Sunday Times Teaser 3331 – Salarium Solace
by Mark Valentine
Published Sunday July 26 2026
During his last campaign in Germany, Centurion Ulpius lost more than one-third of his original 80 legionaries. The remaining number of men was prime. On their return to Rome, the Legate ordered his Prefect to share out the leftover salt stock to Ulpius’s remaining men, as a reward for their bravery.
The Prefect must ensure each man receives an identical amount. Using his scales he could accurately halve any quantity of salt. He divided the initial amount into two equal piles repeatedly until the number of piles exceeded the number of men. The men then received a pile each. The remaining piles were again each halved and distributed in the same way until only one pile remained. The Prefect returned home with this pile, one in 16,384 of the original stock.
How many of Ulpius’s men returned to Rome?
or
Only just over half returned to Rome!
It might have been less.
Of the set of numbers 0 < n < 81 only 16384 % 3 and 16384 % 43 == 1. We can assume that more than 3 returned. The constraint that the number of returnees be prime is superfluous if one asserts that the salt can never be exactly divided up with nothing remaining.