Henry Ernest Dudeney/Puzzles and Curious Problems/121 - Find the Numbers/Solution
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Puzzles and Curious Problems by Henry Ernest Dudeney: $121$
- Find the Numbers
- Can you find $2$ numbers composed only of ones which give the same result by addition and multiplication?
- Of course $1$ and $11$ are very near, but they will not quite do,
- because added they make $12$, and multiplied they make only $11$.
Solution
- $11$ and $1 \cdotp 1$
Proof
Dudeney has already raised this question in his Modern Puzzles: $93$ - Sum Equals Product, where he shows that:
- $y = \dfrac x {x - 1}$
for any pair $x$ and $y$ such that $x y = x + y$.
The only time $x$ and $y$ are both integers is when $x = y = 2$.
So there's a trick to look out for.
Suppose $x = 11$.
We have:
- $y = \dfrac {11} {11 - 1} = \dfrac {11} {10}$
and we notice that:
- $\dfrac {11} {10} = 1 \cdotp 1$
We note that:
- $11 + 1 \cdotp 1 = 1 \cdotp 21 = 11 \times 1 \cdotp 1$
$\blacksquare$
Sources
- 1932: Henry Ernest Dudeney: Puzzles and Curious Problems ... (previous) ... (next): Solutions: $121$. -- Find the Numbers
- 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $140$. Find the Numbers