# 1000.54 – Swapping Places

The Ohio State University Crew Team is taking a rower’s holiday on the Scioto River. In one scull, 4 members of the Black team sit in the bow and 4 members of the Grey team sit in the stern. There is one empty seat in the middle. For the fun of it, the teams decide to switch places. The only way to switch seats without tipping the scull over is either to slide into an empty seat, fore or aft, or to climb over a single person into an empty seat (that is, essentially, to jump over the rower next to you).

How can the Black and the Grey teams arrange to switch?

Solution

Try a simpler problem first (often a good strategy). What if we had just 4 rowers in the canoe, 2 Black in front and 2 Grey in back? To keep track of everybody, we have numbered the rowers in the figure below and given the Greys odd numbers and the Blacks even numbers as in part A. Part B shows how the switch can be accomplished. Look at how the empty seat moves through the boat. In particular, note the symmetry between the the first moves and the last.

Now try the switch with 6 rowers, 3 Black and 3 Grey. It works really well to make a model. Draw the outline of the canoe on a piece of paper, and then use pennies and dimes for the rowers. Once this is solved with 6 rowers, everyone will believe you can do it with 8.