Henry Ernest Dudeney/Puzzles and Curious Problems/86 - Digits and Cubes/Solution

Puzzles and Curious Problems by Henry Ernest Dudeney: $86$

Digits and Cubes

Find all $5$-digit squares such that:

the number formed from the first $2$ digits added to that formed by the last $2$ digits form a cube.

Solution

There are $3$ solutions:

\(\ds 237^2\)

\(=\)

\(\ds 56 \, 169\)

where $56 + 69 = 125 = 5^3$

\(\ds 251^2\)

\(=\)

\(\ds 63 \, 001\)

where $63 + 01 = 64 = 4^3$

\(\ds 152^2\)

\(=\)

\(\ds 23 \, 104\)

where $23 + 04 = 27 = 3^3$

Proof

A computer program can be used to perform an exhaustive search.

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Sources

• 1932: Henry Ernest Dudeney: Puzzles and Curious Problems ... (previous) ... (next): Solutions: $86$. -- Digits and Cubes
• 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $123$. Digits and Cubes