You have a box with n shoelaces in it. You randomly pick two ends and tie them together. You then repeat this process until there are no free ends left.

On average, how many loops will you have created?

How many laces are needed to get (just over) 2 loops on average?

This is a non-trivial extension to the previous blockbuster problem.

Use b blue, 3 green and 3 red blocks, to form a ring. If no reds are allowed to touch each other, and no greens are allowed allowed to touch each other, how many unique patterns can you form?

To keep things simple, rotated versions of a [...]

Under what circumstances is/isn’t C(n,r) divisible by n?

Assume that n and are r integers (include 0 and negatives).

—

C(n,r) = n! / ((n-r)! r!)

You might find it convenient to use e.g. a|b for “a divides b”

and a¦b for “a doesn’t divide b”

On my British keyboard, the | is next to the Z and

the ¦ is next [...]

Here’s a repost from many years ago. I have no idea how to do it or what the answer is.

A boy has four red blocks and eight blue blocks. He arranges the twelve blocks uniformly randomly, in a ring.

What is the probability that no two red blocks are next to each other?

As I strongly suspect [...]

A block of ice measuring 1 foot by 1 foot at the base is placed in a 2 foot by 3 foot aquarium. Water is then added to a level of 9 inches. Once the ice melts, the level of the water remains at 9 inches. How tall was the block of ice?

A 2 foot by 3 foot aquarium is filled with 9 inches of water. A 1 foot ice cube is placed in the corner. How much will the water rise once the ice melts?

A 2 foot by 3 foot aquarium is filled with 9 inches of water. (2) 1 foot metal cubes are placed in opposite corners. How much will the water rise?

A 2 foot by 3 foot aquarium is filled with 9 inches of water. A 1 foot metal cube is placed in the corner. How much will the water rise?