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.