Imagine that there are twelve coins in front of you. They are exactly the same size and shape, but one is either lighter or heavier than the other 11. To determine which coin is the odd one out, you are allowed to use a scale exactly three times. How do you find the unique coin while figuring out whether it is heavier or lighter than the other coins?
The first weigh is simple – divide the coins into three groups of 4, then weigh the first 4 against the second 4. If they balance, the different coin must be in the third group and the first 8 must be normal. From there, you can weigh 3 coins from the third group against 3 normal coins.
- If they balance, the odd one is the 4th coin from the third group and you can use your last weigh to compare that coin to a normal coin.
- If they don’t balance, the odd one must be one of the 3 coins and you also know whether the odd coin is heavier or lighter, as the other side is all normal.
- Therefore, simply weigh one coin out of the 3 against another coin from the 3 – if they balance, the third one is odd, if they don’t balance, you will know which one is odd because you know whether the coin should be lighter or heavier than a normal coin.
If the first weigh doesn’t balance, the riddle becomes much more complicated. To make it less complicated, we will name the three groups as so: the “heavies”, the “lighties” and the “normies”.
The third group of 4 must be normal as the odd coin is in the first or second group – they are the “normies”. Out of the first two groups of 4, either one side has a heavy coin or the other side has a lighter coin. Ergo, we know that one side only has “potentially heavier coins” and the other only has “potentially lighter coins” – respectively “heavies” and “lighties”.
In the second weigh, take two
to make one group, and make another group just the same, leaving 2
“heavies”. Now weigh those groups. If they balance, there is a heavy coin (as they are from the “heavies” group) in the remaining 2 and you can just weigh them against each other to find out which one is heavier.
If they don’t balance, there are two possibilities. Either a heavier coin is making one side happy, or one of the “lighties” coin is making one side lighter. This means that the only possible odd coin is the “heavies” coin from the heavy side or one of the two “lighties” coin from the lighter side.
In the last weigh, weigh the “heavies” coin and one of the “lighties” against two “normies”. If they balance, the second “lighties” is the light coin. If they don’t balance, you’ll know whether the “heavies” or the “lighties” was the odd one out by looking at whether the two coins are lighter or heavier than the “normies”.