## Puzzles based on the concept of symmetry?

For fans of all other kinds of logic puzzles

### Puzzles based on the concept of symmetry?

Hello Sudoku/Puzzle community!

I have a puzzle related question. Are there any known puzzle games that actively make use of the concept of symmetry? I recently made a board game based on mirror-image symmetry myself and I was wondering if maybe anyone here knows if there have been more attempts to use symmetry as a puzzle/game mechanism.

best,

Thomas
thomasbarefootcoders

Posts: 2
Joined: 15 August 2013

### Re: Puzzles based on the concept of symmetry?

Hi Thomas,

thomasbarefootcoders wrote:Are there any known puzzle games that actively make use of the concept of symmetry? I recently made a board game based on mirror-image symmetry myself and I was wondering if maybe anyone here knows if there have been more attempts to use symmetry as a puzzle/game mechanism.

It depends on what you mean by "actively mak[ing] use of the concept of symmetry":
- in Sudoku, Kakuro and other puzzles I've seen, many puzzle creators prefer puzzles with pattern of given cells having some symmetry. But this is merely aestethic; it plays no role in solving.
- when a game is based on a symmetric grid and its constraints are related to the grid structure, symmetry plays a role in the formulation of its resolution rules. In Sudoku, it goes even further: the set of resolution rules has a full row / column / digit symmetry (plus row-col to block-square analogy).
- if you mean the much stronger condition that the solution must have some symmetry, then I know no such game. Indeed, in this interpretation, why not split the board into two and keep only half of it?

denis_berthier
2010 Supporter

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Joined: 19 June 2007
Location: Paris

### Re: Puzzles based on the concept of symmetry?

Hi Denis, yes sure, here is a short description of the game.

In the game, for which, together with two programmers from Germany we made an implementation for on the iPhone, 2 players put in turn black and white stones on the board. The symmetry element comes in when a move by a player causes a square on the board (which could be of any size 2x2, 3x3 etc) to be fully covered by stones. If because of this move the color pattern of this square is also ´mirror symmetric´, horizontally or vertically (I exclude ´diagonal´ symmetries), then all stones in that square change into the color of the player who just put the stone on the board. And of course, in the end, when the board is full with stones, the player with the most stones of her/his color wins.

So In this way the game quite directly uses the concept of symmetry (We also named the game "Symmetry - The Board Game" ) . Now this is a game, but I could imagine that this ´reasoning whith symmetries' would be something that lends itself very well for puzzles, but as far as I know I´ve never come across such puzzle.
thomasbarefootcoders

Posts: 2
Joined: 15 August 2013

thomasbarefootcoders wrote:I could imagine that this ´reasoning whith symmetries' would be something that lends itself very well for puzzles

when a SuDoku happens to present a full "mirror" situation,
this has been used in solving

Pat

Posts: 3653
Joined: 18 July 2005

### Re: Puzzles based on the concept of symmetry?

One could add a constraint to the solving rules, such as whenever x appears in any cell, 10-x must appear in the diametrically opposite cell. This would immediately force the center cell to be 5, and would greatly reduce the number of givens necessary for a unique solution.

Bill Smythe
Smythe Dakota

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Joined: 11 February 2006