SharpBrains

A quick teaser: Imagine you are one of 120 people in a room. Each person in the room is given two lengths of rope and told to chose two of his or her four rope-ends at random and to tie them together. Then each person is told to tie the remaining two rope-ends together.

Then, we count up the loops of rope. How many should there be?

SOLUTION:

When each person prepares to choose his second rope-end, we note that one of the available three rope-ends is the other end of the rope he is holding, and the other two are from the other length of rope.

He is equally likely to pick any of these three rope-ends, so there is a one-in-three chance that he will create a loop at this time, and a two-in-three chance that he will instead simply join two ropes into one. He'll be left with one rope (plus possibly a loop). Either way, he'll tie his final rope into a loop in the second step.

Therefore, we expect one third of the people (40) to have made two loops, and the remaining two-thirds (80) to have made one, for a total of (40 x 2 + 80 = ) 160 loops.

-- Wes Carroll is the head of Do The Math private tutoring services, Puzzle Master for the Ask A Scientist lecture series, and much more. Find out more at wescarroll.com.