When confronted with mathematical problems I always recommend that you should use the APEX method which I propose.

This APEX method provides a simple and effective way to deal with any mathematical problem, no matter how small or large, and no matter how apparently easy or hard.

## (A)NALYSE:

Let us see what we have and what we are supposed to give as the answer.

We are told that there are three boxes, labelled GOLD, SILVER, and LEAD.

We are also told that there is a painting inside one and only one of these boxes.

We are further told that there are some statements written on the boxes:

- GOLD: The painting is inside this box.
- SILVER: The painting is not inside this box.
- LEAD: The painting is not inside the GOLD box.

We are also told that two of the statements are FALSE, and that one is TRUE.

## (P)LAN:

It is clear that this is a logical problem, since problems which involve finding TRUE and FALSE are mostly logical in nature.

One approach is to see which can be TRUE and which can be FALSE, and this means that each of the statements can be either TRUE or FALSE, thus there are 2 choices for each box, and thus there 2x2x2 possibilities, since there are three boxes in total, and this makes 8 total choices.

## (E)XECUTE:

Firstly, since there are two FALSEs and only one TRUE, then the number of choices is actually only three. The TRUE is either GOLD, SILVER, or LEAD, with the other two being FALSE.

Secondly, it is also clear that the statements on the GOLD and LEAD boxes are in conflict, since they cannot both be TRUE. This means that the TRUE and one of the FALSEs is already allocated, and thus the SILVER can only be FALSE.

If SILVER is FALSE then ths statement that the painting is NOT in the SILVER box is FALSE, and thus the **painting IS in the SILVER box**.

## (X)CHECK (crosscheck):

However, we have not yet allocated the TRUE and FALSE to the GOLD and SILVER boxes, so could the painting be in one of these as well? Let us crosscheck to see.

Firstly let us consider that the GOLD box is FALSE, and the LEAD box is TRUE.

- GOLD (FALSE): The painting is in this box – this works since we know that it is in the SILVER box already.
- LEAD (TRUE): The painting is not in the GOLD box – this also works.

Now let us consider that the GOLD box is TRUE and the LEAD box is FALSE.

- GOLD (TRUE) : The painting is in this box – cannot work, since we know it is in the SILVER box
- LEAD (FALSE): The painting is not in the GOLD box – if this is false then the painting should be in the GOLD box, which is also a problem, since the painting is in the SILVER box.

Thus the answer is that the GOLD box is FALSE, the SILVER box is FALSE, and the LEAD box is TRUE, and the painting is in the SILVER box.