### Brain Teasers

# Four Boxes

There are four boxes: one brown, one black, one green, and one white. One box is labelled "15 cents", one is labeled "20 cents", one is labeled "25 cents", and one is labeled "30 cents". One contains three nickels, one contains three dimes, one contains two nickels and a dime, and one contains two dimes and a nickel. All four boxes are mislabeled.

If a perfect logician who knows all of the above...

1. looked at the label on the brown box and saw two nickels in the box, he would know what the third coin is.

2. looked at the label on the black box and saw two dimes in the box, he would not know what the third coin is.

3. looked at the label on the green box and saw one nickel and one dime in the box, he would know what the third coin is.

What are the labels and contents of each box?

If a perfect logician who knows all of the above...

1. looked at the label on the brown box and saw two nickels in the box, he would know what the third coin is.

2. looked at the label on the black box and saw two dimes in the box, he would not know what the third coin is.

3. looked at the label on the green box and saw one nickel and one dime in the box, he would know what the third coin is.

What are the labels and contents of each box?

### Hint

3 nickels=15 cents.2 nickels and 1 dime=20 cents.

2 dimes and 1 nickel=25 cents.

3 dimes=30 cents.

### Answer

The brown box is labeled "20 cents" and contains 15 cents, the black box is labeled "15 cents" and contains 30 cents, the green box is labeled "25 cents" and contains 20 cents, and the white box is labeled "30 cents" and contains 25 cents.From fact 1, we know that there are at least two nickels in the brown box, so there are either 15 cents or 20 cents. If the box was labeled "25 cents" or "30 cents", the value of the third coin would be indeterminate, so the box must be labeled "15 cents" or "20 cents", and the contents of the box are 15 cents if the box is labeled "20 cents", and vice versa.

From fact 2, we know that there are at least two dimes in the black box. But since the third coin is indeterminate, the box must be labeled "15 cents" or "20 cents", implying that there is at most one dime. So the brown box and the black box must be labeled "15 cents" and "20 cents" in some order.

From fact 3, we know that there is at least one nickel and at least one dime in the green box. Since the third coin is determinate, we know that the label of the box is either "20 cents" or "25 cents". But since we have determined that the box labeled "20 cents" is either brown or black, the green box must be labeled "25 cents" and contain 20 cents.

Since the brown box cannot contain 20 cents, it must contain 15 cents and be labeled "20 cents", while the black box must be labeled "15 cents". This leaves only the white box's label: "30 cents".

The box containing 30 cents cannot be the white one, so it must be the black one, and the white box must contain 25 cents.

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