Henry Ernest Dudeney/Puzzles and Curious Problems/235 - An Easter Egg Problem/Solution

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Puzzles and Curious Problems by Henry Ernest Dudeney: $235$

An Easter Egg Problem
I have an easter egg exactly $3$ inches in length, and $3$ other eggs all similar in shape,
having together the same contents as the large egg.
Can you tell me the exact measurements for the lengths of the three smaller ones?


Solution

There is an infinite number of solutions.

From Cubes which are Sum of Three Cubes, we can use any of the sequence:

$6^3, 9^3, 12^3, 18^3, 19^3, 20^3, 24^3, 25^3, \ldots$

For example, take $n = 6$.

Then:

\(\ds 6^3\) \(=\) \(\ds 216\)
\(\ds \) \(=\) \(\ds 27 + 64 + 125\)
\(\ds \) \(=\) \(\ds 3^3 + 4^3 + 5^3\)

and so:

\(\ds 3^3\) \(=\) \(\ds \paren {\frac 6 2}^2\)
\(\ds \) \(=\) \(\ds \dfrac {27 + 64 + 125} 8\)
\(\ds \) \(=\) \(\ds \dfrac {3^3 + 4^3 + 5^3} 8\)
\(\ds \) \(=\) \(\ds \paren {\dfrac 3 2}^3 + 2^3 + \paren {\dfrac 5 2}^3\)

giving the solution $1 \tfrac 1 2$ inches, $2$ inches and $2 \tfrac 1 2$ inches


Or for another example, take $n = 9$.

Then:

\(\ds 9^3\) \(=\) \(\ds 729\)
\(\ds \) \(=\) \(\ds 1 + 216 + 512\)
\(\ds \) \(=\) \(\ds 1^3 + 6^3 + 8^3\)

and so:

\(\ds 3^3\) \(=\) \(\ds \paren {\frac 9 3}^2\)
\(\ds \) \(=\) \(\ds \dfrac {1 + 216 + 512} {27}\)
\(\ds \) \(=\) \(\ds \dfrac {1^3 + 6^3 + 8^3} {27}\)
\(\ds \) \(=\) \(\ds \paren {\dfrac 1 3}^3 + 2^3 + \paren {\dfrac 8 3}^3\)

giving the solution $\tfrac 1 3$ inches, $2$ inches and $2 \tfrac 2 3$ inches.

These are the two solutions provided by Dudeney.

$\blacksquare$


Sources

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