# Sample Problems

True problem solving is as much about the journey as the destination. Math Circle participants are asked to challenge themselves and explain their reasoning as they ponder their way through dozens of problem sets, games, and activities—all composed by professional mathematicians at Stevens.

In a Stevens Math Circle, students are exposed to problems that touch upon a large variety of mathematical topics, including geometry, combinatorics, number theory, and logic. They play mathematical games—some single-player, some two-player, some collaborative—that are rich in strategy. And they engage in hands-on activities that allow them to explore a particular mathematical subject.

The following is a small collection of sample materials intended to give a sense of the questions that Math Circle participants are asked to explore.

- Take a look at the nine interlocking cogs shown here:

Is it possible for these cogs to rotate? Why or why not? What if there were ten cogs instead? - A
*Möbius strip*is a fascinating object obtained by taking a strip of paper, giving it a half-twist, and gluing its ends together:

What happens if you cut a Möbius strip along a line drawn halfway between its edges? What do you think happens if you cut it along a line drawn*one third*of the way between its edges? Can you answer these questions without actually cutting the Möbius strip? - In two years, my little brother will be twice as old as he was two years ago. In three years, my little sister will be three times as old as she was three years ago. Which of them is older?
- Place a coin on a tabletop, and place another coin of the same size next to it. Now roll the second coin around the first one without slipping. How many turns does the second coin make after it goes all the way around?

Be sure to conduct an experiment! How can you explain the result? - In the game
*Bridge It*, Math Circle participants take turns creating continuous, nonintersecting paths—one from left to right along the black dots, another from top to bottom along the white dots:

The first player to bridge opposite sides of the playing field wins. Is it possible for this game to end in a draw?