Multiple choice general knowledge math & puzzles

Long ago, there was a king who had six sons. The king possessed a huge amount of gold, which he hid carefully in a building consisting of a number of rooms. In each room there were a number of chests; this number of chests was equal to the number of rooms in the building. Each chest contained a number of golden coins that equaled the number of chests per room. When the king died, one chest was given to the royal barber. The remainder of the coins had to be divided fairly between his six sons. The Question: Is a fair division possible in all situations?

  1. Yes

  2. No

  3. For some integers

  4. Indeterminable
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A Correct answer
Explanation

Let $r$ be the number of rooms. The number of chests per room is $r$, and coins per chest is $r$, making the total coins $r^3$. One chest ($r$ coins) is given to the barber, leaving $r^3 - r$ coins. Factoring gives $r^3 - r = r(r-1)(r+1)$, which is the product of three consecutive integers. One of these must be divisible by 3, and at least one must be even, making the product always divisible by 6. Thus, a fair division among 6 sons is always possible.

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