# Real MATHEMATICS – Game #8

Judge Seesaw 2.0.1

In the previous post I asked how you can find the 12th box when we don’t know whether it is heavier or lighter than the rest of the boxes.

To start the solution, we will divide the boxes into three groups of 4. But this time I will assign names to these groups and boxes:

In the first try we’ll place the groups A and B to the end of the seesaw. There will be two possible outcomes:

Outcome 1: Seesaw is in balance.

This outcome shows that the boxes of A and B are identical.

Then, we can conclude that oddly-weighted box is among the group C.

In the second try let’s place any three boxes of the group C with any three boxes of the group A. (Why three from A? Since all the boxes of A and B are identical, it doesn’t matter which three boxes I choose among them. I selected three from the group A. I could have selected two boxes from A and one from B as well. It wouldn’t change anything.)

After the second try, we will again left with two possible outcomes:

a. Seesaw is in balance.

It means three boxes of C and A are identical. Then we can conclude that the forth box of C is the oddly-weighted box.

b. Seesaw is tilted.

This concludes that one of the three boxes I choose from C is the oddly-weighted box. In this case the seesaw is either tilted towards C or A which tells us our box is either heavier or lighter than the rest of the boxes.

In other words, after the second try we ended up with the following question: “Two of three boxes have the same weight. If you know whether the odd box is heavier or lighter than the other two, how can you find the odd box?”

This can be solved with exactly one try on the seesaw. Select any two of the three boxes, place them on the seesaw.

If seesaw is in balance, then the third box is the odd one:

In case seesaw is tilted, as we know whether the odd box is heavier or lighter than the others, we can find the box we’ve been looking for:

Outcome 2: Seesaw is tilted.

In this case the box we are looking for is either inside the group A or inside the group B.

Another deduction we can make is that one of these groups is heavier than the other. For the sake of a clearer solution I am going to assume that A is heavier than B. (I could have chosen the other way around; it wouldn’t change anything.)

Before starting the second try I will be swapping a box between A and B. Also I will be replacing the remaining three boxes of B with three boxes of C.

There are three possible outcomes:

a. Seesaw is balanced.

This means that all the boxes sitting on the seesaw are identical. Hence the box we are looking is among the three boxes of group B which we replaced. We can also deduce that oddly-weighted box is lighter than the other boxes since the seesaw is balanced.

At this point all we need to do is to find the lighter one inside three boxes. As we showed in the previous situation, it can be done with comparing any two of those boxes.

b. Seesaw is tilted in the same direction as the first try.

This means that the oddly-weighted box is among the three boxes of A. We can also conclude that this box is heavier than the rest.

Again we have three boxes with one of them heavier than the other two. We can find that box with simply comparing any two of those boxes with the seesaw.

c. Seesaw is tilted in the opposite direction of the first try.

This means that the oddly-weighted box is either one of the boxes which we swapped between A and B. But in this situation we have no clue if the box we are looking for is heavier or lighter.

In order to find our box, we should compare one of those two boxes with any one of the remaining 10 boxes.

If there is a balance on the seesaw, it means our box is the one we didn’t place on the seesaw:

If the seesaw is tilted, then it means our box (in the following cases it is A4) is the one we’ve chosen and placed on the seesaw.

M. Serkan Kalaycıoğlu