13 Away

Task 59 ... Years 2 - 7

Summary

This is a game situation that has an underlying strategy waiting to be discovered. It is easy to state and easy to start, which is one feature that makes it suitable for younger as well as older students.

The starting point is a pile of 13 counters. The person who takes the last one loses. Players take turns to take either 1, 2 or 3 counters on any move.

The unwritten challenge is to find a way to always win.

This cameo includes two Investigation Guides.

 

Materials

  • More than 13 counters

Content

  • reasoning skills
  • patterns in number
  • generalisation
  • algebraic representation
13 Away

Iceberg

A task is the tip of a learning iceberg. There is always more to a task than is recorded on the card.
   

Students will probably play the game randomly to begin with. That's fine. A mathematician has to play with a problem to collect data about it. Gradually they start to organise the data and seek some form of pattern or connection. In this case the strategies of working backwards and trying every possible case lead to knowing how to win. Of course, you must assume that you are playing against the world's best opponent.

If I leave you facing one counter, I win.
If I leave you facing two counters, you win because you will take one and leave me to take the last one.
If I leave you facing three counters, you win because you will take two and leave me to take the last one.
If I leave you facing four counters, you win because you will take three and leave me to take the last one.
If I leave you facing five counters, I win because:
  • if you take 1 I will take 3 and leave you to take the last one.
  • if you take 2 I will take 2 and leave you to take the last one.
  • if you take 3 I will take 1 and leave you to take the last one.
So, if I make you face 1 counter or 5 counters I win.

Continuing this reasoning, try it...

...shows that I also win if I make you face 9 counters or 13 counters.
But the only way I can make you face 13 counters is to invite you to go first. Now I know how to win.

Thinking further about this outcome, 1, 5, 9, 13 make a number pattern with a difference of 4, and 4 is one more than the highest number of counters that could be taken on any move. So, what would happen if, as in the Challenge, there were 21 counters and players could take 1, 2, 3 or 4 on any move? Test your hypothesis.

Extensions

  • If I tell you any number of counters in a pile and that you can take 1, 2, ... up to any number I say, can you tell me how to play to win?
  • What happens if the aim is to take the last counter?
  • What happens if the counters you can take are 2, 3, ... up to any number I say, ie: you are not allowed to take one counter at any point?
Leanne Bateman & Margy Carracher chose to extend their Year 1 & 2 students working on this task with this Investigation Guide.
Paul Cecere, St. Matthew's, Page, used this Investigation Guide with his Year 5 & 6 students.

Whole Class Investigation

Tasks are an invitation for two students to work like a mathematician. Tasks can also be modified to become whole class investigations which model how a mathematician works.
   

Each pair needs a collection of counters. If you have Poly Plug they work very well. Begin the lesson by gathering the class around two volunteer students to whom you introduce the game. Then send pairs off to play the game, and, as information is discovered, record it on the whiteboard. Use the outline above to guide the investigation.

At this stage, 13 Away does not have a matching lesson on Maths300. However, the logic involved is almost identical to that Lesson 27, Game of 31, which is based on Task 86, Thirty-One. This lesson has software to support and extend it. The two investigations would blend well. The essential difference between the two is that in Game of 31 it is possible for a card value to run out before the game is ended. This leads to investigating ways to beat the winning strategy. In 13 Away, however, it is always the case that the specified number can be taken away, so the winning strategy cannot be challenged.

For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 27, Game of 31, which includes an Investigation Guide and is extended by companion software.

Visit 13 Away in Menu Maths Pack A.

Is it in Maths With Attitude?

Maths With Attitude is a set of hands-on learning kits available from Years 3-10 which structure the use of tasks and whole class investigations into a week by week planner.
   

13 Away is not in any MWA kit. However it can be used to enrich the Pattern & Algebra kit at Years 3/4 and Years 7/8.

Green Line
Follow this link to Task Centre Home page.