Sunday Times Teaser 3252 – Family Tree
by Danny Roth
Published Sunday January 19 2025 (link)
George and Martha have five daughters; they are, in order of arrival, Andrea, Bertha, Caroline, Dorothy and Elizabeth. They now have ages that are all prime numbers in the range 28 to 60. The average is also a prime number, different from the other five.
Each daughter has a son, Adam, Brian, Colin, David and Edward respectively. They are all mathematics students, and are studying how to calculate square roots without using a calculator. One of them was given a perfect square with three or four digits (none repeated) and told to work out the square root. This he did accurately, getting his mother’s age, but he noted that the sum of the digits of that perfect square was also prime.
Which boy was it, and what is his mother’s age?
2 Comments
Leave one →
-
BRG permalink1234567891011121314151617181920212223from itertools import combinations# primes below 60prms = {3, 5, 7}prms |= {2} | {n for n in range(11, 60, 2) if all(n % p for p in prms)}# the possible ages of the daughters (oldest first)ps = [p for p in range(59, 28, -1) if p in prms]# consider the five daughters's agesfor p5 in combinations(ps, 5):# their average age is also a (different) primeavg, r = divmod(sum(p5), 5)if not r and avg in prms and avg not in p5:# find a daughter's age whose square has# different digits that sum to a primefor age in p5:dgts = tuple(int(d) for d in str(age * age))if len(dgts) == len(set(dgts)) and sum(dgts) in prms:son = "Adam Brian Colin David Edward".split()[p5.index(age)]print(f"{p5} -> {avg = } -> (Son = {son}, Mother's {age = }).")
-
John Z permalink12345678910111213141516171819202122from itertools import combinationssons = "Adam Brian Colin David Edward".split(' ')# possible ages of mothers, oldest firstp = tuple(i for i in range(59, 28, -2) if all(i % d for d in(3, 5, 7)))# 3 <= sum of three or four digits <= 36: generate primes < 36p36 = tuple(i for i in range(7, 36, 2) if all(i % d for d in(3, 5))) + (3, 5)for p5 in combinations(p, 5):if sum(p5) / 5 in set(p).difference(p5): #avg is prime and not a mother's age# consider mother's agefor n, m in enumerate(p5):if len(str(mm := m * m)) == len(set(str(mm))): #duplicate digits checkif sum(int(d) for d in str(mm)) in p36: #sum digits prime?print(f"{sons[n]}, Mother's age: {m}")
