## John Conway’s Wizards

John Conway sent me a puzzle about wizards, which he invented in the sixties. Here it is:

Last night I sat behind two wizards on a bus, and overheard the following:

β A: “I have a positive integral number of children, whose ages are positive integers, the sum of which is the number of this bus, while the product is my own age.”
β B: “How interesting! Perhaps if you told me your age and the number of your children, I could work out their individual ages?”
β A: “No.”
β B: “Aha! AT LAST I know how old you are!”

Now what was the number of the bus?

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1. #### apurv:

i found the bus number π

2. #### Tanya Khovanova’s Math Blog » Blog Archive » Simplified Wizards Puzzle:

[…] Here is my simplified version of Conway’s wizards puzzle. […]

3. #### Gaurang Shah:

Hey…

where is the solution…

4. #### Tanya Khovanova’s Math Blog » Blog Archive » Conway’s Wizards Generalized:

[…] I repeat the Conway’s Wizards Puzzle from a previous posting: Last night I sat behind two wizards on a bus, and overheard the […]

5. #### Sergey:

Assuming the minimal number of children sufficient to provide a solution, I am unable to get a unique solution. I see six solutions (two of which result in the same bus number). Even if I assume no twins (and all birthdays more than 12 months apart, of course), there are two distinct solutions.

Depending on the level of precision assumed in the wizard’s statement, I can interpret “…ages are positive integers” to mean “all my children were born on the same day of the year and today is that day”, which still admits twins. Of course if we start considering age with sub-day resolution, this requires wizardly powers in timing his wife’s deliveries π

When I looked at the simplified problem, referenced above, I was only able to identify one difference: we are no longer encouraged to assume the wizard’s age falls into a reasonable range. In that case, I see a very large number of solutions… Example: Children aged 48, 42 and 35 or 49, 40 and 36 for bus number 125 and a quite a large pocket.

It seems obvious to me that I am missing something; it is much less obvious what it is, that I am missing π

Any hints or pointers would be greatly appreciated.

6. #### Sergey:

Follow-up:
I read the problem again. Apparently, my definition of “minimal” was not minimal.
Still working on it, but I do not need any hints.

7. #### Sergey:

Got the bus number/age by brute force search, that was limited by maximum number of children and maximum age of a child. It seems possible that the limits are making my solution invalid. For example, it seems possible that wizard B may NOT be able to figure out A’s age, because A could have 12 children, some of which may be 100 years old – highly plausible for a wizard with magically enhanced longevity and reproductive powers.

Is there a solution, that is provably complete and correct without limitations? Or is there a proof that my solution is correct (that is no other solution is possible)?

8. #### Tanya Khovanova:

Sergey,

There is a unique solution.