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Puzzle for Today

Puzzle No. 522 – Friday 12 July

Today's puzzle was originally set in July 1869 — 150 years ago — as part of the Harvard entrance exam. A man bought a watch, a chain, and a locket for $216. The watch and locket together cost three times as much as the chain, and the chain and locket together cost half as much as the watch. What was the price of each?

Today’s #PuzzleForToday has been set by David Baynard, PhD student at Emmanuel college, Cambridge.

The written description of the puzzle can be translated to mathematical expressions — in this case, giving three simultaneous equations, with three unknowns: the prices of the watch, $W, chain, $C and locket, $L.

The equations are as follows:

W + C + L = 216 ①

• W + L = 3 C

⇒ W - 3 C + L = 0 ②

• 2(C + L) = W

⇒ W - 2 C - 2 L = 0 ③

① - ②

4 C = 216

⇒ C = 54

2×① + ③

3 W = 432

⇒ W = 144

Finally, substitute these solutions back into ①.

L = 216 - W - C = 216 - 144 - 54 = 18

So the watch cost $144, the chain $54 and the locket $18.

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