Algebra Middle School

My “Fake World” Task

Dan Meyer’s recent post on “fake world” math tasks has me thinking about many of the openers and games I have used in my classroom.  I have written about The Take-Away Game before, and I still use it often…until the kids learn how to beat me and the strategy is revealed.  This next one is not so much a game, but more of a task, similar in some ways to the Locker Problem.


In this task, chairs are placed in a circle.  Chairs will be removed from the circle using the following rules:

  • Chair #1 is removed first.
  • The next remaining chair is skipped, and the next chair removed.
  • This continues, with chairs skipped and removed until only one chair remains.
  • Once a chair is removed, it is “out” of the circle
  • Whoever is sitting in the last remaining chair “wins”

Here’s a brief Doceri video which shows some game playings:

Like the “Take-Away Game”, I can’t recall where I first encountered this problem.  They have both been sitting in my files for over a dozen years.  If anyone can name a source, I’d be happy to award some credit.

Why I enjoy this problem:

  1. It’s not intimidating.  We have a chance to draw, get out blocks, magnets….whatever we want to use to model the problem.  Great for working in teams.
  2. I can let the problem marinate.  On one day, I may ask the class “Where should I sit if there are 8 chairs?”, and come back the next day with “How about 24 chairs?”  If it seems like discussion is flowing, I can put my foot on the gas.
  3. I can use this problem with all levels of students.  If we need to create a data table and look for a pattern as a class, that can happen.  If my honors kids want to fly with it, that can also happen.
  4. The answer is not obvious, but a clear pattern eventually emerges if you model enough circles.  And there will be some nice vocabulary opportunities as the payoff.

There are a number of ways to express the solution.  Later this week, I will post the “answer”.  Until then, have fun moving around the furniture.


By Bob Lochel

HS Math Teacher. Hatboro-Horsham School District, Horsham, PA.

8 replies on “My “Fake World” Task”

Strange coincidence with the problem I’d just put down when I saw your post.

From Ch 10.4 of Art of Problem Solving’s “Intermediate Counting and Probability” book. It says that it was sourced from the AIME, but I don’t know the year the problem appeared on the AIME:

“I have 2048 letters numbered 1 to 2048. I have to address ever single one. Originally they’re stacked in order with #1 on top. To make the task a bit less mind-numbing, I address ever other one starting with #1. When I address a letter, I put it in my outbox. The ones I skip I stack as I skip them (so #2 is on the bottom of the stack after my first pass). After I finish my first pass, I have 1024 letters which are not addressed; #2048 is on top, #2 is on the bottom. I them repeat my procedure over and over until there’s only one letter left. What number is that letter?”

I wouldn’t be surprised if it was part of an old AIME problem, as I have used them as a basis for many of my tasks. Will have to check my files tomorrow to see when I first wrote up this problem. Not sure I was aware of the AIME 15 years ago.

This problem is a rewording of problem #9 from the 1996 AIME. The idea behind this problem, though, has been left unchanged. Here is problem #9 from the 1996 AIME:
A bored student walks down a hall that contains a row of closed lockers, numbered 1 to 1024. He opens the locker numbered 1, and then alternates between skipping and opening each locker thereafter. When he reaches the end of the hall, the student turns around and starts back. He opens the first closed locker he encounters, and then alternates between skipping and opening each closed locker thereafter. The student continues wandering back and forth in this manner until every locker is open. What is the number of the last locker he opens?

I have been looking for problems with a context for recognizing patterns, generating sequences, and then generalizing. I think this is tremendous fun, and I could see how this could be developed over time as you have unfolded it as a warm up activity, and have students attempting the generalize over a period of time. Great classroom action.

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