16 May 2013

A Better Game With Four Lockers: Upper Secondary Mathematics Competition Question

Let’s return to Mrs Pixey’s class and the game she has been playing with her hapless students. Today is Thursday, and they have been playing this game for the past three days. So far, the students have won just 4 points, with Mrs Pixey having accumulated 8 points. The situation seems hopeless and the students feel that this is a very unfair game. During lunch, the four students sit together to discuss whether they should give up and forget about the prizes on offer. However, Debbie has a bright idea! Before we look at her plan, let’s remind ourselves of the game.

Each of the four students – Alex, Brenda, Chris and Debbie – has a school locker with their name on it. At the end of the day, Mrs Pixey places their homework books, one in each locker, in such a way that every book is inside the wrong student’s locker. The task of the four students is to each pick a locker, open it and see if they have found their own book. They all do this at the same time, having decided in advance who should open which locker. Each student who finds their own homework book gets 1 point, otherwise Mrs Pixey gets 1 point. At the end of the week, if the sum of the points gained by the students is greater than that won by their teacher, each student will receive a small prize.

Now, back to Debbie’s bright idea! Her suggestion is that they should change the rules. Instead of them all picking their chosen locker at the start, they should open the lockers one by one. After each locker is opened, the students can decide who should open the next locker and which one they should pick. Debbie tries to explain the mathematics to her three friends; they look a bit bemused but decide that it is better than losing every day.

So that afternoon, Debbie explains to Mrs Pixey that the game as it stands is not fair and that the students would like to change the rules. Mrs Pixey smiles; my students learn quickly, she thinks to herself. She agrees to Debbie’s new rules and they play their fourth game.

Is Debbie right? Calculate the expected number of points that the students can win in this new game?

Remember, Mrs Pixey places one book in each of four lockers in such a way that every book is in the wrong locker. The first student opens one of the lockers and takes the homework book out. If the book belongs to that student, they keep it and get 1 point. If it belongs to another student, the book is given to its owner and Mrs Pixey gets 1 point. The students then decide who should be next and which locker to open. The game continues until every student has his or her own homework book.

One other thing, the correct calculation depends on having the correct strategy, so feel free to discuss Debbie’s game plan in the comments below before tackling the probabilities. As the number of lockers is small, you may also wish to actually play the game yourself and see what works... and what doesn’t!

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