# Problems of the Week

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# 2018-12-03 Basic

A container with different-sized nuts, including large Brazil nuts, cashews, and peanuts, is packaged at a factory. When the nuts are first packaged, they are evenly distributed.

What will happen to the container of nuts over time if it is shaken?

Hint: Consider how cereal at the bottom of a bag compares to the cereal on top!

You yank on the end of the toilet paper roll fast enough to tear off a small length. Roll A has more paper on it than Roll B.

In which case is it easier to do this?

If I flip a fair coin two times, the probability of obtaining 1 head and 1 tail (in any order) is 50%.

If I flip the coin four times, is the probability of obtaining 2 heads and 2 tails (in any order) 50% as well?

An astronaut stuntman is on a unicycle outside of a space shuttle in Earth's orbit. He jumps from the space shuttle, and as he is falling in space, he starts pedaling.

What will happen next?

Assume that the astronaut uses their thighs to grip the seat.

Suppose you interconnect 6 nodes such that every node has an edge going to all the other nodes. Each edge is colored either red or blue (at random).

Call a "complete triangle" one formed by 3 nodes where all the edges are the same color.

Will there always be at least two complete triangles?

With this example, an all-red triangle and an all-blue triangle are marked.

Note: The triangles are allowed to both be the same color, and they are allowed to re-use the same edges as long as at least one of the edges is different.

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