Counter Escape

Task 6 ... Years 4 - 8

Summary

We have three 'boxes' (A, B, C) and 3 counters. Any number of counters (up to 3) can be placed in any of the boxes. They are removed from a box by the roll of a dice according to the rules:

If the roll is 1, take a counter from A.
If the roll is 2 or 3, take a counter from B.
If the roll is 4, 5, 6, take a counter from C.

Problem: What is the best way to place the counters to remove them all in the least number of rolls?

This cameo has a From The Classroom section which offers very senior students an explanation for calculating the probabilities in the problem.

Materials

6 counters in two colours - 3 for each player
1 dice

Content

probability - experiencing chance
expressing chance as a fraction
combination theory
problem solving - testing all possible combinations
expected number of trials to reach a success
statistical inference

Iceberg

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

The challenge of finding the best placement strategy can be tackled at several levels. At the less sophisticated level students might choose a placement strategy (eg: AAA) and carry out say 20 experiments to get an idea of the number of trials it would take to remove all the counters. The next step might be to compare this strategy to a different one (eg: AAB) with another 20 trials.

At a more sophisticated level students could try to detail all the possible placement strategies (an application of combination theory) and carry out a set of trials for each one. The next step might be to use the probabilities of rolling:

1
2 or 3
4, 5 or 6

to calculate the expected number of rolls to remove all the counters for each strategy.

Once this problem has been explored, What happens if...? questions generate an even deeper investigation:

What happens if we change the number of boxes?
What happens if we change the number of counters?
What happens if we change the dice rules?

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.
Think of the three boxes as Intergalactic Transportation Tubes and represent them on the classroom floor with three large size hula hoops from the physical education department. Identify the tubes with these labels. Select three volunteers to be the Intergalactic Good Guys who have to save the world. Explain that they must race to another galaxy through the tubes to turn off the End Of The World machine before they are captured by the Intergalactic Bad Guys.
The IGGs have to decide which tubes to use given:

Any number of IGGs can be in any tube.

The computer rolls a dice to decide which tube will transport the next person.

Only one person can be transported on each roll.

The aim for the IGGs is to place themselves so that all three are transported in as few rolls as possible. The other students become the computer and take turns to roll the dice.
Record the chosen IGG strategy, for example (ABB or CCC) and run a trial. Discuss whether the number of rolls would be the same if this strategy was trialed again. Do it once more. Ask students to vote on what they think is the most likely number of rolls for this strategy. Check hypotheses by asking each pair to carry out one more trial of the strategy by sketching three tubes and using Poly Plug, or counters, as IGGs. Collect and discuss results using statistics appropriate to the age and experience of the students.
By now students will be wanting to try other strategies, but before they do, challenge them to answer the questions:

How many strategies are there to test?

How do we know when we have found them all?

Trying every possible case gives: 3 in a tube (AAA, BBB, CCC) ... 2 in a tube (AAB, AAC, ABB, BBC, ACC, BCC) ... 1 in a tube (ABC).
Students now choose, or have assigned, a strategy to test. They must run enough trials of their strategy to be able to answer the question:

The most likely number of rolls for my strategy to work is...?

Continue the investigation to explore the best strategy and extend with the What happens if...? questions above as appropriate.
An alternative form of whole class investigation is for each pair to tackle the task at a work station over a week and add to a class display of the outcomes. Each new pair must:

refer to the class record to see if there is a placement strategy still to be tested,

carry out a given number of trials on this new strategy, or,

add a given number of trials to the data for a strategy already tested.

Over time this approach provides whole class first-hand data which becomes the basis for class discussion and analysis and the starting point for the What happens if...? questions.
For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 88, Counter Escape, which also includes an Investigation Guide and companion software.
Visit Counter Escape in Menu Maths Pack C.

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.
The Counter Escape task is an integral part of:

MWA Chance & Measurement Years 3 & 4

MWA Chance & Measurement Years 7 & 8

The Counter Escape lesson is an integral part of:

MWA Chance & Measurement Years 7 & 8

From The Classroom

Högskolan Malmö
Sweden

Barbro Söderberg
Teachers in Training

Barbro's class of teachers in training were exploring Counter Escape as a whole class investigation with another teacher. They had decided on their best strategy and used the Maths300 software to run many trials to check. (Yes, this Swedish teacher training institution is a member of Maths300.) The software showed that the average number of rolls for an AAA strategy was 18·15. This made sense to the students because we can expect 6 rolls to remove each counter in box A, so a theoretical average of 18.

The teacher then commented that the probability theory needed to evaluate the expected number of rolls for most other strategies was too complex for the students they would teach, but at least the common sense result for AAA indicated that the software was doing its job correctly.

Barbro loves probability and the comment set her thinking. This contribution may certainly be for senior high school or university students, but it also certainly reinforces our principle that each task is the tip of an iceberg.

Thanks are extended to Charles Lovitt for assistance with interpreting Barbro's original work which was in Swedish.

Theoretical Analysis of the Expected Value for AAB

The key to the problem is realising that at any stage there are only two options to consider and analyse - whether the counter is drawn from either Cell A or Cell B.

First Step

The first counter can be from A or B.
Pr(A) = ¹/₆ ... Pr(B) = ²/₆ ... so ... Pr(A or B) = ³/₆ = ¹/₂.

Therefore...
Expected Number of Rolls to remove one of A or B (1st Step) = 2.

Second Step

Now you are facing...

Outcome 1 ... OR ...
Outcome 2

To get to Outcome 1 from Step 1 only rolls of 1, 2, or 3 matter. It is twice as likely that 2 or 3 will appear, so:

Pr(Outcome 1) = ²/₃

Then 6 more rolls can be expected to remove the next counter from A.

Therefore, the Expected Number of Rolls to reach the next step = ²/₃ x 6

To get to Outcome 2 from Step 1 only rolls of 1, 2, or 3 matter. It is twice as likely that 2 or 3 will appear, so:

Pr(Outcome 2) = ¹/₃

Then Pr(A or B) = ³/₆ = ¹/₂, so 2 rolls can be expected to remove one of A or B.

Therefore, the Expected Number of Rolls to reach the next step = ¹/₃ x 2

Combining these possibilities gives...
Expected Rolls (2nd Step) = ²/₃ x 6 + ¹/₃ x 2

Third Step

Now you are facing...

How did you get here?
... OR ...
How did you get here?

Pr (Remove B then A) = ⁶/₉ Pr (Remove A then B) = ²/₉

...and Expected Number of Rolls to remove final A = 6

Combining these possibilities gives:
Expected Number of Rolls to reach the next step (all removed) = ⁸/₉ x 6

Pr (Remove A then A) = ¹/₉

...and Expected Number of Rolls to remove B = 3

Therefore:
Expected Number of Rolls to reach the next step (all removed) = ¹/₉ x 3