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N-Dimensional Tic-Tac-Toe


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#1 ITIWBS

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Posted 10 October 2011 - 07:57 AM

N-dimensional tic-tac-toe is an ideal pastime for whiling away the tedium of, for example, a long ride in the back of the family car and is also susceptible to email forum play.

As a kid, passing time in back the family car on long rides, it began, for me, with conventional tic-tac-toe, but that quickly got tiresome on account of its sheer simplicity, even though its simplicity was initially an advantage since it made it easy to visualize and play simply by means of calling off coordinates when the often skimpy supplies of paper (brown paper bags or wrapping paper) were exhausted or the ride was too rough to make play convenient while scribbling on unsupported paper with a pencil stub sharpened by means of chewing it to a point when the lead broke.

Casting about for a game somewhat more challenging, next it was simple 3-d tic-tac-toe, played in a 3 x 3 x 3 cube, which in turn quickly palled on account of the fact that the game allows a trivial win on a basis of a first move advantage, just so long as one makes no mistakes in play. To make the game more interesting it was expanded to play in a 4 x 4 x 4 field, which allows of a draw if neither player makes a mistake, just as does simple 3 x 3 2-d tic-tac-toe.

Calling off the coordinates, where the 'R' number is the number of the round of the game (each player makes one move per round), P1 is player one and P2 is player two, X is the horizontal axis with squares numbered from left to right and Y is the vertical axis with squares numbered from bottom to top, a two-d game might go as follows, with the players calling off their moves by means of giving the coordinates of the square they want to mark:

R1 [P1, X2 x Y2 (the center square); P2, X3 x Y1 (lower right hand square)]

R2 [P1, X3 x Y3; P2, X1 x Y1 (blocking P1 from a win on their next move)

R3 [P1, X2 x Y1, (blocking P2 from a win on their next move); P2, X2 x Y3 (blocking P1)]

R4 [P1, X3 x Y2; P2, X1 x Y2], ending the game in a draw, cats' game.




Expanding the game a dimension, in a 3 x 3 x 3 manifold, Z is from front to back, a trivial first move win follows:

R1 [P1, X2 x Y2 x Z2; P2, X1 x Y1 x Z1]

R2 [P1, X1 x Y3 x Z3; P2, X3 x Y1 x Z1 (blocking a possible win by P1)]

R3 [P1, X2 x Y1 x Z1; P2, X2 x Y3 x Z3]

R4 [P1, X1 x Y2 x Z2... (P1 has created a situation where no matter what P2 does on their next move, P1 wins. As a matter of fact P1 always wins just so long as they make no errors in play.)]




Expanding the game again, this time to a 4 x 4 x 4 manifold, reintroduces the possibility of a taut game draw.




Though a major part of the reason for the game is to build one's capacity for visualization, one shouldn't hesitate to draw diagrams as necessary.

For a 3-d game, a simple way to diagram on paper is a vertical column of tic-tac-toe boards where the lowest board is arbitrarily the one in 'front' and the highest is the board in the 'back' of the cube.




Expanding into a 4-d game, a simple way to arrange the diagram is in the form of a square with as many tic-tac-toe boards on a side as there are squares on each board. (This also works for checkers and chess, in which case each square of the checkerboard is replaced with an entire board, the flat projection making it easy to display on a tabletop.)

For 5-d, one simply takes as many 4-d boards as there are squares on the side of the board and arranges them into a column, horizontally or vertically as one likes. With even numbered dimensions higher than zero, one can always arrange the playing field projected onto a 2-d square, and with odd numbered dimensions, the field can always be arranged in columns.




For four dimensions, its simple and convenient to first specify the given board and then the specific space on that board. Four example, one might say, where 'b' represents a board and 's' represents a space on a board, " 4-d game, 5 spaces on a side of each board, 'b' X3 x Y3, 's' X3 x Y3." to indicate the center space of your 5 x 5 x 5 x 5 four dimensional playing field.

Anybody care to get up a game?
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#2 Mrs Zen

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Posted 10 October 2011 - 08:35 AM

Gulp.

I really want to do this.

My brain has no spare capacity. <wah>
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#3 ITIWBS

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Posted 10 October 2011 - 11:02 AM

I imagine you're probably frantic with overwork at the moment. :D7:

Once I was doing exercises with Cantorian transfinites. The transformations were simplistic, so next I started playing with variations, for example, changing one of his postulates, "X squared does not equal X squared" to 'X squared always equals X squared', which has the effect of restricting the field to the sub-infinite range, though one's numbers can be indefinitely large. Next I started counting categories of topological curves I could visualize in 'N' dimensions of the modified Cantorian manifold. I found that I could achieve a stable visualization of forty categories in six dimension and even extend that to 42 categories in seven dimensions, but that if I tried to extend the visualization to a third category in the seventh dimension, the visualization would collapse, producing a subjective sensation like a a portion of my brain melting and running like molten wax, evolving a hot, malodorous gas as it did so. (I think this may have something to do with close-packing laws operating at the molecular level in the physiology of the brain.) After several essays with the same result, I resolved never again to drive myself beyond forty variables in six dimensions, If the problem requires more than six dimensions, I'll just do what everyone else does and break it down into a more manageable assortment of linears and quadratics.

Between the h2g2 move and working on your Master's thesis I imagine you've more than enough on your mins at the moment.

Some other time perhaps. :G11: :CARD:
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#4 ITIWBS

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Posted 10 October 2011 - 11:07 AM

Amend fourth to last word in next to last sentence to read 'mind'. (Must have blurred behind a smudge on my glasses.
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#5 Mrs Zen

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Posted 10 October 2011 - 12:56 PM

Do you know, I understood what every single word of that means, but none of them in combination with the others! <laugh>

My problem is I am so strongly visual, if I can't see it I struggle to conceptualise it, which limits my topological understanding to 3d and the odd klein bottle.

On the other hand, I knit moebius scarves: I have to cast off, of course, but there's no casting on. <devil>
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#6 ITIWBS

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Posted 11 October 2011 - 10:31 AM

What an idea! How about Klein bottle caps and hand-warmers?
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#7 ITIWBS

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Posted 13 October 2011 - 09:28 AM

Afterthoughts on N- dimensional Tic-tac-toe.

The 4 x 4 x 4 3D game can be played on a standard chess board, just so long as the players are agreed on which of the 4 quadrants is 1st, 2nd, 3rd and 4th from the bottom of the stack.

The 5 x 5 x 5 x 5 4D game, with a total of 625 squares, can be played with more than two players, scoring points every time someone gets 5 in a row.
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#8 Joshuakem

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Posted 25 June 2015 - 07:47 AM

HI can anyone provide me source code of Tic tac Toe......or any link ... .... Thanks Rahul




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