Five friends Andy, Brice, Cal, Danny, and Ethan make the following statements each others' ages.
Andy’s statement:
Ethan is 27.
Brice’s statement:
I am 81.
Cal’s statement:
Brice is 61.
Danny’s statements:
1. Andy is 57.
2. What Cal says is false.
Ethan’s statements:
1. Brice is older than Andy.
2. Danny is 30 years younger than Cal.
It is known that:
- None of the friends is younger than 10 or older than 99.
- Remarks made by anyone 50 or older are true, unless his age is a perfect square.
- Remarks made by anyone who is under 50 are false, unless his age is a perfect cube.
Determine the age of each of the five friends.
Brice can't be telling the truth, as an 81-year-old would lie; he's not 81, but of some other age that lies.
Cal must be speaking falsely, since if Brice were 61 he'd be telling the truth, and he wasn't.
So Danny speaks the truth, and Andy is indeed 57, and thus speaks truthfully when he says that Ethan if 27.
As Ethan is 27, a perfect cube, he speaks the truth: first about Brice being older than Andy. Brice is older than 57, tells lies and is not 81, so his age is a 2-digit perfect square larger than 57 but not 81; he must be 64.
From Ethan's other statement, Danny's age + 30 = Cal's age. We have to add 30 to a truth-speaking age to get a lying age. Let's try some truth-speaking ages below 50 and lying ages above 50, as lying below 50 would require a truth below 50 as well, and truth above 50 would alo require lying above 50:
27 + 30 = 57, these ages both speak truthfully.
That in fact exhausts the truth-speaking under-50's.
64 - 30 = 34, but both these ages lie.
81 - 30 = 51, aha: 51 tells the truth and 81 lies.
Danny is 51 and Cal is 81.
Summary:
Andy is 57
Brice is 64
Cal is 81
Danny is 51
Ethan is 27
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Posted by Charlie
on 2014-05-19 11:03:00 |
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