Think of a Number/Examples/Bachet/1
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Classic Problem
- If it is even, he takes half of it,
- or if it is odd, he adds one and then takes one half.
- Next he multiplies the result by $3$,
- and tells you how many times $9$ will divide into the answer, ignoring the remainder.
- The number he chose is -- what?
Solution
Let $n$ be the number given at the end.
If $n$ is stated as being even, then the number originally chosen was $2 n$.
If $n$ is stated as being odd, then the number originally chosen was $2 n + 1$.
Proof
Let $x$ be the number chosen.
Let $x$ be either $2 n$ or $2 n + 1$, depending on whether it is odd or even.
If $x$ is even, the successive operations do the following:
- $2 n \to 6 n \to 3 n \to 9 n \to n$
If $x$ is odd, the successive operations do the following:
- $2 n + 1 \to 6 n + 3 \to 6 n + 4 \to 3 n + 2 = 9 n + 6 \to n$
$\blacksquare$
Sources
- 1612: Claude-Gaspar Bachet: Problèmes Plaisans et Delectables qui se font par les Nombres
- 1992: David Wells: Curious and Interesting Puzzles ... (previous) ... (next): Bachet: $105$