The Vicar and the Choirmaster
Three members of a family ask the choirmaster if they can join the choir.
“Certainly,” he replies. “First of all, please tell me your ages.”
“I won’t tell you that,” says one of the three, “but I will tell you that the product of our ages is 2450 and the sum of our ages is twice your own age.”
The choirmaster thinks for a minute and says “I’m afraid that’s not enough information.” At this point the vicar walks in and says,
“I know these people, and I am older than all of them.” The choirmaster then says “Ah, then that’s all perfectly clear.”
How old is the vicar?
(The solution is contained in the comments, but I strongly advise you to give the problem a go yourself. Here’s why:
[S]ome people will turn instinctively and all too quickly [to the solution]. These people can’t be called mathematicians for whom the effort required to solve a problem is part of the attraction. That is not to say that sometimes you won’t just give up . . . but defeat should only be admitted after a brave struggle. Reading someone else’s solution too soon gives you no pleasure, and you are unlikely to learn anything, or remember anything you do learn. Reading the solution after battling the problem for some time will often cause you to laugh at yourself — a valuable skill — or realise that you just didn’t have the tools to tackle the problem. Either way you will learn something, and have a good chance of remembering it. But solving the problem yourself and then comparing your solution to someone else’s guarantees not only the maximum enjoyment but also the maximum benefit. You are unlikely to forget anything learnt in this way.
Taken from this review.)
Filed under: mathematics | Tagged: math, mathematics, maths, problem solving, puzzle
[...] easy? Try another puzzle on this [...]
Well, I am not yet figured out the solution, but what I can see at once is the fact it will not be possible to tell the age of the vicar.. I can assume it will be possible to tell ages of all the family members, however, since it is only said that the vicar ir older than each of the family members, we can’t know the age of him for sure - is he one year older or two?
Hi edgarsr, and thanks for thinking about this puzzle!
In fact, it IS possible in this case to determine the age of the vicar precisely. This is a mathematics blog so I’m not interested in puzzles where guessing or vagueness is involved
Good luck.
50
Well, I really can’t know how would this be possible.. So, I figured out that there can be 4 possible ages for those family members to satisfy all the conditions:
1) 5, 10 and 49 years;
2) 7, 10 and 35 years;
3) 7, 14 and 25 years;
4) 5, 14 and 35 years.
So, it is only known vicar ir older than 49. Am I missing something??
Maybe there is some `well known fact` not included in the exercise? For example, maybe vicars are not allowed to be older than 26.. Then the age of the vicar would be known precisely
Sorry, what I meant was it is only known the vicar is older than 25, not certainly 49.
Hi all.
Thanks for your answer, David. Would you like to share how you got it, then we can see if it is correct?
Edgar, no hidden facts are required (at least, no hidden facts about vicars). You might want to look at your list again. Are there other alternatives? For example, 7, 7, 50 . . .
The choirmaster knows his own age, so the only way he couldn’t have enough information is if there are two triples of numbers which multiply to 2450 and which add to the same answer, twice his age.
The only two triples with this property are (49, 10, 5) and (50, 7, 7). (So he’s 32.)
Now we’re told that the choirmaster can distinguish between these triples on hearing a piece of information which bounds the upper age. It can only be because the choirmaster knows that the age of the vicar is 50. Any older and neither triple would be ruled out.
OK, even I could understand this now
I didn”t know the choirmaster knows the age of the vicar though.
Hello everyone.
David, thanks very much for your concise answer which is absolutely correct. Moreover, I’m delighted to have you comment on this blog for a number of intersecting reasons. Firstly, because I think your work is very interesting. Secondly and serendipitously, I note from your webpage that you attended the “Mathematics and Narrative” conference on Mykonos, which has been reported in Nature and which I have been enthusing my colleagues about. Thirdly, and perhaps most trivially (or perhaps not) we both work at a “University of Canterbury”, albeit antipodally (and with a “Kent at” inserted, in your case)
edgarsr, I suppose that the assumption that the choirmaster and the vicar would know each other’s ages is a “hidden fact”. Vicars and choirmasters usually work quite closely and so would know each other reasonably well. But perhaps the church-based setting of this puzzle is too old-fashioned now.
Does anybody have any suggestions for updating this puzzle to a modern setting?
[...] or Surcharge? « The Magic Weblog of EDGARS on 17=evil.Phil Wilson on 17=evil.Phil Wilson on The Vicar and the Choirmasteredgarsr on The Vicar and the ChoirmasterDavid Corfield on The Vicar and [...]
Regarding that Mykonos meeting, there was a successor of a slightly different nature held in Delphi this July, reported here. Some volumes of papers and interviews will appear at some point.
Hi David, and thanks for the link. I’ll be sure to check it out over the coming break (which is also conference season down here in NZ!).