Sunday Times Brain-Teaser 904 – The Six Ages
by Victor Bryant
From The Sunday Times, 18th November 1979 (link)
Problems concerning ages have always proved fruitful and entertaining exercises to both mathematicians and non-mathematicians. Trial and error methods, calculators and normal or esoteric mathematical techniques can all be deployed to find the correct solution. The most elegant or the most economical method is naturally the most commendable, but the correct solution, however obtained, is the desideratum.
Our problem concerns six men whose ages are within the range 21 to 89 and any two of them differ by at least 9. If we take the two digits comprising each of the ages of three of the men, and reverse them, we obtain the ages of the other three men.
What is more, if we take the sum of the ages of the first group, we find that it equals the sum of the ages of the second group of three.
Also the sum of the squares of the three ages of the first group equals the sum of the squares of the ages of the second group of three.
Finally, one of the ages in each group is exactly twice an age in the other group.
What are the ages of the six men (in increasing order)?