by Bill Graham

(Imagine yourself back in eighth grade.) One morning your math teacher has invited many of the students including you to a room for a pizza brunch (or so he said). When you arrive in this room, the door is locked behind you and you discover the terrible truth. All of these students have not been responsible about doing homework. The teacher begins to describe the horrible fate which awaits them. There is a large, circular table with chairs all around, just enough for all the students. He plans to put a dunce's hat on the kid in chair #1 and the Assistant Principal in charge of Severe Discipline will place the hazed kid on math suspension for the rest of the year by providing an hour's detention after school doing long division problems every day. The world's meanest teacher then skips the kid in chair #2 and selects #3 who is also put on math suspension. This process continues, skipping one student, selecting the next all the way around the table. When he gets back to the beginning of the circular table he continues in the same fashion with the remaining students. Finally, there is only one student left. That student gets an A+ and no homework for the rest of the year.

The world's meanest teacher tells everyone to get seated. You count the number of kids in the room and have to quickly decide where to sit so you won't get chosen and will be the one left with the A+ and no homework. How can you come up with a rule so you will know where to sit no matter how many kids there are? Suppose there are 12 kids, where will you sit? Fifty-three kids? One hundred thirty kids? Good luck!

Hint:

Start with a small number of students, make an organized list and look for a pattern. If there are two students, chair #2 is the place to be. For three students, also chair #2. If there are four kids, it’s chair #4. For five students, where should you sit? Keep going and look for a pattern.