# The nicest maths problem I know

I was told this by the father of a family from Hamilton in New Zealand, whom a friend and I met while cycling in NZ in early 1975.

I like this problem because (a) initially it looks like there is not enough information to solve it and (b) I can explain the solution to a moderately bright 12 year old.

I have since seen this or similar problems in a number of places, but I still think it is beautifully constructed, with implicit information that is nevertheless there if you can see it.

**The Bellringers Problem**

A bishop and her friend the vicar were both amateur but well-practised mathematicians, and often set each other problems to solve.

One day, the bishop visited the vicar. While the two were walking in the grounds of the vicarage, three bellringers passed them by.

“Why, they’re a fine bunch of bellringers!” said the bishop. “How old are they?”

“Well, I can tell you two things…” said the vicar.

“Firstly, if you multiply their ages, you get 2450.”

“Secondly, if you add their ages together, you get twice your age.”

The bishop had come prepared for such a challenge. She took pen and paper from her pocket and scribbled down a few numbers, but then looked up and said, “You haven’t given me enough information.”

The vicar added, “Oh, yes, I see. Well, I’m older than any of you.”

The bishop looked down at her scribblings and then said, “Ah! Now I know the bellringers’ ages.”

*The question is: How old is the vicar?*

## Notes:

(1) A solution does exist and is unique for the conversation as described.

(2) This is not a trick problem. For example, no one was born on February 29.

(3) Neither has any tricky wording has been used. For example, the vicar saying that he is “older than any of you” means that the vicar is the eldest of the five people in the story.

(4) You can assume from the course of the conversation that we are dealing with whole number ages.

(5) You can also assume that the bishop was correct about not having enough information to solve *her* problem..

Over time, I’ve developed a shorthand version of the conversation, as follows:

*B: a, b, c = ?*

*V: a x b x c = 2450*

* a + b + c = 2 x B*

*B: Not enough info*

*V: V > a, b, c, B*

*B: a, b, c = !*

This version is very useful if writing it on a restaurant serviette.

I will give the solution in a near future (maybe the next) post.

*Stretch question: If all five people in the story are less than 100 years old, what other numbers could be substituted for 2450 above, which would also provide a (different) unique answer for the age of the vicar? I’ve counted 70 of them (well, an Excel spreadsheet I made has done so).*

See my next post for some answers to the above questions.

Ekspong’s unique problem with no input numerical information about the age of the bishop.

Here it is attached:

The bishop had only three visitors in his church, a mother with her two youngest

children, one in preschool age, the other a schoolchild. He knew them well and

amused himself by calculating the sum and the product of their ages. Later when he

met his friend, a mathematician, he said that certainly you know that I am oldest

among the priests in the diocese, but can you find the ages of my visitors from the

knowledge of the sum and the product of their ages. He gave his friend these

numbers.

After some thoughts the mathematician declared: Well, really not, since the solution

is ambiguous. But if you were the oldest person in church that Sunday the solution

becomes unique.

So it was. The bishop was amused and impressed, so he told the story at the next

gathering of the priests in the diocese. Surprisingly, a group among them had the

same Sunday each on his own experienced exactly the same miracle with a unique

solution but still completely different ages of the three visitors.

Questions:

How old was the bishop?

How many priests could maximally have experienced this miracle?

How old were those priests and all their visitors?

G.Ekspong, Stockholm, Sweden

ekspong@fysik.su.se

2016-12-12