Seniority - Futility Closet

Annie, Betty, Carrie, Darla, and Eve all have the same birthday, but all are different ages. On their shared birthday:

Darla said to Betty: “I’m 9 years older than Eve.”
Eve said to Betty: “I’m 7 years older than Annie.”
Annie said to Betty: “Your age is exactly 70 percent greater than mine.”
Betty said to Carrie: “Eve is younger than you.”
Carrie said to Darla: “The difference between our ages is 6 years.”
Carrie said to Annie: “I’m 10 years older than you.”
Carrie said to Annie: “Betty is younger than Darla.”
Betty said to Carrie: “The difference between your age and Darla’s is the same as the difference between Darla’s and Eve’s.”

Whenever one of them spoke to someone older than herself, everything she said was true, but when she spoke to someone younger, everything she said was false. How old is each person?

	Select Click for Answer
Annie – 30 Betty – 51 Carrie – 55 Darla – 46 Eve – 37 Carrie tells Annie she’s older than her by 10 years. If Carrie were really younger than Annie, this statement would be true, and that’s impossible, so Carrie must be older than Annie, just not by 10 years. Record that as a fact: Carrie is older than Annie (but not by 10 years). Carrie also lies to (younger) Annie that Betty is younger than Darla. Fact: Darla is younger than Betty. Darla tells the truth to (older) Betty that she’s 9 years older than Eve. Fact: Darla is 9 years older than Eve. Eve tells the truth to (older) Betty that she’s 7 years older than Annie. Fact: Eve is 7 years older than Annie. Annie tells the truth to (older) Betty that Betty’s age is 70 percent greater than her own. For Betty’s age to be a whole number, Annie’s age must be a multiple of 10. Since Betty is older than Darla, and Darla is 7 + 9 = 16 years older than Annie, that means Betty has to be more than 16 years older than Annie. The lowest multiple of 7 greater than 16 is 21. Fact: Annie is at least 30 years old (and definitely a multiple of 10). At this point, it appears that either Carrie or Betty is oldest and thus lying to everyone else. Let’s assume that Carrie is oldest and see whether that works. In that case, Carrie is lying to Darla that the difference in their ages is 6 years, but Betty is telling the truth to (older) Carrie that the difference between Carrie’s age and Darla’s is the same as the difference between Darla’s and Eve’s, namely 9 years. Let’s test this scenario, assuming Annie’s age is 30. Then we get, from youngest to oldest: Annie = 30, Eve = 37, Darla = 46, Betty = 51, Carrie = 55 Checking all statements and the age relations shows that this is an answer. Is it the only answer? If Annie’s age were 40, then Betty’s age would be 68, and Carrie’s age would be 65, so Carrie would not be the oldest, and that would be a fatal flaw. If Annie is older than 30, then Betty is older than Carrie, and Carrie is not the oldest. Hence the values above form the only consistent answer. From the Wikibooks puzzles collection (see that solution for a mathematical argument).

Click for Answer

Annie – 30
Betty – 51
Carrie – 55
Darla – 46
Eve – 37

Carrie tells Annie she’s older than her by 10 years. If Carrie were really younger than Annie, this statement would be true, and that’s impossible, so Carrie must be older than Annie, just not by 10 years.

Record that as a fact: Carrie is older than Annie (but not by 10 years).

Carrie also lies to (younger) Annie that Betty is younger than Darla.

Fact: Darla is younger than Betty.

Darla tells the truth to (older) Betty that she’s 9 years older than Eve.

Fact: Darla is 9 years older than Eve.

Eve tells the truth to (older) Betty that she’s 7 years older than Annie.

Fact: Eve is 7 years older than Annie.

Annie tells the truth to (older) Betty that Betty’s age is 70 percent greater than her own. For Betty’s age to be a whole number, Annie’s age must be a multiple of 10. Since Betty is older than Darla, and Darla is 7 + 9 = 16 years older than Annie, that means Betty has to be more than 16 years older than Annie. The lowest multiple of 7 greater than 16 is 21.

Fact: Annie is at least 30 years old (and definitely a multiple of 10).

At this point, it appears that either Carrie or Betty is oldest and thus lying to everyone else. Let’s assume that Carrie is oldest and see whether that works.

In that case, Carrie is lying to Darla that the difference in their ages is 6 years, but Betty is telling the truth to (older) Carrie that the difference between Carrie’s age and Darla’s is the same as the difference between Darla’s and Eve’s, namely 9 years. Let’s test this scenario, assuming Annie’s age is 30. Then we get, from youngest to oldest:

Annie = 30, Eve = 37, Darla = 46, Betty = 51, Carrie = 55

Checking all statements and the age relations shows that this is an answer. Is it the only answer?

If Annie’s age were 40, then Betty’s age would be 68, and Carrie’s age would be 65, so Carrie would not be the oldest, and that would be a fatal flaw. If Annie is older than 30, then Betty is older than Carrie, and Carrie is not the oldest. Hence the values above form the only consistent answer.

From the Wikibooks puzzles collection (see that solution for a mathematical argument).