporting Specific Periods

Postby A for awesome » June 1st, 2018, 2:17 pm

Macbi wrote:Does anyone think it might be possible to get any families of oscillators with three cells? I'm imagining that two of the cells are close together and create a spacefiller which evolves until it encounters the third cell. The third cell causes a collapse which sends the spacefiller back to its original state.

Here's something that comes fairly close:
x = 10, y = 3, rule = B2cei3aq4aeir5y6i/S01c2i3jry4ent5aj6c7e
o$9bo$o!

To fix it, the left side would have to evolve into a spaceship travelling to the right that collides with the leftover dot to produce another quadruple wickstretcher in the original location. It's probably a tall order, but you never know.

Also, an incidental unusual replicator:
x = 10, y = 3, rule = B2cei3aq4aeir5y6i/S01c2i3jry4entw5aj6c7e
o$9bo$o!
x₁=ηx
V ⃰_η=c²√(Λη)
K=(Λu²)/2
Pₐ=1−1/(∫^∞_t₀(p(t)ˡ⁽ᵗ⁾)dt)

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

http://conwaylife.com/wiki/A_for_all

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Re: Smallest Oscillators Supporting Specific Periods

Postby Redstoneboi » June 2nd, 2018, 3:11 am

A for awesome wrote:Here's something that comes fairly close:
x = 10, y = 3, rule = B2cei3aq4aeir5y6i/S01c2i3jry4ent5aj6c7e
o$9bo$o!


hey a logarithmic almost sawtooth
c(>^x^<c)~
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Re: Smallest Oscillators Supporting Specific Periods

Postby bprentice » June 2nd, 2018, 12:00 pm

To demonstrate that a two cell oscillator can be constructed with any period, first install this Golly rule:

@RULE SO
@TREE
num_states=50
num_neighbors=4
num_nodes=201
1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49
2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
4 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
1 0 1 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 49
2 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
3 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
4 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
1 0 2 1 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 49
2 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8
3 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9
4 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10
1 0 2 3 1 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 49
2 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12
3 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13 13
4 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14 14
1 0 2 3 4 1 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 49
2 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16 16
3 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17
4 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18 18
1 0 2 3 4 5 1 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 49
2 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20
3 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21 21
4 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22
1 0 2 3 4 5 6 1 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 49
2 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24 24
3 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25
4 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26
1 0 2 3 4 5 6 7 1 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 49
2 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28
3 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29
4 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30
1 0 2 3 4 5 6 7 8 1 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 49
2 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32
3 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33 33
4 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34 34
1 0 2 3 4 5 6 7 8 9 1 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 49
2 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36
3 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37 37
4 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38 38
1 0 2 3 4 5 6 7 8 9 10 1 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 49
2 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40 40
3 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41 41
4 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42 42
1 0 2 3 4 5 6 7 8 9 10 11 1 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 49
2 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44 44
3 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45
4 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46 46
1 0 2 3 4 5 6 7 8 9 10 11 12 1 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 49
2 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48 48
3 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49 49
4 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50
1 0 2 3 4 5 6 7 8 9 10 11 12 13 1 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 49
2 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52 52
3 53 53 53 53 53 53 53 53 53 53 53 53 53 53 53 53 53 53 53 53 53 53 53 53 53 53 53 53 53 53 53 53 53 53 53 53 53 53 53 53 53 53 53 53 53 53 53 53 53 53
4 54 54 54 54 54 54 54 54 54 54 54 54 54 54 54 54 54 54 54 54 54 54 54 54 54 54 54 54 54 54 54 54 54 54 54 54 54 54 54 54 54 54 54 54 54 54 54 54 54 54
1 0 2 3 4 5 6 7 8 9 10 11 12 13 14 1 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 49
2 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56 56
3 57 57 57 57 57 57 57 57 57 57 57 57 57 57 57 57 57 57 57 57 57 57 57 57 57 57 57 57 57 57 57 57 57 57 57 57 57 57 57 57 57 57 57 57 57 57 57 57 57 57
4 58 58 58 58 58 58 58 58 58 58 58 58 58 58 58 58 58 58 58 58 58 58 58 58 58 58 58 58 58 58 58 58 58 58 58 58 58 58 58 58 58 58 58 58 58 58 58 58 58 58
1 0 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 49
2 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60
3 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61
4 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62 62
1 0 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 49
2 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64 64
3 65 65 65 65 65 65 65 65 65 65 65 65 65 65 65 65 65 65 65 65 65 65 65 65 65 65 65 65 65 65 65 65 65 65 65 65 65 65 65 65 65 65 65 65 65 65 65 65 65 65
4 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66
1 0 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 1 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 49
2 68 68 68 68 68 68 68 68 68 68 68 68 68 68 68 68 68 68 68 68 68 68 68 68 68 68 68 68 68 68 68 68 68 68 68 68 68 68 68 68 68 68 68 68 68 68 68 68 68 68
3 69 69 69 69 69 69 69 69 69 69 69 69 69 69 69 69 69 69 69 69 69 69 69 69 69 69 69 69 69 69 69 69 69 69 69 69 69 69 69 69 69 69 69 69 69 69 69 69 69 69
4 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70
1 0 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 49
2 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72 72
3 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73 73
4 74 74 74 74 74 74 74 74 74 74 74 74 74 74 74 74 74 74 74 74 74 74 74 74 74 74 74 74 74 74 74 74 74 74 74 74 74 74 74 74 74 74 74 74 74 74 74 74 74 74
1 0 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 1 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 49
2 76 76 76 76 76 76 76 76 76 76 76 76 76 76 76 76 76 76 76 76 76 76 76 76 76 76 76 76 76 76 76 76 76 76 76 76 76 76 76 76 76 76 76 76 76 76 76 76 76 76
3 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77 77
4 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78 78
1 0 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 49
2 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80
3 81 81 81 81 81 81 81 81 81 81 81 81 81 81 81 81 81 81 81 81 81 81 81 81 81 81 81 81 81 81 81 81 81 81 81 81 81 81 81 81 81 81 81 81 81 81 81 81 81 81
4 82 82 82 82 82 82 82 82 82 82 82 82 82 82 82 82 82 82 82 82 82 82 82 82 82 82 82 82 82 82 82 82 82 82 82 82 82 82 82 82 82 82 82 82 82 82 82 82 82 82
1 0 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 1 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 49
2 84 84 84 84 84 84 84 84 84 84 84 84 84 84 84 84 84 84 84 84 84 84 84 84 84 84 84 84 84 84 84 84 84 84 84 84 84 84 84 84 84 84 84 84 84 84 84 84 84 84
3 85 85 85 85 85 85 85 85 85 85 85 85 85 85 85 85 85 85 85 85 85 85 85 85 85 85 85 85 85 85 85 85 85 85 85 85 85 85 85 85 85 85 85 85 85 85 85 85 85 85
4 86 86 86 86 86 86 86 86 86 86 86 86 86 86 86 86 86 86 86 86 86 86 86 86 86 86 86 86 86 86 86 86 86 86 86 86 86 86 86 86 86 86 86 86 86 86 86 86 86 86
1 0 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 1 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 49
2 88 88 88 88 88 88 88 88 88 88 88 88 88 88 88 88 88 88 88 88 88 88 88 88 88 88 88 88 88 88 88 88 88 88 88 88 88 88 88 88 88 88 88 88 88 88 88 88 88 88
3 89 89 89 89 89 89 89 89 89 89 89 89 89 89 89 89 89 89 89 89 89 89 89 89 89 89 89 89 89 89 89 89 89 89 89 89 89 89 89 89 89 89 89 89 89 89 89 89 89 89
4 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90
1 0 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 1 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 49
2 92 92 92 92 92 92 92 92 92 92 92 92 92 92 92 92 92 92 92 92 92 92 92 92 92 92 92 92 92 92 92 92 92 92 92 92 92 92 92 92 92 92 92 92 92 92 92 92 92 92
3 93 93 93 93 93 93 93 93 93 93 93 93 93 93 93 93 93 93 93 93 93 93 93 93 93 93 93 93 93 93 93 93 93 93 93 93 93 93 93 93 93 93 93 93 93 93 93 93 93 93
4 94 94 94 94 94 94 94 94 94 94 94 94 94 94 94 94 94 94 94 94 94 94 94 94 94 94 94 94 94 94 94 94 94 94 94 94 94 94 94 94 94 94 94 94 94 94 94 94 94 94
1 0 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 1 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 49
2 96 96 96 96 96 96 96 96 96 96 96 96 96 96 96 96 96 96 96 96 96 96 96 96 96 96 96 96 96 96 96 96 96 96 96 96 96 96 96 96 96 96 96 96 96 96 96 96 96 96
3 97 97 97 97 97 97 97 97 97 97 97 97 97 97 97 97 97 97 97 97 97 97 97 97 97 97 97 97 97 97 97 97 97 97 97 97 97 97 97 97 97 97 97 97 97 97 97 97 97 97
4 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98 98
1 0 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 1 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 49
2 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100
3 101 101 101 101 101 101 101 101 101 101 101 101 101 101 101 101 101 101 101 101 101 101 101 101 101 101 101 101 101 101 101 101 101 101 101 101 101 101 101 101 101 101 101 101 101 101 101 101 101 101
4 102 102 102 102 102 102 102 102 102 102 102 102 102 102 102 102 102 102 102 102 102 102 102 102 102 102 102 102 102 102 102 102 102 102 102 102 102 102 102 102 102 102 102 102 102 102 102 102 102 102
1 0 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 1 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 49
2 104 104 104 104 104 104 104 104 104 104 104 104 104 104 104 104 104 104 104 104 104 104 104 104 104 104 104 104 104 104 104 104 104 104 104 104 104 104 104 104 104 104 104 104 104 104 104 104 104 104
3 105 105 105 105 105 105 105 105 105 105 105 105 105 105 105 105 105 105 105 105 105 105 105 105 105 105 105 105 105 105 105 105 105 105 105 105 105 105 105 105 105 105 105 105 105 105 105 105 105 105
4 106 106 106 106 106 106 106 106 106 106 106 106 106 106 106 106 106 106 106 106 106 106 106 106 106 106 106 106 106 106 106 106 106 106 106 106 106 106 106 106 106 106 106 106 106 106 106 106 106 106
1 0 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 1 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 49
2 108 108 108 108 108 108 108 108 108 108 108 108 108 108 108 108 108 108 108 108 108 108 108 108 108 108 108 108 108 108 108 108 108 108 108 108 108 108 108 108 108 108 108 108 108 108 108 108 108 108
3 109 109 109 109 109 109 109 109 109 109 109 109 109 109 109 109 109 109 109 109 109 109 109 109 109 109 109 109 109 109 109 109 109 109 109 109 109 109 109 109 109 109 109 109 109 109 109 109 109 109
4 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110 110
1 0 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 1 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 49
2 112 112 112 112 112 112 112 112 112 112 112 112 112 112 112 112 112 112 112 112 112 112 112 112 112 112 112 112 112 112 112 112 112 112 112 112 112 112 112 112 112 112 112 112 112 112 112 112 112 112
3 113 113 113 113 113 113 113 113 113 113 113 113 113 113 113 113 113 113 113 113 113 113 113 113 113 113 113 113 113 113 113 113 113 113 113 113 113 113 113 113 113 113 113 113 113 113 113 113 113 113
4 114 114 114 114 114 114 114 114 114 114 114 114 114 114 114 114 114 114 114 114 114 114 114 114 114 114 114 114 114 114 114 114 114 114 114 114 114 114 114 114 114 114 114 114 114 114 114 114 114 114
1 0 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 1 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 49
2 116 116 116 116 116 116 116 116 116 116 116 116 116 116 116 116 116 116 116 116 116 116 116 116 116 116 116 116 116 116 116 116 116 116 116 116 116 116 116 116 116 116 116 116 116 116 116 116 116 116
3 117 117 117 117 117 117 117 117 117 117 117 117 117 117 117 117 117 117 117 117 117 117 117 117 117 117 117 117 117 117 117 117 117 117 117 117 117 117 117 117 117 117 117 117 117 117 117 117 117 117
4 118 118 118 118 118 118 118 118 118 118 118 118 118 118 118 118 118 118 118 118 118 118 118 118 118 118 118 118 118 118 118 118 118 118 118 118 118 118 118 118 118 118 118 118 118 118 118 118 118 118
1 0 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 1 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 49
2 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120 120
3 121 121 121 121 121 121 121 121 121 121 121 121 121 121 121 121 121 121 121 121 121 121 121 121 121 121 121 121 121 121 121 121 121 121 121 121 121 121 121 121 121 121 121 121 121 121 121 121 121 121
4 122 122 122 122 122 122 122 122 122 122 122 122 122 122 122 122 122 122 122 122 122 122 122 122 122 122 122 122 122 122 122 122 122 122 122 122 122 122 122 122 122 122 122 122 122 122 122 122 122 122
1 0 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 1 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 49
2 124 124 124 124 124 124 124 124 124 124 124 124 124 124 124 124 124 124 124 124 124 124 124 124 124 124 124 124 124 124 124 124 124 124 124 124 124 124 124 124 124 124 124 124 124 124 124 124 124 124
3 125 125 125 125 125 125 125 125 125 125 125 125 125 125 125 125 125 125 125 125 125 125 125 125 125 125 125 125 125 125 125 125 125 125 125 125 125 125 125 125 125 125 125 125 125 125 125 125 125 125
4 126 126 126 126 126 126 126 126 126 126 126 126 126 126 126 126 126 126 126 126 126 126 126 126 126 126 126 126 126 126 126 126 126 126 126 126 126 126 126 126 126 126 126 126 126 126 126 126 126 126
1 0 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 1 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 49
2 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128
3 129 129 129 129 129 129 129 129 129 129 129 129 129 129 129 129 129 129 129 129 129 129 129 129 129 129 129 129 129 129 129 129 129 129 129 129 129 129 129 129 129 129 129 129 129 129 129 129 129 129
4 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130 130
1 0 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 1 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 49
2 132 132 132 132 132 132 132 132 132 132 132 132 132 132 132 132 132 132 132 132 132 132 132 132 132 132 132 132 132 132 132 132 132 132 132 132 132 132 132 132 132 132 132 132 132 132 132 132 132 132
3 133 133 133 133 133 133 133 133 133 133 133 133 133 133 133 133 133 133 133 133 133 133 133 133 133 133 133 133 133 133 133 133 133 133 133 133 133 133 133 133 133 133 133 133 133 133 133 133 133 133
4 134 134 134 134 134 134 134 134 134 134 134 134 134 134 134 134 134 134 134 134 134 134 134 134 134 134 134 134 134 134 134 134 134 134 134 134 134 134 134 134 134 134 134 134 134 134 134 134 134 134
1 0 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 1 36 37 38 39 40 41 42 43 44 45 46 47 48 49 49
2 136 136 136 136 136 136 136 136 136 136 136 136 136 136 136 136 136 136 136 136 136 136 136 136 136 136 136 136 136 136 136 136 136 136 136 136 136 136 136 136 136 136 136 136 136 136 136 136 136 136
3 137 137 137 137 137 137 137 137 137 137 137 137 137 137 137 137 137 137 137 137 137 137 137 137 137 137 137 137 137 137 137 137 137 137 137 137 137 137 137 137 137 137 137 137 137 137 137 137 137 137
4 138 138 138 138 138 138 138 138 138 138 138 138 138 138 138 138 138 138 138 138 138 138 138 138 138 138 138 138 138 138 138 138 138 138 138 138 138 138 138 138 138 138 138 138 138 138 138 138 138 138
1 0 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 1 37 38 39 40 41 42 43 44 45 46 47 48 49 49
2 140 140 140 140 140 140 140 140 140 140 140 140 140 140 140 140 140 140 140 140 140 140 140 140 140 140 140 140 140 140 140 140 140 140 140 140 140 140 140 140 140 140 140 140 140 140 140 140 140 140
3 141 141 141 141 141 141 141 141 141 141 141 141 141 141 141 141 141 141 141 141 141 141 141 141 141 141 141 141 141 141 141 141 141 141 141 141 141 141 141 141 141 141 141 141 141 141 141 141 141 141
4 142 142 142 142 142 142 142 142 142 142 142 142 142 142 142 142 142 142 142 142 142 142 142 142 142 142 142 142 142 142 142 142 142 142 142 142 142 142 142 142 142 142 142 142 142 142 142 142 142 142
1 0 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 1 38 39 40 41 42 43 44 45 46 47 48 49 49
2 144 144 144 144 144 144 144 144 144 144 144 144 144 144 144 144 144 144 144 144 144 144 144 144 144 144 144 144 144 144 144 144 144 144 144 144 144 144 144 144 144 144 144 144 144 144 144 144 144 144
3 145 145 145 145 145 145 145 145 145 145 145 145 145 145 145 145 145 145 145 145 145 145 145 145 145 145 145 145 145 145 145 145 145 145 145 145 145 145 145 145 145 145 145 145 145 145 145 145 145 145
4 146 146 146 146 146 146 146 146 146 146 146 146 146 146 146 146 146 146 146 146 146 146 146 146 146 146 146 146 146 146 146 146 146 146 146 146 146 146 146 146 146 146 146 146 146 146 146 146 146 146
1 0 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 1 39 40 41 42 43 44 45 46 47 48 49 49
2 148 148 148 148 148 148 148 148 148 148 148 148 148 148 148 148 148 148 148 148 148 148 148 148 148 148 148 148 148 148 148 148 148 148 148 148 148 148 148 148 148 148 148 148 148 148 148 148 148 148
3 149 149 149 149 149 149 149 149 149 149 149 149 149 149 149 149 149 149 149 149 149 149 149 149 149 149 149 149 149 149 149 149 149 149 149 149 149 149 149 149 149 149 149 149 149 149 149 149 149 149
4 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150 150
1 0 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 1 40 41 42 43 44 45 46 47 48 49 49
2 152 152 152 152 152 152 152 152 152 152 152 152 152 152 152 152 152 152 152 152 152 152 152 152 152 152 152 152 152 152 152 152 152 152 152 152 152 152 152 152 152 152 152 152 152 152 152 152 152 152
3 153 153 153 153 153 153 153 153 153 153 153 153 153 153 153 153 153 153 153 153 153 153 153 153 153 153 153 153 153 153 153 153 153 153 153 153 153 153 153 153 153 153 153 153 153 153 153 153 153 153
4 154 154 154 154 154 154 154 154 154 154 154 154 154 154 154 154 154 154 154 154 154 154 154 154 154 154 154 154 154 154 154 154 154 154 154 154 154 154 154 154 154 154 154 154 154 154 154 154 154 154
1 0 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 1 41 42 43 44 45 46 47 48 49 49
2 156 156 156 156 156 156 156 156 156 156 156 156 156 156 156 156 156 156 156 156 156 156 156 156 156 156 156 156 156 156 156 156 156 156 156 156 156 156 156 156 156 156 156 156 156 156 156 156 156 156
3 157 157 157 157 157 157 157 157 157 157 157 157 157 157 157 157 157 157 157 157 157 157 157 157 157 157 157 157 157 157 157 157 157 157 157 157 157 157 157 157 157 157 157 157 157 157 157 157 157 157
4 158 158 158 158 158 158 158 158 158 158 158 158 158 158 158 158 158 158 158 158 158 158 158 158 158 158 158 158 158 158 158 158 158 158 158 158 158 158 158 158 158 158 158 158 158 158 158 158 158 158
1 0 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 1 42 43 44 45 46 47 48 49 49
2 160 160 160 160 160 160 160 160 160 160 160 160 160 160 160 160 160 160 160 160 160 160 160 160 160 160 160 160 160 160 160 160 160 160 160 160 160 160 160 160 160 160 160 160 160 160 160 160 160 160
3 161 161 161 161 161 161 161 161 161 161 161 161 161 161 161 161 161 161 161 161 161 161 161 161 161 161 161 161 161 161 161 161 161 161 161 161 161 161 161 161 161 161 161 161 161 161 161 161 161 161
4 162 162 162 162 162 162 162 162 162 162 162 162 162 162 162 162 162 162 162 162 162 162 162 162 162 162 162 162 162 162 162 162 162 162 162 162 162 162 162 162 162 162 162 162 162 162 162 162 162 162
1 0 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 1 43 44 45 46 47 48 49 49
2 164 164 164 164 164 164 164 164 164 164 164 164 164 164 164 164 164 164 164 164 164 164 164 164 164 164 164 164 164 164 164 164 164 164 164 164 164 164 164 164 164 164 164 164 164 164 164 164 164 164
3 165 165 165 165 165 165 165 165 165 165 165 165 165 165 165 165 165 165 165 165 165 165 165 165 165 165 165 165 165 165 165 165 165 165 165 165 165 165 165 165 165 165 165 165 165 165 165 165 165 165
4 166 166 166 166 166 166 166 166 166 166 166 166 166 166 166 166 166 166 166 166 166 166 166 166 166 166 166 166 166 166 166 166 166 166 166 166 166 166 166 166 166 166 166 166 166 166 166 166 166 166
1 0 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 1 44 45 46 47 48 49 49
2 168 168 168 168 168 168 168 168 168 168 168 168 168 168 168 168 168 168 168 168 168 168 168 168 168 168 168 168 168 168 168 168 168 168 168 168 168 168 168 168 168 168 168 168 168 168 168 168 168 168
3 169 169 169 169 169 169 169 169 169 169 169 169 169 169 169 169 169 169 169 169 169 169 169 169 169 169 169 169 169 169 169 169 169 169 169 169 169 169 169 169 169 169 169 169 169 169 169 169 169 169
4 170 170 170 170 170 170 170 170 170 170 170 170 170 170 170 170 170 170 170 170 170 170 170 170 170 170 170 170 170 170 170 170 170 170 170 170 170 170 170 170 170 170 170 170 170 170 170 170 170 170
1 0 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 1 45 46 47 48 49 49
2 172 172 172 172 172 172 172 172 172 172 172 172 172 172 172 172 172 172 172 172 172 172 172 172 172 172 172 172 172 172 172 172 172 172 172 172 172 172 172 172 172 172 172 172 172 172 172 172 172 172
3 173 173 173 173 173 173 173 173 173 173 173 173 173 173 173 173 173 173 173 173 173 173 173 173 173 173 173 173 173 173 173 173 173 173 173 173 173 173 173 173 173 173 173 173 173 173 173 173 173 173
4 174 174 174 174 174 174 174 174 174 174 174 174 174 174 174 174 174 174 174 174 174 174 174 174 174 174 174 174 174 174 174 174 174 174 174 174 174 174 174 174 174 174 174 174 174 174 174 174 174 174
1 0 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 1 46 47 48 49 49
2 176 176 176 176 176 176 176 176 176 176 176 176 176 176 176 176 176 176 176 176 176 176 176 176 176 176 176 176 176 176 176 176 176 176 176 176 176 176 176 176 176 176 176 176 176 176 176 176 176 176
3 177 177 177 177 177 177 177 177 177 177 177 177 177 177 177 177 177 177 177 177 177 177 177 177 177 177 177 177 177 177 177 177 177 177 177 177 177 177 177 177 177 177 177 177 177 177 177 177 177 177
4 178 178 178 178 178 178 178 178 178 178 178 178 178 178 178 178 178 178 178 178 178 178 178 178 178 178 178 178 178 178 178 178 178 178 178 178 178 178 178 178 178 178 178 178 178 178 178 178 178 178
1 0 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 1 47 48 49 49
2 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180 180
3 181 181 181 181 181 181 181 181 181 181 181 181 181 181 181 181 181 181 181 181 181 181 181 181 181 181 181 181 181 181 181 181 181 181 181 181 181 181 181 181 181 181 181 181 181 181 181 181 181 181
4 182 182 182 182 182 182 182 182 182 182 182 182 182 182 182 182 182 182 182 182 182 182 182 182 182 182 182 182 182 182 182 182 182 182 182 182 182 182 182 182 182 182 182 182 182 182 182 182 182 182
1 0 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 1 48 49 49
2 184 184 184 184 184 184 184 184 184 184 184 184 184 184 184 184 184 184 184 184 184 184 184 184 184 184 184 184 184 184 184 184 184 184 184 184 184 184 184 184 184 184 184 184 184 184 184 184 184 184
3 185 185 185 185 185 185 185 185 185 185 185 185 185 185 185 185 185 185 185 185 185 185 185 185 185 185 185 185 185 185 185 185 185 185 185 185 185 185 185 185 185 185 185 185 185 185 185 185 185 185
4 186 186 186 186 186 186 186 186 186 186 186 186 186 186 186 186 186 186 186 186 186 186 186 186 186 186 186 186 186 186 186 186 186 186 186 186 186 186 186 186 186 186 186 186 186 186 186 186 186 186
1 0 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 1 49 49
2 188 188 188 188 188 188 188 188 188 188 188 188 188 188 188 188 188 188 188 188 188 188 188 188 188 188 188 188 188 188 188 188 188 188 188 188 188 188 188 188 188 188 188 188 188 188 188 188 188 188
3 189 189 189 189 189 189 189 189 189 189 189 189 189 189 189 189 189 189 189 189 189 189 189 189 189 189 189 189 189 189 189 189 189 189 189 189 189 189 189 189 189 189 189 189 189 189 189 189 189 189
4 190 190 190 190 190 190 190 190 190 190 190 190 190 190 190 190 190 190 190 190 190 190 190 190 190 190 190 190 190 190 190 190 190 190 190 190 190 190 190 190 190 190 190 190 190 190 190 190 190 190
1 0 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 1 49
2 192 192 192 192 192 192 192 192 192 192 192 192 192 192 192 192 192 192 192 192 192 192 192 192 192 192 192 192 192 192 192 192 192 192 192 192 192 192 192 192 192 192 192 192 192 192 192 192 192 192
3 193 193 193 193 193 193 193 193 193 193 193 193 193 193 193 193 193 193 193 193 193 193 193 193 193 193 193 193 193 193 193 193 193 193 193 193 193 193 193 193 193 193 193 193 193 193 193 193 193 193
4 194 194 194 194 194 194 194 194 194 194 194 194 194 194 194 194 194 194 194 194 194 194 194 194 194 194 194 194 194 194 194 194 194 194 194 194 194 194 194 194 194 194 194 194 194 194 194 194 194 194
1 0 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 1
2 196 196 196 196 196 196 196 196 196 196 196 196 196 196 196 196 196 196 196 196 196 196 196 196 196 196 196 196 196 196 196 196 196 196 196 196 196 196 196 196 196 196 196 196 196 196 196 196 196 196
3 197 197 197 197 197 197 197 197 197 197 197 197 197 197 197 197 197 197 197 197 197 197 197 197 197 197 197 197 197 197 197 197 197 197 197 197 197 197 197 197 197 197 197 197 197 197 197 197 197 197
4 198 198 198 198 198 198 198 198 198 198 198 198 198 198 198 198 198 198 198 198 198 198 198 198 198 198 198 198 198 198 198 198 198 198 198 198 198 198 198 198 198 198 198 198 198 198 198 198 198 198
5 3 7 11 15 19 23 27 31 35 39 43 47 51 55 59 63 67 71 75 79 83 87 91 95 99 103 107 111 115 119 123 127 131 135 139 143 147 151 155 159 163 167 171 175 179 183 187 191 195 199


To construct a period N oscillator (1 <= N < 50) set a cell to state 1 and its northern neighbor to state N.

Brian Prentice
bprentice
 
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Re: Smallest Oscillators Supporting Specific Periods

Postby BlinkerSpawn » June 2nd, 2018, 1:28 pm

bprentice wrote:To demonstrate that a two cell oscillator can be constructed with any period, first install this Golly rule:

rule


To construct a period N oscillator (1 <= N < 50) set a cell to state 1 and its northern neighbor to state N.

Brian Prentice

It's easy to construct rules like this so we're restricting ourselves to 2-state rules
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BlinkerSpawn
 
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Re: Smallest Oscillators Supporting Specific Periods

Postby Macbi » June 2nd, 2018, 1:33 pm

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Posts: 587
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Re: Smallest Oscillators Supporting Specific Periods

Postby Macbi » June 12th, 2018, 12:45 pm

I'll write this up later, but for now enjoy some oscillators:
Period,Minrule,Maxrule
1,B2a/S0,B2-ci345678/S012345678
2,B2ac/S,B2-i345678/S12345678
5,B2aci4ej5i/S1e2i3e,B2-ek3-einy45678/S1e2-ac3-y4-c56-i78
6,B2aci4c5y/S1e2k3kq,B23-aeij4-i56-i78/S1e2ekn34-cq5-i678
7,B2-kn3y4c5y/S1e2k3k,B2-k3cknry4-iq56-i78/S1e2ekn3-eiq4-c5678
8,B2ai3e4irw5i/S012i3aeiky4ej5e6i8,B2-ce3-ai4-e5-y6-c7c8/S012-ac34-cn5-iy678
9,B2aci4aez5y6c7e/S1e2k3eik4cey,B2-ek3ckqry4-cijk5-e6-i78/S1e2kn34-iqt5-eij6-ci78
10,B2ai3en4ikrw5iy6c8/S012i3aeik4eik5e6i8,B2ain3-ai4-et5678/S012-ac3-jy4-cw5-iy678
11,B2ai3eq4irw5in6ci/S012ei3aeir4aejy5ejn6i8,B2-ce3-aijy4-ae5-cey67c8/S012ein3-ky4-cikn5-ci678
12,B2ai3er4irw5i7e/S012i3-cjnq4eijnt5ey6ik7e8,B2-ce3-aij4-aet5-cey6-ci7/S012ikn34-ck5-ijn6-c78
13,B2ai3eq4jr5i6i/S012ei3aeiky4e5eqr6ci8,B2-ce3-ai4-aeiw5-ey67c/S012-ac34-cir5-ainy67c8
14,B2ai3en4inrw5i6ci8/S012ei3aeiry4eijrw5acejr6ik8,B2-ce3-aij4-aetz5-y67c8/S012-ac3-kq4-cn5-iny6-ac7c8
15,B2ai3eqr4jr5ei6ik/S012ei3aeiky4et5eq6ci8,B2-ce3-ai4-aeitw5-y67c/S012-ac34-cir5-ainy6-a7c8
16,B2ai3ejkn4inrtw5cin6c7e8/S012i3aeiky4eiw5ej6i8,B2ain3-airy4-ae5-ky678/S012ikn34-cjknt5-iy6-ac78
17,B2ai3eknr4irtw5i6c7/S012i3aeiky4eijnwy5e6ci8,B2ain3-aijy4-ae5-cey67/S012ikn34-ck5-ijry67c8
18,B2ai3-acij4iknrw5iq6a/S012i3aeiqy4eint5en6aik8,B2ai3-aij4-aet5-cey6aen7c/S012ikn3-knr4-cky5-cijr6-c7c8
19,B2aik3enqy4aikrwy5ciy78/S012ei3-cnr4aej5ejn6i8,B2-ce3-aijr4-cet5-q6aen78/S012eik3-cr4-ciny5cejnq6-ac78
20,B2ai3en4aknrwy5ceiny6ak8/S012ei3eij4aeir5eijnq6i7e8,B2ain3ceknq4-cei5-jr6-ci78/S012eik3eijnr4-ckt5-ry6-ck78
21,B2ai3enq4-ceqtyz5ijry6c8/S012-ac3-cjqy4eijkw5ek6ci8,B2ain3eknqy4-etz5-ceq67c8/S012-ac3-qy4-cnrty5-ijy6-k7c8
22,B2ai3ekn4cjkr5ainy6cek7e/S012ei3-cqr4eiry5ekqy6i8,B2ain3ceknq4-aeitw5-ceq6-i7/S012-ac3-q4-acjnqt5-ijn6-c7c8
23,B2ai3enq4aijnrw5ijy6ci8/S012ei3aeik4aejktw5ery6ik7e8,B2-ce3ceknq4-etz5-e6-k8/S012eik3-cry4-cinqy5-aijq6-ac78
24,B2ai3er4irw5i6i7e/S012i3-cjnq4eijn5ey6ik7e8,B2-ce3-aij4-aet5-cey6-c7/S012ikn34-ckt5-ijn6-c78
25,B2ain3eny4ajkry5-ejqr6aik7e8/S012eik3-ckry4ei5e6-en8,B2ain3-aijr4-eitw5-ejr6-c78/S012eik3-cky4-acjkt5acekq68
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138,B2aei3ceqy4-ackqyz5ejny6-an/S01e2ek3akr4ejnrty5-ackr6ac7e8,B2-ck3ceqy4-acky5ejnry6-a7c/S01e2ek3akr4-acikqw5-c6-ik78
140,B2aei3eqy4-ckqwy5acn6i8/S01e3kr4-iq5-akq6a7c8,B2aei3eqy4-ckq5acnr6ei8/S01e3kr4-iq5-a6an7c8
142,B2aei3cekq4-actwyz5cejy6-kn/S01e2ek3cr4ejkt5-anqr6-in78,B2aei3cekq4-act5-inr67c/S01e2ek3cr4ejktz5-nqr6-i78
144,B2aei3cnqy4-acjtwy5-aijq6ci8/S01e2ek3ackr4-aeijz5ainry6ce,B2-ck3cnqy4-acjt5-ij6-k7c8/S01e2ek3ackr4-aeij5-cejk6cen7c
146,B2-ck3ey4-acqtyz5-aik6-kn8/S01e2k3r4cknr5jkn6ac,B2-ck3ey4-acty5-ik67c8/S01e2k3akr4-aeiqty5ajknq6acn7c
148,B2-ck3eky4-ckwz5aekny6cei/S01e3akr4cejky5aceq6e7c8,B2-ck3ekqy4-ck5aekny6-k7c/S01e3akr4-ainrt5aceq6ekn7c8
150,B2-ck3ek4-ackwy5ejny6cik8/S01e3ar4ejqrty5-cnqr6ace78,B2-ck3cek4-acky5-cikr6-ae7c8/S01e3ar4-acikn5-cnq6-i78
152,B2aei3ceqy4-actwyz5cejnq6ikn8/S01e2ek3akr4ejnt5-ckr6-in78,B2aei3ceqy4-acty5-iky6-ac7c8/S01e2ek3akr4ejntwz5-ck6-i78
154,B2-ck3ceq4-acqwz5-aci6aci8/S01e2k3kr4enrtyz5-acn6-in7e8,B2-ck3ceq4-ac5-aci6-ek7c8/S01e2k3kr4-acijk5-acn6-i78
156,B2aei3ceqy4-actwyz5cejnq6ikn8/S01e2ek3akr4ejnt5-cr6-in78,B2aei3ceqy4-acty5-iky6-ac7c8/S01e2ek3akr4ejntwz5-c6-i78
158,B2aei3cekq4-ckqwz5cny6ik7c/S01e3r4acejny5-iqr6ce7e8,B2aei3cekq4-ck5cnqry6ikn7c/S01e3ar4-ikrtz5-ir6-ai7e8
160,B2-ck3eky4-acwyz5-ijkn6cei8/S01e2e3qr4-aiqwz5air6ack7e,B2-ck3ekqy4-acy5-ijn6-k8/S01e2e3aqr4-aiq5aiqr6ack7
162,B2-ck3cek4-acwyz5-ainr6cik8/S01e3kqr4-ait5-ek6ekn7e8,B2-ck3cek4-acw5-in6-a7c8/S01e3kqr4-ait5-ek6ekn78
164,B2aei3ceqy4-ackwyz5-ikqy6ci/S01e2e3kqr4cekqry5cery6ac78,B2-ck3ceqy4-ackyz5-ikqy6-e7c/S01e2e3kqr4-aijntw5acery6acn78
166,B2aei3ceqy4-actwyz5cejn6-ac8/S01e2ek3akr4ejnt5-ckr6-in78,B2aei3ceqy4-acty5-ikqy6-ac7c8/S01e2ek3akr4ejntwz5-c6-i78
168,B2-ck3eqy4eijnrt5acekn6-k8/S01e2ek3qr4ceknqr5-jkqr6ace7c8,B2-ck3eqy4-ackz5-ijqy6-k7c8/S01e2ek3qr4-aijtyz5-jkr6-ik7c8
170,B2aei3ceqy4-acqw5ejr6cik7c8/S01e2k3kr4cejkrt5-ckq6ack8,B2aei3ceqy4-ac5aejr6-ae7c8/S01e2k3kr4-ainqyz5-cq6ack7c8
172,B2-ck3eqy4eijnrt5acekn6-k8/S01e2ek3qr4ceknqr5-jkqr6ac7c8,B2-ck3eqy4-ackz5-ijqy6-k7c8/S01e2ek3qr4-aijtyz5-jkr6acn7c8
174,B2aei3eknq4-acqwyz5aceny6-en/S01e2ek3qr4cenqry5ejny6ek7e,B2aei3eknq4-acqy5-ijk6-n7c/S01e2ek3qr4-aijktw5ejny6-ci7
176,B2aei3ceknq4-ackwz5ckny6cik7c/S01e2e3qr4acenqy5aeqry6ace7e8,B2aei3ceknq4-ack5cknqy6-an7c/S01e2e3qr4-ijkrtw5aeqry6-ik78
178,B2-ck3cenqy4ijknrt5-aikq6-n8/S01e2k3cqr4-aijkn5-jkqr6c8,B2-ck3cenqy4-aceyz5-aiq68/S01e2k3cqr4-aijkn5-jkqr6cen7c8
180,B2aei3ceqy4-acqwz5ejr6-e8/S01e3kr4-ainwyz5-ckq6ack8,B2aei3ceqy4-acz5aejqr6-e7c8/S01e3kr4-ainy5-ckq6-ei7c8
182,B2-ck3cekny4-actwy5cjnqy6-ek8/S01e2e3ckr4-aikqyz5-cknq6ack7e8,B2-ck3-aijr4-actwy5-eir6-e7c8/S01e2e3ckr4-aikqy5-cknq6-ei7e8
184,B2aei3ceqy4-acqwz5ejkr6-e8/S01e3kr4-ainwyz5-ckq6ack8,B2aei3ceqy4-acz5-ciny6-e7c8/S01e3kr4-ainy5-ckq6-ei7c8
186,B2-ck3eknqy4-actwyz5-ijkq6aci7c/S01e2e3akqr4-aiqrwz5-cjkq6ak7e8,B2-ck3-aijr4-actwz5-ijq6-e7c/S01e2e3akqr4-aiqrw5-cjkq6akn7e8
188,B2-ck3cenq4-ackz5ejnqy6-an8/S01e2e3kr4-inw5ceiry6ce8,B2-ck3cenq4-ackz5-aci6-a8/S01e2e3kr4-inw5-ajnq6-ai7c8
190,B2aei3cekn4-ackwz5cejn6cik8/S01e3kqr4-ikntwz5-jr6c7e8,B2-ck3cekn4-ack5-iky6-e8/S01e3kqr4-iknt5-jr6acn7e8
192,B2aei3eknq4-acqwyz5ceny6-en/S01e2e3r4-ikrw5-jkn6aek7e8,B2aei3ceknq4-acqyz5-ijqr6-e7c/S01e2e3r4-ikrw5-jkn6-ci78
194,B2aei3ceknq4-acktwz5-eik6ci7c8/S01e2ek3akr4-aiqr5aeiny6ak7e8,B2-ck3ceknq4-ackt5-ei6-ae7c8/S01e2ek3akr4-aiqr5-cjqr6akn7e8
196,B2aei3eknq4-acqwyz5ceny6cik/S01e2e3r4-ikrw5-jkn6aek7e8,B2aei3ceknq4-acqyz5-ijqr6-ae7c/S01e2e3r4-ikrw5-jkn6-ci78
198,B2aei3ceknq4-acktwz5-aiqr6aci8/S01e2e3ar4-aikqr5aeiy6ak7e8,B2aei3ceknq4-ackt5-i6-e7c8/S01e2e3ar4-aikqr5aeiqy6akn78
200,B2-ck3cenq4ijknry5ckny6-en8/S01e2ek3kry4acekrt5aceiy6ace,B2-ck3cenq4-acet5cknry6-e7c8/S01e2ek3ckry4-ijnqwy5aceiy6-ik7c
202,B2aei3eknq4-acqyz5acnry6aci7c/S01e2e3kry4-aijqwz5-ijkq6ck8,B2aei3ceknq4-acqy5-eik6-k7c/S01e2e3kry4-aijw5-ijk6ckn7c8
204,B2aei3ceknq4-acktwz5-aiqr6aci8/S01e2e3ar4-aikqrz5aeiy6ak7e8,B2aei3ceknq4-ackt5-i6-e7c8/S01e2e3ar4-aikqrz5aeiqy6akn78
208,B2aei3eknqy4-acwyz5-ij6ci/S01e2en3ry4cejry5cey6cek8,B2aei3eknqy4-acy5-ij6-ak7c/S01e2en3kry4-aikntw5cey6-ai8
212,B2aei3eknqy4-acqwyz5-ij6ci/S01e2en3ry4cejry5cey6cek8,B2aei3eknqy4-acqy5-ij6-ak7c/S01e2en3kry4-aikntw5cey6-ai8
All of them have the same starting state.

Period 212:
x = 3, y = 1, rule = B2aei3eknqy4-acqy5-ij6-ak7c/S01e2en3kry4cejqryz5cey6-ai8
obo!
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Re: Smallest Oscillators Supporting Specific Periods

Postby Majestas32 » June 12th, 2018, 2:22 pm

Did I read that right? 2 cell for everything up to 102?
Please, stop spam searching Snowflakes.
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Re: Smallest Oscillators Supporting Specific Periods

Postby Macbi » June 12th, 2018, 2:24 pm

Majestas32 wrote:Did I read that right? 2 cell for everything up to 102?
Yep. I basically did what I described here, although I only searched within the 7x7 oscillators, and didn't even manage to exhaust all of those.
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Re: Smallest Oscillators Supporting Specific Periods

Postby Macbi » June 14th, 2018, 6:41 am

4 cell, period 4N+3 (N>1):
x = 6, y = 4, rule = B2ei3ir4r/S012ce3j
5bo$2bo$o$2bo!
This completes the proof that all periods can be achieved in at most 4 cells.

AbhpzTa wrote:4N: 4 cells (N >= 2)
x = 9, y = 24, rule = B2ei3i/S012ce3j
2bo$o4bo$2bo5$2bo$o5bo$2bo5$2bo$o6bo$2bo5$2bo$o7bo$2bo!

4N+2: 4 cells (N >= 2)
x = 9, y = 24, rule = B2ei3i/S012ce3j4t
2bo$o4bo$2bo5$2bo$o5bo$2bo5$2bo$o6bo$2bo5$2bo$o7bo$2bo!
2718281828 wrote:4 cells for 4N+1 (N>2):
x = 9, y = 22, rule = B2ei3i4r/S012cek3j4t
5bo$2bo$o$2bo3$6bo$2bo$o$2bo3$7bo$2bo$o$2bo3$8bo$2bo$o$2bo!


EDIT: BlinkerSpawn points out that 2718281828's alleged 4N+1 is actually a 4N+3. The following 4N+1 is needed to complete the proof.
x = 15, y = 4, rule = B2ein3aciq4jkny5any6ck/S012ce3ejnr4jknrwy5-aiky6ci7e
14bo$2bo$o2bo$2bo!
Last edited by Macbi on February 4th, 2019, 11:51 am, edited 1 time in total.
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Re: Smallest Oscillators Supporting Specific Periods

Postby Macbi » January 2nd, 2019, 4:23 pm

I promised to write up an explanation of how I found all those oscillators and then I never did. But in fact the code is pretty much what I described in this comment.
Macbi wrote:Start with a fixed 100 by 100 grid with two cells one cell apart horizontally in the middle of it (actually use symmetry to only simulate one quadrant). Store a partial rule which starts with B2a turned on, B01ce turned off and all other transitions unknown.

Then repeat the following procedure recursively. List all the transitions in the current pattern that aren't yet assigned. For each possible assignment to them evolve the pattern one generation. Then if the pattern has returned to its original state (or its rotation) you've found an oscillator. The current partial rule tells you exactly all the rules it works in. Save the rule for this oscillator and backtrack. If instead you reach a pattern that you've seen in a previous generation (or it's rotation) you have some other oscillator and you should backtrack immediately. You should also backtrack immediately if you create a pattern with two adjacent cells on the outer edge. Because of 2a it will explode.

I believe that this will quite quickly produce a list of all possible 2-cell oscillators. It's searching through the whole 2^transitions sized rulespace, but it can eliminate multiple rules at a time because it keeps track for each evolution of only the transitions needed for that evolution to work.

The code I used was the following:
#include <iostream>
#include <algorithm>
#include <string>
#include <fstream>

const int max_x = 4;
const int max_y = 4;
const int max_t = 512;
const int number_of_transitions = 102;
const std::string output_filename = "oscillators-7x7.csv";
const std::string log_filename = "log-7x7";
const int reporting_interval = 5000000;

int transition_from_cells(int parent_0,
                          int parent_1,
                          int parent_2,
                          int parent_3,
                          int parent_4,
                          int parent_5,
                          int parent_6,
                          int parent_7,
                          int parent_8){
  int index = (parent_0<<8)|(parent_1<<7)|(parent_2<<6)|(parent_3<<5)|(parent_4<<4)|(parent_5<<3)|(parent_6<<2)|(parent_7<<1)|parent_8;
  static const int lookup_table[512] = {0,1,2,6,1,3,6,13,2,5,4,17,6,14,12,22,1,8,5,16,3,9,14,24,6,16,17,30,13,24,22,35,2,5,7,18,5,15,18,29,4,11,10,27,17,21,28,37,6,16,18,31,14,25,23,41,12,26,28,39,22,40,36,45,1,3,5,14,8,9,16,24,5,15,11,21,16,25,26,40,3,9,15,25,9,19,25,33,14,25,21,34,24,33,40,43,6,14,18,23,16,25,31,41,17,21,27,38,30,34,39,44,13,24,29,41,24,33,41,46,22,40,37,44,35,43,45,49,2,6,4,12,5,14,17,22,7,18,10,28,18,23,28,36,5,16,11,26,15,25,21,40,18,31,27,39,29,41,37,45,4,17,10,28,11,21,27,37,10,27,20,32,27,38,32,42,17,30,27,39,21,34,38,44,28,39,32,47,37,44,42,48,6,13,17,22,16,24,30,35,18,29,27,37,31,41,39,45,14,24,21,40,25,33,34,43,23,41,38,44,41,46,44,49,12,22,28,36,26,40,39,45,28,37,32,42,39,44,47,48,22,35,37,45,40,43,44,49,36,45,42,48,45,49,48,50,51,52,53,57,52,54,57,64,53,56,55,68,57,65,63,73,52,59,56,67,54,60,65,75,57,67,68,81,64,75,73,86,53,56,58,69,56,66,69,80,55,62,61,78,68,72,79,88,57,67,69,82,65,76,74,92,63,77,79,90,73,91,87,96,52,54,56,65,59,60,67,75,56,66,62,72,67,76,77,91,54,60,66,76,60,70,76,84,65,76,72,85,75,84,91,94,57,65,69,74,67,76,82,92,68,72,78,89,81,85,90,95,64,75,80,92,75,84,92,97,73,91,88,95,86,94,96,100,53,57,55,63,56,65,68,73,58,69,61,79,69,74,79,87,56,67,62,77,66,76,72,91,69,82,78,90,80,92,88,96,55,68,61,79,62,72,78,88,61,78,71,83,78,89,83,93,68,81,78,90,72,85,89,95,79,90,83,98,88,95,93,99,57,64,68,73,67,75,81,86,69,80,78,88,82,92,90,96,65,75,72,91,76,84,85,94,74,92,89,95,92,97,95,100,63,73,79,87,77,91,90,96,79,88,83,93,90,95,98,99,73,86,88,96,91,94,95,100,87,96,93,99,96,100,99,101};
  // B0-0
  // B1c-1 B1e-2
  // B2c-3 B2e-4 B2k-5 B2a-6 B2i-7 B2n-8
  // B3c-9 B3e-10 B3k-11 B3a-12 B3i-13 B3n-14 B3y-15 B3q-16 B3j-17 B3r-18
  // B4c-19 B4e-20 B4k-21 B4a-22 B4i-23 B4n-24 B4y-25 B4q-26 B4j-27 B4r-28 B4t-29 B4w-30 B4z-31
  // B5c-32 B5e-33 B5k-34 B5a-35 B5i-36 B5n-37 B5y-38 B5q-39 B5j-40 B5r-41
  // B6c-42 B6e-43 B6k-44 B6a-45 B6i-46 B6n-47
  // B7c-48 B7e-49
  // B8-50
  // S0-51
  // S1c-52 S1e-53
  // S2c-54 S2e-55 S2k-56 S2a-57 S2i-58 S2n-59
  // S3c-60 S3e-61 S3k-62 S3a-63 S3i-64 S3n-65 S3y-66 S3q-67 S3j-68 S3r-69
  // S4c-70 S4e-71 S4k-72 S4a-73 S4i-74 S4n-75 S4y-76 S4q-77 S4j-78 S4r-79 S4t-80 S4w-81 S4z-82
  // S5c-83 S5e-84 S5k-85 S5a-86 S5i-87 S5n-88 S5y-89 S5q-90 S5j-91 S5r-92
  // S6c-93 S6e-94 S6k-95 S6a-96 S6i-97 S6n-98
  // S7c-99 S7e-100
  // S8-101
  return lookup_table[index];
}

std::string rulestring_from_rule(int rule[number_of_transitions]){
  static const std::string letters = "cekainyqjrtwz";
  static const int numbers_of_letters[9] = {1,2,6,10,13,10,6,2,1};
  std::string rulestring = "";
  int transition = 0;
  for(int BS=0; BS<2; BS++){
    if(BS){
      rulestring += "/S";
    } else {
      rulestring += "B";
    }
    for(int number_of_neighbours = 0; number_of_neighbours <= 8; number_of_neighbours++){
      int number_of_letters = numbers_of_letters[number_of_neighbours];
      std::string alive = "";
      std::string dead = "";
      for(int letter_number=0; letter_number < number_of_letters; letter_number++){
        char letter = letters[letter_number];
        if(rule[transition]){
          alive += letter;
        } else {
          dead += letter;
        }
        transition++;
      }
      int number_alive = alive.size();
      if(number_alive){
        rulestring += std::to_string(number_of_neighbours);
        if(number_alive != number_of_letters){
          if(number_alive*2 <= number_of_letters){
            std::sort(alive.begin(), alive.end());
            rulestring += alive;
          } else {
            std::sort(dead.begin(), dead.end());
            rulestring += "-" + dead;
          }
        }
      }
    }
  }
  return rulestring;
}

std::string minrulestring(int rule[number_of_transitions]){
  int minrule[number_of_transitions];
  for(int transition = 0; transition < number_of_transitions; transition++){
    if(rule[transition] == 1){
      minrule[transition] = 1;
    } else {
      minrule[transition] = 0;
    }
  }
  return rulestring_from_rule(minrule);
}

std::string maxrulestring(int rule[number_of_transitions]){
  int maxrule[number_of_transitions];
  for(int transition = 0; transition < number_of_transitions; transition++){
    if(rule[transition] == 0){
      maxrule[transition] = 0;
    } else {
      maxrule[transition] = 1;
    }
  }
  return rulestring_from_rule(maxrule);
}

int cells[max_t][max_y+3][max_x+3];

int main(){
  std::ofstream output_file;
  output_file.open(output_filename);
  std::ofstream log_file;
  log_file.open(log_filename);

  // Grid of cells
  for(int x = 0; x < max_x+3; x++){
    for(int y = 0; y < max_y+3; y++){
      for(int t = 0; t < max_t; t++){
        cells[t][y][x] = 0;
      }
    }
  }
  cells[0][0][1] = 1;

  // Rules which apply at that generation (-1 if undetermined)
  int rules[max_t][number_of_transitions];
  for(int t = 0; t < max_t; t++){
    for(int transition = 0; transition < number_of_transitions; transition++){
      rules[t][transition] = -1;
    }
    rules[t][0] = 0; // B0
    rules[t][1] = 0; // B1c
    rules[t][2] = 0; // B1e
    rules[t][6] = 1; // B2a
  }

  // Bounding Box
  int x_bound[max_t];
  int y_bound[max_t];
  for(int t = 0; t < max_t; t++){
    x_bound[t] = y_bound[t] = 0;
  }
  x_bound[0] = 2;
  y_bound[0] = 1;

  // Which transitions need to be set
  int new_transitions[max_t][number_of_transitions];
  for(int t = 0; t < max_t; t++){
    for(int transition = 0; transition < number_of_transitions; transition++){
      new_transitions[t][transition] = -1;
    }
  }

  // Number of transitions needing to be set
  int number_of_new_transitions[max_t];
  for(int t = 0; t < max_t; t++){
    number_of_new_transitions[t] = -1;
  }

  // Integer < pow(2,number_of_new_transitions) to specify in binary which transitions to set
  int transitions_to_set[max_t];
  for(int t = 0; t < max_t; t++){
    transitions_to_set[t] = -1;
  }

  // Current generation
  int t = 0;

  // output_file << "Period,Minrule,Maxrule" << std::endl;

  bool  completion_flag = false;
  int number_of_reports = 0;
  int number_of_oscillators = 0;
  while(!completion_flag){
    for(int count = 0; count < reporting_interval; count++){
      if(transitions_to_set[t] == -1){
        transitions_to_set[t] = 0;

        for(int x=0; x<=x_bound[t]; x++){
          for(int y=0; y<=y_bound[t]; y++){
            int transition = transition_from_cells(
              cells[t][y][x],
              cells[t][y][x+1],
              cells[t][y+1][x+1],
              cells[t][y+1][x],
              x==0 ? cells[t][y+1][1] : cells[t][y+1][x-1],
              x==0 ? cells[t][y][1] : cells[t][y][x-1],
              x==0 ? (y==0 ? cells[t][1][1] : cells[t][y-1][1]) : (y==0 ? cells[t][1][x-1] : cells[t][y-1][x-1]),
              y==0 ? cells[t][1][x] : cells[t][y-1][x],
              y==0 ? cells[t][1][x+1] : cells[t][y-1][x+1]
            );
            if((rules[t][transition] == -1) && (new_transitions[t][transition] == -1)){
              new_transitions[t][transition] = transition;
            }
          }
        }

        //Rearrange new_transitions[t] to get new transitions at the front, and count them
        number_of_new_transitions[t] = 0;
        for(int transition = 0; transition < number_of_transitions; transition++){
          if(new_transitions[t][transition]!=-1){
            new_transitions[t][transition] = -1;
            new_transitions[t][number_of_new_transitions[t]] = transition;
            number_of_new_transitions[t]++;
          }
        }
        if(number_of_new_transitions[t]>30){
          log_file << "Possible sign error caused by number_of_new_transitions > 30 (t=" << t << ", " << minrulestring(rules[t]) << ", " << maxrulestring(rules[t]) << ")" << std::endl;
        }
      }

      // Copy rule to next generation
      for(int transition = 0; transition < number_of_transitions; transition++){
        rules[t+1][transition] = rules[t][transition];
      }

      // std::cout << "Parent rule:" << minrulestring(rules[t]) << ", " << maxrulestring(rules[t]) << std::endl;
      // for(int transition = 0; transition < number_of_transitions; transition++){
      //   std::cout << rules[t][transition];
      // }
      // std::cout << std::endl;
      // std::cout << "Parent pattern: (t=" << t << ")" << std::endl;
      // for(int y = 0; y<y_bound[t]; y++){
      //   for(int x = 0; x<x_bound[t]; x++){
      //     std::cout << cells[t][y][x];
      //   }
      //   std::cout << std::endl;
      // }

      // Set rule for next generation
      for(int i = 0; i < number_of_new_transitions[t]; i++){
        rules[t+1][new_transitions[t][i]] = !!(transitions_to_set[t] & (1 << i)); // Check ith bit of transitions_to_set[t]
      }
      transitions_to_set[t]++;

      // Create new Grid
      for(int y=0; y<=y_bound[t]; y++){
        for(int x=0; x<=x_bound[t]; x++){

          int transition = transition_from_cells(
            cells[t][y][x],
            cells[t][y][x+1],
            cells[t][y+1][x+1],
            cells[t][y+1][x],
            x==0 ? cells[t][y+1][1] : cells[t][y+1][x-1],
            x==0 ? cells[t][y][1] : cells[t][y][x-1],
            x==0 ? (y==0 ? cells[t][1][1] : cells[t][y-1][1]) : (y==0 ? cells[t][1][x-1] : cells[t][y-1][x-1]),
            y==0 ? cells[t][1][x] : cells[t][y-1][x],
            y==0 ? cells[t][1][x+1] : cells[t][y-1][x+1]
          );
          // std::cout << transition << ",";
          cells[t+1][y][x] = rules[t+1][transition];
          if(cells[t+1][y][x]){
            x_bound[t+1] = std::max(x+1,x_bound[t+1]);
            y_bound[t+1] = std::max(y+1,y_bound[t+1]);
          }
        }
        // std::cout << std::endl;
      }

      // Increase current time
      t++;

      // std::cout << "Child rule:" << minrulestring(rules[t]) << ", " << maxrulestring(rules[t]) << std::endl;
      // for(int transition = 0; transition < number_of_transitions; transition++){
      //   std::cout << rules[t][transition];
      // }
      // std::cout << std::endl;
      // std::cout << "Child pattern: (t=" << t << ")" << std::endl;
      // for(int y = 0; y<y_bound[t]; y++){
      //   for(int x = 0; x<x_bound[t]; x++){
      //     std::cout << cells[t][y][x];
      //   }
      //   std::cout << std::endl;
      // }

      bool backtrack_flag = false;

      // Check if pattern is empty
      if(!x_bound[t]){
        // std::cout << "Empty" << std::endl;
        backtrack_flag = true;
      }

      //Check if pattern is too big
      if(!backtrack_flag){
        if(t == max_t - 1){
          log_file << "Out of time, t=" << t << ", " << minrulestring(rules[t]) << ", " << maxrulestring(rules[t]) << std::endl;
          backtrack_flag = true;
        }
      }
      if(!backtrack_flag){
        if(x_bound[t] > max_x){
          // std::cout << "Out of space (x), t=" << t << ", " << minrulestring(rules[t]) << ", " << maxrulestring(rules[t]) << std::endl;
          backtrack_flag = true;
        }
      }
      if(!backtrack_flag){
        if(y_bound[t] > max_y){
          // std::cout << "Out of space (y), t=" << t << ", " << minrulestring(rules[t]) << ", " << maxrulestring(rules[t]) << std::endl;
          backtrack_flag = true;
        }
      }
      //Check if pattern is exploding
      // x
      if(!backtrack_flag){
        if(rules[t][13] == 1){
          // >= 2 adjacent
          for(int x = 0; x+1<x_bound[t]; x++){
            if(
              (cells[t][y_bound[t]-1][x] == 1)
              && (cells[t][y_bound[t]-1][x+1] == 1)
            ){
              // std::cout << "Exploding 1" << std::endl;
              backtrack_flag = true;
              break;
            }
          }
        } else if(rules[t][3] == 0){
          // Exactly 2 adjacent
          for(int x = 0; x+2<x_bound[t]; x++){
            if(
              (cells[t][y_bound[t]-1][x] == 0)
              && (cells[t][y_bound[t]-1][x+1] == 1)
              && (cells[t][y_bound[t]-1][x+2] == 1)
              && ((x+3 >= x_bound[t]) || (cells[t][y_bound[t]-1][x+3] == 0))
            ){
              // std::cout << "Exploding 2" << std::endl;
              backtrack_flag = true;
              break;
            }
          }
        } else {
          //dead,dead,alive,alive,dead,dead
          for(int x = 0; x+3<x_bound[t]; x++){
            if(
              (cells[t][y_bound[t]-1][x] == 0)
              && (cells[t][y_bound[t]-1][x+1] == 0)
              && (cells[t][y_bound[t]-1][x+2] == 1)
              && (cells[t][y_bound[t]-1][x+3] == 1)
              && ((x+4 >= x_bound[t]) || (cells[t][y_bound[t]-1][x+4] == 0))
              && ((x+5 >= x_bound[t]) || (cells[t][y_bound[t]-1][x+5] == 0))
            ){
              // std::cout << "Exploding 3" << std::endl;
              backtrack_flag = true;
              break;
            }
          }
        }
      }
      // y
      if(!backtrack_flag){
        if(rules[t][13] == 1){
          // >= 2 adjacent
          for(int y = 0; y+1<y_bound[t]; y++){
            if(
              (cells[t][y][x_bound[t]-1] == 1)
              && (cells[t][y+1][x_bound[t]-1] == 1)
            ){
              // std::cout << "Exploding 1" << std::endl;
              backtrack_flag = true;
              break;
            }
          }
        } else if(rules[t][3] == 0){
          // Exactly 2 adjacent
          for(int y = 0; y+2<y_bound[t]; y++){
            if(
              (cells[t][y][x_bound[t]-1] == 0)
              && (cells[t][y+1][x_bound[t]-1] == 1)
              && (cells[t][y+2][x_bound[t]-1] == 1)
              && ((y+3 >= y_bound[t]) || (cells[t][y+3][x_bound[t]-1] == 0))
            ){
              // std::cout << "Exploding 2" << std::endl;
              backtrack_flag = true;
              break;
            }
          }
        } else {
          //dead,dead,alive,alive,dead,dead
          for(int y = 0; y+3<y_bound[t]; y++){
            if(
              (cells[t][y][x_bound[t]-1] == 0)
              && (cells[t][y+1][x_bound[t]-1] == 0)
              && (cells[t][y+2][x_bound[t]-1] == 1)
              && (cells[t][y+3][x_bound[t]-1] == 1)
              && ((y+4 >= y_bound[t]) || (cells[t][y+4][x_bound[t]-1] == 0))
              && ((y+5 >= y_bound[t]) || (cells[t][y+5][x_bound[t]-1] == 0))
            ){
              // std::cout << "Exploding 3" << std::endl;
              backtrack_flag = true;
              break;
            }
          }
        }
      }
      // Check if pattern is repeating
      // Flipped
      if(!backtrack_flag){
        for(int t_past = t; t_past >= 0; t_past--){
          if(!backtrack_flag){
            if(x_bound[t]==y_bound[t_past] && y_bound[t]==x_bound[t_past]){
              int equality_flag = true;
              for(int x=0;x<x_bound[t];x++){
                for(int y=0;y<y_bound[t];y++){
                  if(cells[t][y][x] != cells[t_past][x][y]){
                    equality_flag = false;
                    goto outside_of_loop_0;
                  }
                }
              }
              outside_of_loop_0:
              if(equality_flag){
                backtrack_flag = true;
                if(t_past == 0){
                  output_file << (2*t) << "," << minrulestring(rules[t]) << "," << maxrulestring(rules[t]) << std::endl;
                  number_of_oscillators++;
                } else {
                  // std::cout << "Repeating" << std::endl;
                }
              }
            }
          }
        }
      }
      // Not Flipped
      if(!backtrack_flag){
        for(int t_past = t-1; t_past >= 0; t_past--){
          if(!backtrack_flag){
            if(x_bound[t]==x_bound[t_past] && y_bound[t]==y_bound[t_past]){
              int equality_flag = true;
              for(int x=0;x<x_bound[t];x++){
                for(int y=0;y<y_bound[t];y++){
                  if(cells[t][y][x] != cells[t_past][y][x]){
                    equality_flag = false;
                    goto outside_of_loop_1;
                  }
                }
              }
              outside_of_loop_1:
              if(equality_flag){
                backtrack_flag = true;
                if(t_past == 0){
                  output_file << t << "," << minrulestring(rules[t]) << "," << maxrulestring(rules[t]) << std::endl;
                  number_of_oscillators++;
                } else {
                  // std::cout << "Repeating" << std::endl;
                }
              }
            }
          }
        }
      }
      if(backtrack_flag){
        while(true){
          for(int x = 0; x < x_bound[t]; x++){
            for(int y = 0; y < y_bound[t]; y++){
              cells[t][y][x] = 0;
            }
          }
          for(int transition = 0; transition < number_of_transitions; transition++){
            rules[t][transition] = -1;
          }
          rules[t][0] = 0; // B0
          rules[t][1] = 0; // B1c
          rules[t][2] = 0; // B1e
          rules[t][6] = 1; // B2a
          x_bound[t] = y_bound[t] = 0;
          for(int transition = 0; transition < number_of_transitions; transition++){
            new_transitions[t][transition] = -1;
          }
          number_of_new_transitions[t] = -1;
          transitions_to_set[t] = -1;
          if(t){
            t--;
          } else {
            std::cout << std::endl;
            std::cout << "Search complete" << std::endl;
            completion_flag = true;
            break;
          }
          if(transitions_to_set[t] < (1 << number_of_new_transitions[t])){
            break;
          }
        }
      }
      // std::cout << std::endl;
      if(completion_flag){
        break;
      }
    }
    if(!completion_flag){
      number_of_reports++;
      std::string progress_string = "";
      float progress_float = 0;
      float denominator = 1;
      for(int t_output = 0; t_output<max_t; t_output++){
        if(number_of_new_transitions[t_output]!=-1){
          if(t_output){
            progress_string += ", ";
          }
          progress_string += std::to_string(transitions_to_set[t_output] - 1) + "/" + std::to_string(1 << number_of_new_transitions[t_output]);
          denominator = denominator/(1 << number_of_new_transitions[t_output]);
          progress_float += (transitions_to_set[t_output] - 1)*denominator;
        } else {
          break;
        }
      }
      std::cout << std::endl;
      std::cout << "Number of iterations = " << (number_of_reports*(reporting_interval/1000000)) << " million" << std::endl;
      std::cout << "Number of oscillators = " << number_of_oscillators << std::endl;
      std::cout << "Progress = " << (100*progress_float) << "\% (" << progress_string << ")" << std::endl;
    }
  }
  output_file.close();
  return 0;
}
It's written in C++, although I can't remember now why plain C wasn't good enough.
Last edited by Macbi on February 4th, 2019, 5:34 am, edited 1 time in total.
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Posts: 587
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Re: Smallest Oscillators Supporting Specific Periods

Postby wildmyron » February 4th, 2019, 4:37 am

@Macbi: Thanks for posting your code. I thought I'd have a go at running it for different size boards. No new periods as yet. It surprised me how dense the results are for odd periods up to the highest odd period found. There's a similar effect for even periods too, but not quite so pronounced.

I had to make a change to the code, as shown in this patch:
@@ -254,7 +254,7 @@
             y==0 ? cells[t][1][x+1] : cells[t][y-1][x+1]
           );
           // std::cout << transition << ",";
-          cells[t+1][y][x] = rules[t+1][transition];
+          cells[t+1][y][x] = (rules[t+1][transition]>0 ? 1 : 0);
           if(cells[t+1][y][x]){
             x_bound[t+1] = std::max(x+1,x_bound[t+1]);
             y_bound[t+1] = std::max(y+1,y_bound[t+1]);

I'm sure the change I made is resulting in incorrect behaviour - I just haven't figured out precisely how the rules[][] array works yet. To be honest though, I can't see how you could run the program, let alone get results, without that change or something to the same effect. Without it the program segfaults early on - after several out of bounds accesses to lookup_table[] in transition_from_cells() return bogus values (which area then used to index into rules[][]).

I haven't investigated why, but the minimum rules are nearly all wrong too - almost all of them cause the pattern to die out. This may be related to my fix to the bug above.

Edit:
Results from a 5x9 search:

p103
x = 3, y = 1, rule = B2aei3eknq4ejknqt5cy6c7e/S01e2ein3-acn4cinrz5jqr6ek7e
obo!
Last edited by wildmyron on February 4th, 2019, 5:49 am, edited 1 time in total.
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Posts: 948
Joined: August 9th, 2013, 12:45 am

Re: Smallest Oscillators Supporting Specific Periods

Postby Macbi » February 4th, 2019, 5:48 am

You're right. My fix was to change the line:
        rules[t+1][new_transitions[t][i]] = (transitions_to_set[t] & (1 << i)); // Check ith bit of transitions_to_set[t]
to
        rules[t+1][new_transitions[t][i]] = !!(transitions_to_set[t] & (1 << i)); // Check ith bit of transitions_to_set[t]
since the double exclamation mark has the effect of setting any nonzero int to 1. I've now edited this fix into my above post.

The reason I made this mistake was that I had removed one of those exclamation marks in my code. This causes the rulespace to be searched in reverse order, meaning that I could run two searches simultaneously and wait for them to meet in the middle. When I came to copy the code to this thread I remembered that I needed to restore it to the "forward" direction, but I foolishly deleted the exclamation mark rather than adding a second.

Incidentally, I never did manage to get the search to "meet in the middle". I took some screenshots to record how far I had got, which I've attached. Somebody might want to search some or all of the missing space, although much of it might be very slow.

screenshot-top.png
screenshot-top.png (179.48 KiB) Viewed 464 times
screenshot-bottom.png
screenshot-bottom.png (182.91 KiB) Viewed 464 times
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Re: Smallest Oscillators Supporting Specific Periods

Postby wildmyron » February 4th, 2019, 5:59 am

Ahh, thanks for the explanation. I'm currently running 5x9 and 9x5 searches separately (in the forward direction only). I edited the first result into my post above, just after you replied.

I suppose that resuming the 7x7 searches would require some code modification to set the search state. I don't think I'd try tackling it though - it looks to me as though the search space is pretty big.
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Re: Smallest Oscillators Supporting Specific Periods

Postby Macbi » February 4th, 2019, 7:15 am

I don't think I ever exhaustively searched 5x7 and 7x5 so I'm doing that now. Of course these results will be included in your searches and the 7x7 search, but I fear that none of those will ever be completed.

EDIT: By the way, "sort filename -gu" is a good way to handle the file of saved oscillators. It shows one oscillator of each period found, sorted into order.

EDIT2: 5x7 done, no new periods.
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