Jigsaw Ingredients

Let $x, y$ be the horizontal and vertical dimensions of the puzzle. Then we have

• total number of pieces: $N = xy$ ;

• interior pieces: $I = (x-2)(y-2)$ ;

• edge pieces: $E = N - I - 4 = xy - (x-2)(y-2) - 4 = 2(x+y) - 8$ .

We are told that $x = 1\tfrac12 y\ \ \ E = 0.13\cdot N,$ which gives the following equation for $y$ : $0.13\cdot 1\tfrac12y \cdot y = 2(1\tfrac12y + y) - 8 \\ 0.195y^2 = 5y - 8 \\ 0.195y^2 - 5y + 8 = 0.$ The solutions of this quadratic equation are $y = \frac{5 \pm \sqrt{5^2 - 4\cdot 0.195\cdot 8}}{2\cdot 0.195} = \frac{5\pm\sqrt{18.76}}{0.39} \approx 23.93\ \ \text{or}\ \ 1.71.$ Clearly the latter solution makes no sense. We conclude that $y = 24$ , so that $x = \boxed{36}$ .

Check: A $24\times36$ puzzle has

• $N = 24\cdot 36 = 864$ pieces in total,

• of which $I = 22\cdot 34 = 748$ interior pieces,

• and $E = 864 - 748 - 4 = 112$ edge pieces,

so that the percentage of edge piece is $E/N = 112/864 \approx 12.96\%.$ .