# Problems to Ponder (July edition)

Welcome to this month’s edition of Problems to Ponder! Pose them in your classroom as a challenge or try them out yourself. Have an interesting solution? Send it to thevariable@smts.ca for publication in a future issue of The Variable, our monthly periodical.

The sixth cent

You toss a fair coin 6 times, and I toss a fair coin 5 times. What is the probability that you get more heads than I do?

Adapted from Barbeau, E. J., Klamkin, M. S., & Moser, W. O. J. (1995). Five hundred mathematical challenges. USA: The Mathematical Association of America.

Dueling dice

Consider the following four dice, which have the following numbers on their faces:

• Red : 0, 1, 7, 8, 8, 9
• Blue: 5, 5, 6, 6, 7, 7
• Green: 1, 2, 3, 9, 10, 11
• Black: 3, 4, 4, 5, 11, 12

The dice are used to play the following game for two people. Player 1 chooses a die, then Player 2 chooses a die. Then, each player rolls their die. The player with the highest number showing gets a point. The first player to get 7 points wins the game. If you are Player 1, which die should you choose? If you are Player 2, which die should you choose?

Two too many dice

Suppose you have a clear, sealed cube containing three smaller, indistinguishable six-sided dice. How can you use this three-in-one die to simulate a single, six-sided die? (Bonus: How can you use the three-in-one die to simulate two six-sided dice?)