Henry Ernest Dudeney/Modern Puzzles/115 - The Carpenter's Puzzle/Solution

Modern Puzzles by Henry Ernest Dudeney: $115$

The Carpenter's Puzzle

A ship's carpenter had to stop a hole $12$ inches square,

and the only piece of wood that was available measured $9 \ \mathrm{in.}$ in breadth by $16 \ \mathrm{in.}$ length.

How did he cut it into only two pieces that would exactly fit the hole?

The answer is based on the "step principle", as shown in the diagram.

If you move the piece marked $B$ up one step to the left,

it will exactly fit on $A$ and form a perfect square measuring $12$ inches on every side.

This is very simple and obvious.

But nobody has ever attempted to explain the general law of the thing.

As a consequence, the notion seems to have got abroad that the method will apply to any rectangle where the proportion of length to breadth is within reasonable limits.

This is not so, and I have had to expose some bad blunders in the case of published puzzles that were supposed to be solved by an application of this step principle,

but were really impossible of solution.$^*$

Let the reader take different measurements, instead of $9 \ \mathrm{in.}$ by $16 \ \mathrm{in.}$,

and see if he [or she] can find other cases in which this trick will work in two pieces and form a perfect square.

Solution

There is only one general situation in which it is possible to dissect a rectangle into two pieces using this specific technique in order to make a square.

That is when the sides of the rectangle are in the proportion of consecutive square numbers.

Thus for a rectangle whose sides are in the ratio $n^2 : \paren {n + 1}^2$, a cut of $n$ steps is used to dissect it into a square whose sides are $n \paren {n + 1}$.

Proof

This theorem requires a proof. In particular: Dudeney provides no proof as such, merely a page of discussion and assertion. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sources

• 1926: Henry Ernest Dudeney: Modern Puzzles ... (previous) ... (next): Solutions: $115$. -- The Carpenter's Puzzle
• 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $338$. The Carpenter's Puzzle