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