Bottleneck

In this puzzle we invite you to explore how repeated random events can lead to predictable outcomes.

For this puzzle you will need:

six cups
a pile of 100 or so paper clips (or coins, or match sticks, or any objects that you have a lot of)
a six-sided die

Put ten clips in each cup and the extras in a pile. The idea is to move clips from one cup to the next according to the following rules:

Start with cup #1. Roll the die and see what number you get. Move that many clips from cup #1 to cup #2.
Roll the die again and see what number comes up. Move that number of clips from cup #2 to cup #3.
Continue this way down the line. When you get to cup #6 move the clips to the pile of extras, since there’s no 7th cup.
Go back to the first cup and start again with step 1 above.

One more rule: Cup #1 never runs out of clips. If you need more for a move to cup #2 than you have in cup #1, just get what you need from the pile of extras.

What happens after you’ve done this for a while? Start again and go through the process a few times to see if you get the same general results.

Here are some variants to try:

Use more than six cups or fewer than six.
Put more than ten clips in each cup or fewer than ten. Try starting with
zero clips in each cup.

Why did we call this puzzle “Bottleneck”?

Course:

Math [1]

Result/Solution(s)

We found that after a while one of the cups would end up with zero clips. This could happen to any of the cups except the first one. Then, the number of clips in the next cup in the sequence would drop because there were still clips going out but none coming in from the previous, now empty, cup.

This simulates a production line in a manufacturing process, where random fluctuations at various stages can cause a drop in throughput further down the line. Hence, the name "Bottleneck" for this puzzle.

We ran a few computer simulations of 1,000 rolls of the die and found that even when the number of clips in one cup drops to zero, the process recovers and the overall throughput is pretty much the same regardless of fluctuations along the way.

This puzzle has ended up being more complicated than we originally thought, and we’re still working on it. We’ll post more results in the future. In the meantime, why don’t you try it and tell us what you’ve found?

Math Puzzle [2]