# Solving the 9-Piece Animal Puzzle

In summary, a group of friends presented a puzzle during a vacation, involving arranging 9 square pieces with half an animal on each edge into a 3x3 square by rotating them correctly. The puzzle sparked the question of whether it could be solved mathematically, and a possible representation was suggested. With 4 rotations for each piece and 9! ways to arrange them, there are 95,126,814,720 possible arrangements.
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,

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)

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.

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

Dragonfall said:
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!

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

## 1. How do I start solving the 9-piece animal puzzle?

To start solving the 9-piece animal puzzle, first examine the pieces and familiarize yourself with the different shapes and images on each piece. Then, try to identify which pieces may fit together based on their edges and images.

## 2. What strategies can I use to solve the 9-piece animal puzzle?

One strategy is to start by assembling the border pieces to create a frame for the puzzle. Then, focus on grouping together pieces with similar images or colors. You can also try solving smaller sections of the puzzle at a time and then connecting them together.

## 3. How long does it typically take to solve the 9-piece animal puzzle?

The time it takes to solve the 9-piece animal puzzle can vary depending on the individual's puzzle solving skills and experience. It can range from a few minutes to several hours.

## 4. Can I use a reference image while solving the 9-piece animal puzzle?

Yes, using a reference image of the completed puzzle can be helpful in guiding your progress and confirming that the pieces are placed correctly.

## 5. What should I do if I am unable to solve the 9-piece animal puzzle?

If you are unable to solve the puzzle, take a break and come back to it later with a fresh perspective. You can also try rearranging the pieces in a different way or asking for help from a friend or family member.

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