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A man had three pieces of beautiful wood, measuring 12 inches, 15 inches, and 16 inches square respectively. He wanted to cut these into the fewest pieces possible that would fit together and form a small square table top 25 inches by 25 inches. How was he to do it? I have found several easy solutions in six pieces, very pretty, but have failed to do it in five pieces. Perhaps the latter is not possible. I know it will interest my readers to examine the question.

3 squares with side lengths 12, 15 and 16 inches

This puzzle was created by Henry Dudeney.

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  • $\begingroup$ Easy solutions in 6 pieces? My best one is 7 so far... $\endgroup$
    – Florian F
    Commented Apr 12 at 22:09
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    $\begingroup$ @FlorianF Maybe easy by Henry Dudeney’s standards. After all, according to his Wikipedia article, “Some of Dudeney's most famous innovations were his 1903 success at solving the Haberdasher's Puzzle (Cut an equilateral triangle into four pieces that can be rearranged to make a square) …” $\endgroup$
    – Will.Octagon.Gibson
    Commented Apr 13 at 0:39
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    $\begingroup$ OK, I got 6. A bit tricky though. Now 5 looks difficult. $\endgroup$
    – Florian F
    Commented Apr 13 at 8:19

2 Answers 2

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For a start, here is a 6-piece solution.

enter image description here

From there you can find some variations.

Here is a less boring one.

enter image description here

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    $\begingroup$ That 15x15 to 9x25 seems potential to be converted into two pieces $\endgroup$
    – justhalf
    Commented Apr 13 at 15:16
  • $\begingroup$ @justhalf How so? the step method works only for $n(n+1)$ by $n(n+1)$ to $(n+1)^2$ by $n^2$. It doesn't work if the squares are not in the ratio of the squares of consecutive integers. $\endgroup$
    – Rosie F
    Commented Apr 13 at 15:39
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    $\begingroup$ @RosieF That's why I said "potential". I don't know the answer. $\endgroup$
    – justhalf
    Commented Apr 13 at 15:45
  • $\begingroup$ +1 I have seen only one solution by Henry Dudeney and yours is quite similar. You divided the 15x15 square into a rectangle and two other pieces; he divided it into a rectangle and two identical pieces. $\endgroup$
    – Will.Octagon.Gibson
    Commented Apr 13 at 17:50
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    $\begingroup$ I like your second dissection with its 180 degree symmetry even more than Dudeney’s. $\endgroup$
    – Will.Octagon.Gibson
    Commented Apr 14 at 9:11
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My (boring) 6 solution; and (cool) 7 piece solution for inspiration.

enter image description here

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    $\begingroup$ You call your 6-piece solution boring. I call it simple and elegant. $\endgroup$
    – Will.Octagon.Gibson
    Commented Apr 14 at 6:35
  • $\begingroup$ @WillOctagonGibson I think minimal straight cuts beast minimal pieces in elegance. A matter of taste of course. I created a new question for it. (of course there is nothing wrong with this question, nor Florian's and my later, almost identical answer.) $\endgroup$
    – Retudin
    Commented Apr 14 at 8:12
  • $\begingroup$ If I were a carpenter who needs to do the cutting I know what dissection I would choose. Not my second one. $\endgroup$
    – Florian F
    Commented Apr 14 at 19:43

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