A jigsaw puzzle company wants to design a new rectangular puzzle with a specific quirk: it must have the same number of edge pieces and interior pieces. Each piece begins as a square of the same size, and the typical rounded “blanks” and “tabs” (the holes and the parts that stick out, respectively) are then either cut from or glued to its sides. If the company wants the puzzle to be as large as possible, how many pieces will it have?
The puzzle will have 60 pieces. Note that the exact locations of the blanks and tabs don’t affect the number of pieces, so we can treat the pieces as though they’re all equal-sized squares.
Call the number of pieces along the length of the rectangle L and the number of pieces along the width of the rectangle W. The total number of pieces in the puzzle is L × W. The number of interiorpieces is (L – 2) × (W – 2) because the length of the interior rectangle is L minus the two edge pieces flanking it on either side; the same is true for the width of the interior rectangle. Everything else is an edge piece, so we can express the number of edge pieces as the following: (the total number of pieces) – (the number of interior pieces), or LW – (L – 2)(W – 2). Setting the number of edge pieces equal to the number of interior pieces gives:
(L – 2)(W – 2) = LW – (L – 2)(W – 2)
And this simplifies to:
LW – 4L – 4W + 8 = 0
There are a few ways to make sense of this expression. A slick option notices that the lefthand side would become easy to factor if the 8 were a 16. Adding 8 to both sides allows us to factor as follows:
(L – 4)(W – 4) = 8
L and W are both positive integers, so L – 4 and W – 4 must be, too. To make their product equal 8, the lefthand side must therefore either be 4 × 2 or 8 × 1. (Although they are typically valid answers, the lefthand side cannot be equal to –4 × –2 or –8 × –1 because these will lead to impossible values for L and W.) In the first case, (L – 4) would equal 4, and (W – 4) would equal 2, implying L = 8 and W = 6. This results in L × W = 48 total pieces. In the second case, (L – 4) would equal 8, and (W – 4) would equal 1, implying L = 12 and W = 5. This results in L × W = 60 total pieces. These are the only two possible dimensions for the desired jigsaw puzzle, so 60 pieces is the most it can have.
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