Problems

You have a 3-quart bucket, a 5-quart bucket, and an infinite supply of water. How can you measure out exactly 4 quarts?

If you are on a boat and toss a suitcase overboard, will the water level rise or fall?

Suppose you had eight billiard balls, the recruiter began. One of them is slightly heavier, but the only way to tell is by put-ting it on a scale against the others. What’s the fewest number of times you’d have to use the scale to find the heavier ball?

How many times a day do a clock’s hands overlap?

How do you cut a rectangular cake into two equal pieces when someone has already removed a rectangular piece from it? The removed piece can be of any size or orientation. You are allowed just one straight cut.

You have 26 constants, labeled \(A\) through \(Z\). Let \(A\) equal 1. The other constants have values equal to the letter’s position in the alphabet, raised to the power of the previous constant. That means that \(B\) (the second letter) = \(2^A=2^1= 2\), \(C = 3^B=3^2= 9\), and so on. Find the exact numerical value for this expression: \[(X-A)(X-B)(X-C)\dots (X-Y)(X-Z).\]

Why does a mirror reverse right and left instead of up and down?

Remove a \(1 \times 1\) square from the corner of a \(4 \times 4\) square. Can this shape be dissected into \(3\) congruent parts?

Max asked Emily how old she was. She replied that she was 13 years old the day before yesterday, and will be 16 next year. Then, Max asked her brother, whether it was true, and he said yes. How is it possible if nobody was lying?

Ten little circles are drawn on a squared board \(4\times4\).

Cut the board into identical parts in such a way that each part contains 1, 2, 3, and 4 drawn circles correspondingly.