Henry Ernest Dudeney/Modern Puzzles/143 - The Stone Pedestal/Solution

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Modern Puzzles by Henry Ernest Dudeney: $143$

The Stone Pedestal
In laying the base and cubic pedestal for a certain public memorial,
the stonemason used cubic blocks of stone all measuring one foot on each side.
There was exactly the same number of these blocks (all uncut) in the pedestal as in the square base on the centre of which it stood.
Look at the sketch and try to determine the total number of blocks actually used.
Dudeney-Modern-Puzzles-143.png
The base is only a single block in depth.


Solution

$1458$ blocks.


Proof

Ridiculously messy question.

Let $2 x$ be the number of blocks used in total.

Then we have that $x$ are used in the base and $x$ are used in the pedestal.

Let $y$ be the number of blocks on along one edge of the pedestal.

We have that:

$x = y^3$

It looks as though the length of one side of the base is $3$ times the length of one edge of the pedestal.

On that assumption, we have that:

\(\ds \paren {3 y}^2\) \(=\) \(\ds y^3\)
\(\ds \leadsto \ \ \) \(\ds y\) \(=\) \(\ds 9\)
\(\ds \leadsto \ \ \) \(\ds x\) \(=\) \(\ds y^3 = 729\)
\(\ds \leadsto \ \ \) \(\ds 2 x\) \(=\) \(\ds 1458\)

$\blacksquare$


Sources

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