Solving the 9-Piece Animal Puzzle

[adoado]

Hello all,

Recently while on a 'schoolies' vacation a few friends presented a puzzle of 9 squares pieces that can be arranged into a larger 3x3 square. Each small piece had half an animal on each edge (either the head or tail, of say, a goat for a seahorse). The aim was to rotate each piece correctly so across adjacent edges an appropriate animal formed, and with this constraint arrange them all (validly!) into the 3x3 square...


It took a good 10 minutes before I completed the puzzle, more or less based on luck. It got me wondering - it is definitely a mathematical combination - so is their any way to solve this?

Is there any way of representing such a problem mathematically, whereby one can solve for the order and rotation of each piece?

Cheers,
Adrian ;)

 

[Com3t]

There are many representations possible, such as a 3x3 matrix of two-dimensional elements, the first being dimension being the set of integers 1 through 9 representing each piece, and the second being the set of integers 1 through 4 representing the four rotational possibilities.

You can choose, without loss of generality, that the final solution has rotation 1:
(1,1)(2,1)(3,1)
(4,1)(5,1)(6,1)
(7,1)(8,1)(9,1)
and have 2 mean a piece is rotated clockwise from the ideal position, 3 mean a piece is upside down, and 4 mean a piece is rotated counterclockwise.

Now, if you mix up the pieces and spin them around, any starting position can be represented, such as:

(2,1)(9,2)(4,3)
(7,4)(5,1)(3,2)
(6,3)(1,4)(8,1)

 

[pbandjay]

It took a good 10 minutes before I completed the puzzle, more or less based on luck.

If luck had something to do with it, it seems to me like quite a bit of a statistical improbability! I am curious, are each of the squares unique? If so, then there being 4 rotations for each of the 9 squares and 9! ways place the squares in the 3x3 grid, there are 4⁹*9! = 95,126,814,720 possible arrangements of the squares . If there is only one configuration that solves the puzzle then, arranging the squares in a random order gives 1/95,126,814,720 * 100 = 0.00000000105% chance of solving the puzzle for that random configuration.

 

[Dragonfall]

1/95,126,814,720! * 100 should be WAY less than 0.00000000105%. Assuming that's a 95,126,814,720 factorial.

 

[pbandjay]

1/95,126,814,720! * 100 should be WAY less than 0.00000000105%. Assuming that's a 95,126,814,720 factorial.

Sorry, I accidentally inserted the factorial. I meant just 95,126,814,720. Thank you!

 

[Dragonfall]

I wish we chose a better symbol for factorials. As it is now we can't express shock & awe in math without ambiguity.