### [GAME] TREE function games

Posted:

**April 9th, 2019, 5:41 pm**after posting this, I have a compelling desire to play a "game" of TREE(3) with someone.

symbols:

1: () -- parentheses

2: [] --brackets

3: {} --braces

an example game:

P1: ()

P2: []

P1: {{}}

P2: {}

P1 wins as they have no legal moves.

note that if () and [] have been played but not {{}} or {} you should play {{}}, forcing the other player to play {}.

In this example, P2 made a mistake. by playing [] instead of [{}], {[]}, or [[]] he sentenced himself to losing as it gave P1 the winning strategy above.

Another winning strategy is to play [] after () and {{}} have been played as it forces the opponent to play {} and lose.

or {} after () and [[]] have been played to force the other player to play []. This means that if P1 plays (), your only options are to play [{}] or {[]}.

so here's a better game.

P1: ()

P2: [{}]

P1: {[]}

P2: [[]]

P1: [] <-fatal blunder

P2: {{}}

P1: {}

NEVER EVER make that mistake. If you play [] after () has been played, but not {{}}, your opponent wins by playing {{}} and forcing you to play {}, leaving them with no legal moves.

Note that any game is equivalent -- has the same outcome -- if you swap a pair or parentheses/brackets/braces (how about we call they openy-closies).

for instance, swapping brackets with braces, you get:

P1: ()

P2: {[]}

P1: [{}]

P2: {{}}

P1: {} <- fatal blunder

P2: [[]]

P1: []

this is because our symbols are rather arbitrary. you could use fore/backslashes -- /\ -- as well (anyone up for a game of TREEgame(4)?)

or angle brackets: <>

we have at least five symbols on the qwerty keyboard, so we can play TREEgame(5):

1: () -- parentheses

2: [] --brackets

3: {} --braces

4: /\ -- slashes

5: <> -- angle brackets

Of course, playing these games won’t give TREE(n) -- after all, we’re not trying to optimize the length of the game but rather are trying to make the parity of the length of the game even for one player and odd for the other.

EDIT:

In fakeTREE, playing [{()}] after [()] is legal— it’s not in actual tree but it makes the game more fun.

symbols:

1: () -- parentheses

2: [] --brackets

3: {} --braces

an example game:

P1: ()

P2: []

P1: {{}}

P2: {}

P1 wins as they have no legal moves.

note that if () and [] have been played but not {{}} or {} you should play {{}}, forcing the other player to play {}.

In this example, P2 made a mistake. by playing [] instead of [{}], {[]}, or [[]] he sentenced himself to losing as it gave P1 the winning strategy above.

Another winning strategy is to play [] after () and {{}} have been played as it forces the opponent to play {} and lose.

or {} after () and [[]] have been played to force the other player to play []. This means that if P1 plays (), your only options are to play [{}] or {[]}.

so here's a better game.

P1: ()

P2: [{}]

P1: {[]}

P2: [[]]

P1: [] <-fatal blunder

P2: {{}}

P1: {}

NEVER EVER make that mistake. If you play [] after () has been played, but not {{}}, your opponent wins by playing {{}} and forcing you to play {}, leaving them with no legal moves.

Note that any game is equivalent -- has the same outcome -- if you swap a pair or parentheses/brackets/braces (how about we call they openy-closies).

for instance, swapping brackets with braces, you get:

P1: ()

P2: {[]}

P1: [{}]

P2: {{}}

P1: {} <- fatal blunder

P2: [[]]

P1: []

this is because our symbols are rather arbitrary. you could use fore/backslashes -- /\ -- as well (anyone up for a game of TREEgame(4)?)

or angle brackets: <>

we have at least five symbols on the qwerty keyboard, so we can play TREEgame(5):

1: () -- parentheses

2: [] --brackets

3: {} --braces

4: /\ -- slashes

5: <> -- angle brackets

Of course, playing these games won’t give TREE(n) -- after all, we’re not trying to optimize the length of the game but rather are trying to make the parity of the length of the game even for one player and odd for the other.

EDIT:

In fakeTREE, playing [{()}] after [()] is legal— it’s not in actual tree but it makes the game more fun.