Tuesday, July 7, 2015

How Old is the Vicar?

There once was a choirmaster.
One day three people came in and asked to join the choir.
The choirmaster, who believes that there should be age for his choir's members, asks their ages.

To that question, one of them replied: "We can't tell you our ages, but we can tell you the following: the product of our ages is 2450, and the sum of our ages is twice your age."

The choirmaster is puzzled: "That's not enough information!"

Just then, the vicar walked in and said: "But I'm older than all of them"

The choirmaster, who knew the vicar's age, then exlaimed: "Ah! Now I know."

How old is the vicar?

How Old is the Vicar? Puzzle Solution

The vicar is 50.

The way to solve this puzzle, is to first of all write down all the possible permutations of three numbers whose product is 2450.
Starting Numbers Product Sum Choirmaster
1, 1, 2450 2450 2452 1226
1, 2, 1225 2450 1228 614
1, 5, 490 2450 496 248
1, 7, 350 2450 358 179
1, 10, 245 2450 256 128
1, 14, 175 2450 190 95
1, 25, 98 2450 124 62
1, 35, 70 2450 106 53
1, 49, 50 2450 100 50
2, 5, 245 2450 252 126
2, 7, 175 2450 184 92
2, 25, 49 2450 76 38
2, 35, 35 2450 72 36
5, 5, 98 2450 108 54
5, 7, 70 2450 82 41
5, 10, 49 2450 64 32
5, 14, 35 2450 54 27
7, 7, 50 2450 64 32
7, 10, 35 2450 52 26
7, 14, 25 2450 46 23

Since the choirmaster, after being told that the product of the ages is 2450 and that the sum is twice his age, still can't work out the ages, we can deduce that there are two (or more) combinations with the same sum. Those combinations have been highlighted in the table above.

The vicar then claims to be older than all of them. The oldest of the three is 49 in the first remaining combination, and 50 in the other. The choirmaster knows the vicar's age, and after his claim, he deduces everyone's age. The only way he's able to do so is if the vicar is 50, leaving the combination 7, 7, 50 logically impossible (the vicar has to be older, that is at least 1 year older than the others), and therefore learning that the people's ages are 5, 10, and 49.