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

Modern Puzzles by Henry Ernest Dudeney: $115$

The Carpenter's Puzzle

Here is a well-known puzzle, given in all the old books.

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.

Historical Note

Martin Gardner points out that Dudeney, in Problem $150$ in his Amusements in Mathematics, catches Sam Loyd out in exactly such a blunder.

He does this in 536 Puzzles & Curious Problems, his $1968$ repackaging of Dudeney's Modern Puzzles and Puzzles and Curious Problems.

Sources

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