It was Saturday evening in the Croesus Club, and two of its members—both bankers—were discussing the previous week's business.
"We're thinking of opening seven days a week," said Werther Packet.
"We did that last week," replied his friend Ty Koon.
"Did any of your customers use that facility?"
"Just one," said Ty. "Funny little chap, name of Leo Fibonacci… Always deposits his cash in one-pound coins. Last Sunday he made a deposit, and he made another one on Monday. On Tuesday he made a deposit equal to the sum of the amounts he had deposited on Sunday and Monday. And on every day thereafter, the amount he deposited was the sum of the amounts he had deposited on the previous two days."
"Weird."
"He's a bit eccentric, yes. Anyway, the amount he deposited today was exactly £100. You're good with figures, Werther: can you work out how much he deposited on Sunday and Monday?"
Werther thought for a moment. "£4 and £10," he said. "Then the subsequent deposits were £14, £24, £38, £62, and finally £100."
"No, that wasn't it," said Ty. "I remember noticing that he deposited less on Monday than he did on Sunday."
What sums did Leo Fibonacci deposit on Sunday and Monday?
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The answer
Leo deposited £12 on Sunday and £5 on Monday.
Let the amounts deposited on those two days be s and m respectively. Both are whole numbers greater than zero (since he made a deposit). The subsequent deposits are: Tuesday s+m, Wednesday s+2m, Thursday 2s+3m, Friday 3s+5m, Saturday 5s+8m.
Therefore 5s+8m=100.
To solve this observe that 8m=100-5s is a multiple of 5, so m is a multiple of 5. But
8m is greater than 0 and at most 100, so m=5 or 10.
If m=5 then s=12; otherwise m=10 and s=4.
The second choice is ruled out by Ty's remark.
The first leads to the deposits 12, 5, 17, 22, 39, 61, 100.
The winner was Matthew Bloomer
