### Brain Teasers

# Is it Fair

At the fairground, Ben was watching a cup and balls trick. The stall owner had labeled the cups A-D. There was a total of four balls under the cups. The owner made the following claims:

"There is 1 ball under cup A, zero or 1 ball under B, 2 balls under C, and zero or 2 balls under D."

In reality, no cup had more than 2 balls underneath and exactly three of his four claims were lies. If we know that cup A had no balls underneath, how many balls did each cup have?

"There is 1 ball under cup A, zero or 1 ball under B, 2 balls under C, and zero or 2 balls under D."

In reality, no cup had more than 2 balls underneath and exactly three of his four claims were lies. If we know that cup A had no balls underneath, how many balls did each cup have?

### Answer

A=0B=2

C=0

D=2

We know that claim 1 is a lie because A has no balls under it.

If claim 3 is true, then C has 2 balls under it, which would mean that either B or D would have to have 2 balls under them, or they would each have 1 ball. This means that if claim 3 is true, then either claim 2 or claim 4 would also be true. Also, if C had 1 ball, then B has 1 and D has 2 which makes claims 1 and 4 true, or B has 2 and D has 1 which makes all claims lies. As only one claim is true, then this means that claim 3 must be false and C must have 0 balls.

This means that both B and D must have 2 balls, which means that claim 2 is a lie and claim 4 is true.

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