Sunday Times Brain-Teaser 351 – Birthday Party
by J S Rowley
Published 28th January 1968 (link)
Some years ago the Bell family were holding their usual annual special birthday party. Four members of the family, of four different generations, had birthdays on the same day of the year. They were old Adam, his son Enoch, Enoch’s son Joseph and Joseph’s son David. On this occasion David remarked that the sum of any three of their four ages was a perfect square
Some years later old Adam died on his birthday, but it so happened that on the very same day David’s son Samuel was born, and the annual party was continued in subsequent years.
In 1967 at the usual party Samuel made exactly the same remark that David had made, on the previous occasion.
In what year did Adam die and how old was he then?
(Perhaps I should mention that no one survived to be 100!).
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BRG permalink1234567891011121314151617181920212223242526272829# the set of positive squares below 300sqs = set(n * n for n in range(1, 18))# find sets of four different values, all less# than 100, for which any three add to a squarequads = []for a in range(1, 100):for b in range(a + 1, 100):for s1 in sqs:if b < (c := s1 - (a + b)) < 100:for s2 in sqs:if c < (d := s2 - (b + c)) < 100:if len({a + b + d, a + c + d} & sqs) == 2:quads.append((a, b, c, d))# consider the two events mentionedfor p in quads:for q in quads:if p == q:continue# ages in 1967(d, j, e, a), (s, d_, j_, e_) = p, q# the time between the two events must# be the same for David, Joseph and Enochif (gap := d_ - d) == j_ - j == e_ - e:if a < (ad := a + gap - s) < 100:print(f"Adam died in {1967 - s} at age {ad}.")
