### Brain Teasers

# Last Time we Met

Said Alf to Bert, "The last time we met, our ages were both prime numbers, and when I was a quarter of the age I am now, you were that age plus half the age your father would have been thirty years previous to when he was six times the age you would have been when I was half your age". Said Bert to Alf "I'm off down the pub". How old were Alf and Bert the last time they met?

### Answer

Alf and Bert were 2 and 5 years old respectively the last time they met. If A and B are Alf and Bert's present ages and a and b are their ages when Alf was half Bert's age thenB - 3A/4 = (A/4) + (6b - 30)/2

Since their ages have always had the same difference we have

B - A = 3b - 15 = b - a

and since 2a = b then a = 3 and b = 6. So their ages differ by 3. Since all primes except 2 are odd and differ by an even number we can only have 2 and 5, Bert being the older of the two. The father plays no part in the problem!

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## Comments

That's some pretty advanced math for a 3-year old and a pub-crawling 6-year old, don't you think?

Did this conversation take place at the School for the Unnaturally Precocious?

I derived the geneneral solution to be: A=7s-4, B=4s+2, where s is a freely-chosen parameter.

If you set s=1, you get your solution, but many of the intermediate ages are fractions.

If you make s a multiple of four, then all intermediate ages are integers, but some of them may be negative or zero.

So, for example, if s=4, then A=24 and B=18, which is a little more plausible in terms of the conversation,

but gives the father's age as 18 before you subtract the 30 years. That solution gives you multiple prime possibilities:

they could have been 11 and 5, 13 and 7, 17 and 11, 19 and 13, or 23 and 17. Or if 18 is still too young for your local

drinking age, you could try s=8, which gives you A=52, B=34, and again multiple prime solutions, e.g. 23 and 5, or 29 and 11.

The puzzle is a nice idea, but I'm afraid I've got to downgrade this one on the fun catagory - if I'm going to do that much work, I want a nice crisp answer as

the payoff, each of which is flawed with some degree of implausibility.

Did this conversation take place at the School for the Unnaturally Precocious?

I derived the geneneral solution to be: A=7s-4, B=4s+2, where s is a freely-chosen parameter.

If you set s=1, you get your solution, but many of the intermediate ages are fractions.

If you make s a multiple of four, then all intermediate ages are integers, but some of them may be negative or zero.

So, for example, if s=4, then A=24 and B=18, which is a little more plausible in terms of the conversation,

but gives the father's age as 18 before you subtract the 30 years. That solution gives you multiple prime possibilities:

they could have been 11 and 5, 13 and 7, 17 and 11, 19 and 13, or 23 and 17. Or if 18 is still too young for your local

drinking age, you could try s=8, which gives you A=52, B=34, and again multiple prime solutions, e.g. 23 and 5, or 29 and 11.

The puzzle is a nice idea, but I'm afraid I've got to downgrade this one on the fun catagory - if I'm going to do that much work, I want a nice crisp answer as

the payoff, each of which is flawed with some degree of implausibility.

That should read "I want a nice crisp answer as the payoff, not such a muddled mess of different solutions, each of which is flawed with some degree of implausibility.

This one took some thought. I figured it out but it took about an half hour. It was nicely done.

So simple. The answer I had was earned.

had my share of fun here. i just don't understand what's mr. dewtell there complained about. the answers are VERY valid.

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