Problem of the Week

Good morning from the Center of Math! To start off your week, we have a Problem of the Week that we've posted to Facebook, Twitter, and Google+. I learned about this statistics/logic problem from our summer intern, Or. We've decided to share it with you this week!

Click on any image in the post to expand.

There are several different ways to approach this problem. The method I chose was...
Pretty much brute force.

I organized a few charts, as seen below. The chart in the top left represents the possible strategies that Alice and Beatrice can use. I listed out all seven strategies- the first is that Alice and Beatrice both always guess "heads." The second requires Alice to always guess "heads," while Beatrice always guesses "tails." In the fifth strategy, Alice and Beatrice will both guess the same result that they tossed. In the sixth, the girls both guess the opposite of their result. And the final strategy (VII) is that Alice will guess the same result as she tossed, while Beatrice guesses the opposite.

In the chart on the top right, I've listed a through d, all of the possible results of the coin toss. Alice and Beatrice will both get heads, one will get tails and the other will get heads, or both girls will get tails.

The chart on the bottom is the important one. I've taken the theoretical results of the toin coss, a through d, and examined each of the strategies I through VII and how they would stand up to the results. The only strategy which worked on all four possible outcomes is strategy VII. Alice and Beatrice should use this strategy to ensure a win every time they play!

This is just one of a multitude of strategies that can bring you to this answer. Tweet us a picture of your solution, or reply with it on our Facebook/Google+ posts, or send us a link in the blog comments! I'd like to host a solution by one of our viewers right here on the blog.

-Tori