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
- 1932: Henry Ernest Dudeney: Puzzles and Curious Problems ... (previous) ... (next): Solutions: $232$. -- A Pavement Puzzle
- 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $244$. A Pavement Puzzle