Sunday Times Teaser 3301 – Round the Table
by Andrew Skidmore
Published Sunday December 28 2025 (link)
I recently attended a celebration with fewer than 100 family and friends. We sat round a large oval table with one seat for each person. I said to Liam that he could pick any seat, but it might be interesting to work out how many different arrangements were possible for people round the table. He replied “I’ll leave that to you Grandad”, then (mischievously) added “but remember I always sit next to you and to Granny”. The restriction that I must sit next to Liam divided the number of possible arrangements by a certain prime number. The further restriction that he must also be next to Granny caused further division by a prime number with a different number of digits.
What, in ascending order, are the number of people round the table and the two prime numbers?
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BRG permalink1234567891011121314# The number of arrangements for N people:# Unconstrained: (N - 1)!# Liam next to grandad: 2.(N - 2)!# Liam next to both grandparents: 2.(N - 3)!# Hence:# (N - 1) / 2 and (N - 2) are respectively one and two digit primes# N = 2.p1 + 1 <= 15 ==> p2 = N - 2 <= 13for p2 in (11, 13):N = p2 + 2if (p1 := (N - 1) // 2) in {2, 3, 5, 7}:print(f"{p1 = }, {p2 = }, {N} around the table.")
