The great weights puzzle
You have twelve weights, one of which is different from the rest.
Using just 3 weighings, can you identify which weight is the odd
one out, and whether it is heavier or lighter than the rest?
Age
14 to 16
Challenge level
Problem
This problem follows on from 9 Weights
You have twelve weights, one of which is heavier or lighter than the rest, but you don't know which one is heavier or lighter, and a balance. Using just 3 weighings on the balance, can you find a way to identify which weight is the odd one out, and whether it is heavier or lighter?
Getting Started
For the first weighing you need to put 4 weights on each tray.
Now look at each possible outcome of the first weighing and decide what you would do in each case.
You might like to first consider the case when the scales balance on the first weighing - what do you know about all the weights on the balance?
You might also like to think about the problem with fewer weights first, and use ideas you come up with there for the 12 weight problem.
Student Solutions
Quan Pham from Vietnam discovered the following method.
We divide the weights into three groups of four, group A (weights 1-4, say), group B (5-8) and group C (9-12).
1. Weigh group A against group B.
Case: A = B
The odd weight must be in group C.
2. Weigh 9, 10 and 11 against 1, 2 and 3.
If it balances then the odd one is weight 12.
3. Weigh 12 against 1
This will determine whether 12 is lighter or heavier.
If weight (9 + 10 + 11) > weight (1 + 2 + 3) then we know that the odd weight is one of 9, 10, 11 and is heavier. Similarly if weight(9 + 10 + 11) < weight(1 + 2 + 3) then one of 9, 10, 11 is lighter. Either way, our final weighing will be
3. Weigh 9 against 10
If the scales are uneven we can work out which one of 9, 10 is the odd weight, and if they are even then clearly our odd weight is 11.
Case: A > B
The odd one is either in group A, in which case it is heavier, or in group B and is lighter.
2. Weigh 1, 5 and 9 against 6, 7 and 2
If the scales balance then the odd one is 3, 4 or 8. In this case we weigh
3. 3 and 8 against 9 and 10
If the scales balance then the odd one is 4. If weight(3 + 8) > weight(9 + 10) then the odd one is heavier, so must be in group A, which means it is 3. Similarly if weight(3 + 8) < weight(9 + 10) the odd one is 8.
If weight(1 + 5 + 9) > weight(6 + 7 + 2) then either 1 is heavier or 6 or 7 are lighter. In this case we weigh
3. 1 and 6 against 9 and 10.
We can find the odd weight using a similar method to the previous step 3.
If weight(1 + 5 + 9) < weight(6 + 7 + 2) then either 5 is lighter or 2 is heavier. Weight
3. 5 against 9
and we will discover which is the odd one.
Case: A < B
We argue in a way analogous to the case A > B.
Teachers' Resources
This problem puts emphasis on trying to gain as much information as
possible from each weighing, whatever the outcome. So if a weighing
has one outcome which gives you a lot of information, but another
outcome which doesn't gives you much new information, it is
probably not going to be a useful way of identifying the odd
weight.