Henry Ernest Dudeney/Puzzles and Curious Problems/161 - Blocks and Squares/Solution

Puzzles and Curious Problems by Henry Ernest Dudeney: $161$

Blocks and Squares

Three children each possess a box containing similar cubic blocks, the same number of blocks in every box.

The first girl was able, using all her blocks, to make a hollow square, as indicated by $A$.

The second girl made a still larger square, as $B$.

The third girl made a still larger square, as $C$ but had four blocks left over for the corners, as shown.

What is the smallest number of blocks that each box could have contained?

Solution

The smallest number of blocks in each box appears to be $1344$.

The innermost hollow square is $34^2$.

Square $A$ is $50^2$.

Square $B$ is $62^2$.

Square $C$ is $72^2$, with four blocks left over at the corners.

We have:

$50^2 - 34^2 = 2500 - 1156 = 1344$

$62^2 - 50^2 = 3844 - 2500 = 1344$

$72^2 - 62^2 = 5184 - 3844 = 1344 - 4$

Proof

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Sources

• 1932: Henry Ernest Dudeney: Puzzles and Curious Problems ... (previous) ... (next): Solutions: $161$. -- Blocks and Squares
• 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $229$. Blocks and Squares