You have 8 oranges – one is heavier than the others. How many weighings on a justice scale do you need to determine which one it is?
It’s tempting to split the oranges in the middle, and weigh 4 and 4, which would take 3 weighings. But you can do it in two weighings. Take 3 and 3 and weigh these. If equal, you know it’s in the two remaining and you can see which it is with another weighing. If not, take the heavier bunch and weigh 2 of the oranges – one will either be heavier, or if equal you know it’s the one remaining.
Two weighings will also work for nine oranges (it’s normally asked with eight as it makes it more tempting to split them evenly). Three weighings can get you to 27, which is 3 to the power of 3 – so you see the pattern.
Another variant of this puzzle is that you have 12 oranges with one being a different weight, but don’t know whether it’s heavier or lighter. This requires more weighings – and is a bit tricker.
The first step is to group the oranges in three groups – let’s call them group A, B and C – with 4 oranges each. Then within each group, you divide the oranges into a single orange (let’s call it A1 for the A group) and a group of 3 oranges (let’s call this A3 for the A group).
Before we start, let’s remember from the first part of the puzzle that if you have 3 oranges and you know if one is heavier or lighter, then you can find it in one weighing. We’ll call this the “known 3 orange case” in the text below.
A1, A3 <--> B1, B3 (leaving out C1 and C3)
Let’s take the case where they balance. You know the orange is in C1 or C3. You can then weigh:
C3 <--> A3
If they balance it’s C1. If they don’t you can go into a third weighing of C3 with a “known three orange case”.
If the first weighing doesn’t balance (let’s assume that A1, A3 are heavier), then of course you know it’s not in C1 or C3 and you go into a second weighing as follows:
A1, B3 <--> B1, C3 (leaving out A3)
If they balance then we go into a third weighing on A3 with a “known three orange case” (we know A3 has the heavy orange from the first weighing).
If A1, B3 is heavier, then you know that either A1 is heavier or B1 is lighter (it can’t be in B3 as we said that was in the lighter side in weighing one) – and you can find out which one by comparing one of them against C1.
If B1, C3 is heavier then you go into a “known three orange case” with B3 knowing that it contains a lighter orange (it can’t be A1 or B1 as they were on the contradictory side of the scale on the first weighing).
Note: We assumed A1, A3 heavier after the first weighing – if it was B1, B3 you just do the above replacing A groups for B groups.
So there you have it – tricky to follow but you can do the 12 orange case where you don’t know if the one orange is heavier or lighter in 3 weighings.