The Best Fit Jigsaw

We wish to design a jigsaw puzzle with square pieces, whose length/width ratio is as close as possible to \(\sqrt 2\).

For a small puzzle, we could make the puzzle \(17\times12\), because \[\left|\left(\dfrac{17}{12}\right)^2 - 2\right| = \dfrac{1}{144}\] is very small. Let us call a \(\ell\times w\) puzzle good if the difference \[\left|\left(\dfrac \ell w\right)^2 - 2\right|\] is less than that same expression for any puzzle with fewer pieces.

How many pieces is the largest good puzzle with less than 10000 pieces?

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