Problems

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.

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?

The date 21.02.2012 reads the same forwards and backwords (such numbers are called palindromes). Are there any more palindrome dates in the twenty first centuary?

Do there exist three natural numbers such that neither of them divide each other, but each number divides the product of the other two?

Find all the solutions of the puzzle and prove there are no others. Different letters denote different digits, while the same letters correspond to the same digits. \[M+MEEE=BOOO.\]

Place coins on a \(6\times 6\) chequered board (one coin on one square), so that all the horizontal lines contain different number of coins, and all vertical lines contain the same number of coins.