Problem: Dissecting a Mittre


The figure that is perplexing the carpenter in the illustration represents a mitre. It will be seen that its proportions are those of a square with one quarter removed. The puzzle is to cut it into five pieces that will fit together and form a perfect square. I show an attempt, published in America, to perform the feat in four pieces, based on what is known as the "step principle," but it is a fallacy.


We are told first to cut oft the pieces $1$ and $2$ and pack them into the triangular space marked off by the dotted line, and so form a rectangle.

So far, so good. Now, we are directed to apply the old step principle, as shown, and, by moving down the piece $4$ one step, form the required square. But, unfortunately, it does not produce a square: only an oblong. Call the three long sides of the mitre $84$ in. each. Then, before cutting the steps, our rectangle in three pieces will be $84\times63.$ The steps must be $10\frac 12$ in. in height and $12$ in. in breadth. Therefore, by moving down a step we reduce by $12$ in. the side $84$ in. and increase by $10\frac 12$ in. the side $63$ in. Hence our final rectangle must be $72$ in. $\times 73\frac 12$ in., which certainly is not a square! The fact is, the step principle can only be applied to rectangles with sides of particular relative lengths. For example, if the shorter side, in this case, were $61 \frac57$ (instead of $63$), then the step method would apply. For the steps would then be $10\frac 27$ in. in height and $12$ in. in breadth. Note that $61 \frac 57 \times 84=72^2.$

At present, no solution has been found in four pieces, and I do not believe one possible.

Solutions: 1

Thank you to the contributors under CC BY-SA 4.0!



Project Gutenberg

  1. Dudeney, H. E.: "Amusements in Mathematics", The Authors' Club, 1917

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