Henry Ernest Dudeney/Modern Puzzles/100 - Odds and Evens/Solution

by : $100$

 * Odds and Evens

Solution
If the friend gives an odd total, the right hand holds the odd number.

Otherwise the left hand holds the odd number.

Proof
Let $r$ be the number of coins in the right hand, and $l$ be the number of coins in the left hand.

If $r$ is odd, then $7 r$ is also odd, while $6 l$ is always even.

Hence the total $7 r + 6 l$ is in this case odd.

If $r$ is even, then $7 r$ is also even, while again $6 l$ is always even.

Hence the total $7 r + 6 l$ is in this case even.

So if the friend gives an odd total, the right hand holds the odd number.

Otherwise the left hand holds the odd number.

It is to be noted that with the low emphasis currently placed on ability to perform mental arithmetic, it is likely to be the case that a friend who is capable of multiplying the number of coins by $7$ and $6$ is also probably sufficiently mathematically sophisticated as to understand immediately how the trick works.