Jigsaw puzzle ratio question.

Question

I was doing a puzzle when this question popped into my head: Assuming a puzzle had an equal amount of pieces in each row, and an equal amount of pieces in each column, what would be the least amount of pieces the puzzle can have if the ratio of edge pieces to middle pieces is 1:5. I already attempted to solve this problem but it was impossible for me to get a number. The best I was able to do was get the number closest to the answer. It won’t be a whole number and I’m not sure if it is even possible to get the number. My equation was “2x + 2y - 4 + 5(2x + 2y - 4) = xy” I’m not sure if this is the correct equation although I think it is. If it’s not, can someone tell me the correct one, and if it is, can someone attempt to solve it for me.

Answer 1

Your equation looks correct to me, notice that it can be further rewritten $12(x+y-2) = xy$. You want to minimize the number $xy$ of pieces, which amounts to minimizing $x+y$.

Let's say we want to simplify our equation : $$xy - 12x - 12y + 24 = 0$$ by removing the linear terms. We can do this by letting $u = x - 12$ and $v = y - 12$. Now compute : $$uv = (x-12)(y-12) = xy - 12x - 12y + 144$$ hence $(x, y)$ is a valid solution if, and only if, $uv = 120$. Notice that we want integer solutions, and $x$ and $y$ are integers exactly when $u$ and $v$ are.

Now the solutions to $uv = 120$ are well-known, since we know how to factor : $$120 = 2^3 \cdot 3 \cdot 5.$$ We get $(3+1) \times (1+1) \times (1+1) = 16$ possible $(u, v)$ solutions, each of which yield a valid $(x, y)$ puzzle. Removing redundancies ($u$ and $v$ play symmetrical roles), we get eight :

$$\begin{aligned}u && v && x && y && xy\\ 1 && 120 && 13 && 132 && 1716\\ 2 && 60 && 14 && 72 && 1008\\ 3 && 40 && 15 && 52 && 780\\ 4 && 30 && 16 && 42 && 672\\ 5 && 24 && 17 && 36 && 612\\ 6 && 20 && 18 && 32 && 576\\ 8 && 15 && 20 && 27 && 540\\ 10 && 12 && 22 && 24 && 528\end{aligned}$$

We can list them, and notice that the minimal $u+v$ (equivalently, the minimal $x+y$, and equivalently the minimal number $xy$ of pieces of the puzzle) is at $\{x, y\} = \{22, 24\}$. Your minimal puzzle has $528$ pieces, $88$ of which are on the edge and $440$ of which are interior.