This question appeared in the 2015 Singapore and Asian Schools Math Olympiad, a competition for 14-year-old students from Singapore, Thailand, Vietnam, China, and the U.K. (I’ve amended the language a bit):
Albert and Bernard have just become friends with Cheryl, and they want to know when her birthday is. Cheryl gives them a list of 10 possible dates:
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May
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15
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16
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19
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June
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17
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18
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July
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14
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16
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August
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14
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15
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17
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Cheryl then tells Albert and Bernard separately the month and the day of her birthday, respectively.
Albert: I don’t know when Cheryl’s birthday is, but I know that Bernard doesn’t know it either.
Bernard: At first I didn’t know when Cheryl’s birthday is, but I know now.
Albert: Then I also know when Cheryl’s birthday is.
When is Cheryl’s birthday?
Singapore TV presenter Kenneth Kong posted the question online, and it went viral in a matter of days. The competition organizers had intended it to “sift out the better students” and expected that 40 percent of the competitors would find the solution. What is it?
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SelectClick for Answer |
Cheryl’s birthday is July 16.
Albert’s first statement is unsurprising — he knows only the month of Cheryl’s birthday, and each month on the list includes several possible days. But the fact that he knows that Bernard doesn’t know it is important — it tells us that the month can’t be May or June, because both of those months contain unambiguous dates. For example, if Cheryl’s birthday were May 19, then Bernard would know immediately that she was born in May, as that’s the only month whose 19th day appears on the list. Similarly, if Bernard knew the day of Cheryl’s birthday was the 18th, he’d know immediately that she was born in June. The fact that Albert can exclude these possibilities shows that he knows she wasn’t born in May or June; she must have been born in July or August.
Bernard’s statement shows that he’s followed this and now knows that the month must be either July or August. And the fact that he now knows the full birthday shows that Cheryl was born on the 15th, 16th, or 17th — if she’d been born on the 14th then he’d still be uncertain, as both July 14 and August 14 fall on the list.
Albert has narrowed the list of possibilities to July 16, August 15, and August 17. As he now declares that he knows the birthday unambiguously, it must be July 16 — if it fell in August he’d have no way to distinguish between August 15 and August 17.
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