Three friends Alan, Ben and Cal, the age of each of whom is a positive integer number of years, made the following statements in response to a query posited by Don, who is a mutual acquaintance. (It is known to Don that the age of each of the three friends exceeds 20 years.)

Alan: "If you take the two digits of my age and subtract 62, the result is 9 less than Ben's age."

Ben: "If you reverse the two digits of my age and subtract 12, the result is Cal's age with the digits reversed."

Cal: "If you reverse the two digits of my age and add 47, the result is Alan's age with the digits reversed."

Don was unable to determine their ages, and requested for additional clarification, when Alan replied, "If I tell you whether my age is prime or composite, you will be able to determine our ages."

Thereupon, Don accurately deduced their ages.

Determine the age of each of the three friends in conformity with the abovementioned statements.

Note: Don is a logician and math wizard.

Alan is 97, Ben is 44, and Cal is 23.

First we are given that each of Don's friends are older than 20.
Second, from Alan's statement, we know his age is a two-digit number. With this restriction, we know from Alan's statement that his age is inclusively between 74 and 99 and Ben's age, correspondingly, is inclusively between 21 and 46.
Ben effectively stated that Cal is 21 years younger  than himself(21 being the reversal of 12). Therefore, with Cal at least 21, the possible ages for Alan would be inclusively between 95 and 99; Ben, 42 and 46; and Cal, 21 and 25.
As the range of Alan's age possibilities contain four composite numbers and one prime, we know from Alan's last statement his age is 97 and, thus, Ben's and Cal's ages are 44 and 23, respectively.

Posted by Dej Mar on 2010-04-03 21:04:51