This Article deals with the following problem.
Problem: Find positive integers x, y and z with x < y < z such that
x + y, x + z, y + z and x + y + z are squares. (1)
The interest in this problem arose from a conversation between the first two authors at The Times Listener Crossword Setters’ Dinner in March 2017. The Listener was a weekly magazine published by the British Broadcasting Corporation (BBC) from 1929 until 1991. Its main purpose was to reproduce the text of broadcast talks. However, from 1930 onwards it also contained a crossword, which was acknowledged to be the hardest cryptic crossword in a national weekly. When the magazine ceased publication, the crossword was ‘rescued’ by The Times newspaper and has appeared therein on Saturdays for over 25 years.