The birthdays of the five friends Al, Ben, Cal, Dan, and Ethan falls on the same day, though they are different ages.
On their mutual birthday, the five friends were discussing their respective current ages. The discussion was as follows:
Dan said to Ben: "I'm nine years older than Ethan."
Ethan said to Ben: "I'm seven years older than Al."
Al said to Ben: "Your age is exactly 1.7 times that of mine."
Ben said to Cal: "Ethan is younger than you."
Cal said to Dan: "The difference between our ages is six years."
Cal said to Al: "I'm ten years older than you."
Cal said to Al: "Ben is younger than Dan."
Ben said to Cal: "The difference between your age and Dan's is the
same as the difference between Dan's and Ethan's."
It is known that when one of them spoke to someone older, everything they said was true. However, when speaking to someone younger, everything they said was false.
What are the current ages of each person?
Statement 6: Cal said to Al: "I'm ten years older than you."
Assume this is true.
Then Older Cal would lie to Al, which is an immediate contradiction.
Therefore, this is a lie.
Therefore, Cal is older than Al, but not by 10 years.
Statement 7: Cal said to Al: "Ben is younger than Dan."
Cal is known to be older than Al, so this is a lie.
Therefore, Ben is older than Dan.
Statement 1: Dan said to Ben: "I'm nine years older than Ethan."
Ben is known to be older than Dan, so this is the truth.
Therefore, Ben is older than Dan who is 9 years older than Ethan.
Therefore, Ben is older than Ethan.
Statement 2: Ethan said to Ben: "I'm seven years older than Al."
Ethan is known to be younger than Ben, so this statement is true.
Therefore, Al is younger than Ethan.
The age sequence (youngest to oldest) is AEDB, with C somewhere older than A.
Statement 4: Ben said to Cal: "Ethan is younger than you."
Assume this is false.
Then Cal is younger than Ethan, so he is also younger than Dan.
Then statement 5 (Cal said to Dan: "The difference between our ages is six years.") is true.
But Dan is 9 years older than Ethan, so Cal is at least 3 years older than Ethan.
This is a contradiction, so statement 4 is true, so Ben is younger than Cal.
The age sequence therefore must be AEDBC.
Statement 8: Ben said to Cal: "The difference between your age and Dan's is the same as the difference between Dan's and Ethan's."
Ben is younger than Cal, so this is necessarily true, which makes Cal 9 years older than Dan and 18 years older than Ethan.
Statement 3: Al said to Ben: "Your age is exactly 1.7 times that of mine."
Al is younger than Ben, so this is true also.
Just to recap, we know know that:
A < E < D < B < C
E = A + 7
B = A * 1.7
C = D + 9 = E + 18
Therefore, C = A + 25
D = A + 16
A + 16 < A*1.7 < A + 25
16 < A*0.7 < 25
22.9 < A < 35.7
Since A must be a multiple of 10 (to make B a whole number), it can only be 30
That makes the ages:
Al 30
Ethan 37
Dan 46
Ben 51
Cal 55
Edited on September 10, 2013, 11:05 pm
Exactly enough info (analytical spoiler)
