Liber Abaci/Problems/Orchard with Seven Gates
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Classic Problem
- A man entered an orchard with $7$ gates, and there took a certain number of apples.
- When he left the orchard he gave the first guard $\dfrac 1 2$ the apples and $1$ apple more.
- To the second guard he gave $\dfrac 1 2$ of his remaining apples and $1$ apple more.
- He did the same for all the remaining $5$ guards, and left the orchard with $1$ apple.
- How many apples did he gather in the orchard?
Solution
$382$ apples.
Proof
After his encounter with the $7$th guard he had one apple left.
After giving the guard $\dfrac 1 2$ his apples, and before giving him $1$ more, he would have had $2$ apples.
Hence after leaving the $6$th guard, he had $4$ apples.
Similarly:
- after leaving the $5$th guard he had $2 \paren {4 + 1} = 10$ apples
- after leaving the $4$th guard he had $2 \paren {10 + 1} = 22$ apples
- after leaving the $3$rd guard he had $2 \paren {22 + 1} = 46$ apples
- after leaving the $2$nd guard he had $2 \paren {46 + 1} = 94$ apples
- after leaving the $1$st guard he had $2 \paren {94 + 1} = 190$ apples
- before meeting the $1$st guard he had $2 \paren {190 + 1} = 382$ apples.
$\blacksquare$
Sources
- 1202: Leonardo Fibonacci: Liber Abaci
- 1976: Howard Eves: Introduction to the History of Mathematics (4th ed.): p. $231$
- 1992: David Wells: Curious and Interesting Puzzles ... (previous) ... (next): Liber Abaci: $91$