Henry Ernest Dudeney/Puzzles and Curious Problems/232 - A Pavement Puzzle/Solution

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Puzzles and Curious Problems by Henry Ernest Dudeney: $232$

A Pavement Puzzle
Two square floors had to be paved with stones each $1$ foot square.
The number of stones in both together was $2120$, but each side of one floor was $12$ feet more than each side of the other floor.
What were the dimensions of the two floors?


Solution

$26$ and $38$ feet on one side.


Proof

Let $a$ and $b$ feet be the lengths of the sides of the floors.

We have:

\(\ds a^2 + b^2\) \(=\) \(\ds 2120\) The number of stones in both together was $2120$,
\(\ds b = a + 12\) \(=\) \(\ds 2120\) but each side of one floor was $12$ feet more than each side of the other floor.
\(\ds \leadsto \ \ \) \(\ds a^2 + \paren {a + 12}^2\) \(=\) \(\ds 2120\)
\(\ds \leadsto \ \ \) \(\ds a^2 + 12 a - 988\) \(=\) \(\ds 0\) after simplification
\(\ds \leadsto \ \ \) \(\ds a\) \(=\) \(\ds \dfrac {-12 \pm \sqrt {12^2 + 4 \times 988} } 2\) Quadratic Formula
\(\ds \) \(=\) \(\ds \dfrac {-12 \pm \sqrt {4096} } 2\) Quadratic Formula
\(\ds \) \(=\) \(\ds -6 \pm 32\) after evaluation
\(\ds \) \(=\) \(\ds 26 \text { or } -38\)

As it is the positive root we want here, we have that $a = 26$ and hence $b = 26 + 12 = 38$.

$\blacksquare$


Sources

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