# 1000.92 – Students Change Seats

In the History classroom there are 25 seats, arranged in 5 rows of 5 chairs. The teacher, Ms. Druid, announces that everyone in the class is to change seats, by moving one seat either to the right, to the left, to the front, or to the back. While it is a good thing for students to change their seats from time to time, one might wonder whether these complicated directions can actually be carried out. Your job is to decide, one way or the other, and then prove your answer. (No diagonal or wrap-around moves are allowed.)

Solution

Ms. Druid's directions are clear enough, but, alas, they can't be carried out. Think of the 25 seats as a small checkerboard. Notice that, as drawn, there are 13 black squares and 12 white ones. According to her directions, a seat on either color must move to the opposite color. Since there aren't equally many squares of each color, the plan can't work. That is, the students on the 13 black squares would have to go to 13 white squares, and there aren't enough.