# Division Boxes

### Task 210 ... Years 4 - 10

#### Summary

This fascinating puzzle involves divisibility tests. However, it's not just about being presented with a number and asked if it is divisibility by X. The rules on the card require 3 divisibility tests each time. For a number to 'pass the test' all three tests must be true. There are many solutions, so success at the first level - 3 digit numbers - is easy. But what happens if we use 4 digit numbers and 4 corresponding divisibility tests? Or 5, or 6, or...? Now the investigation is opening up. Perhaps the most wonderful thing about the investigation is that there actually is a solution for 10 digit numbers!

#### Materials

• 10 tiles numbers 0 - 9
• Calculator

#### Content

• arithmetic, multiplication / division
• concept of proof
• divisibility tests
• division
• factors, multiples & primes
• mental arithmetic
• numbers, odd & even
• numbers, properties & laws
• tree diagrams #### Iceberg

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

This task starts simply but through the challenge of finding all the solutions for a given number of boxes it turns into an extended investigation. The main content knowledge is tests for divisibility. However, the investigation involves many aspects of the investigative process such as using different strategies and particular tools such as tree diagrams and the (possible) use of a computer simulation.

With just one box, the number of solutions is 9. The digit 0 being the exception.
For two boxes, the first digit has to be divisible by 1 and the 2-digit number divisible by 2. This produces the following 41 solutions - and some noticeable patterns to assist the search.
• 10, 12, 14, 16, 18
• 20, 24, 26, 28
• 30, 32, 34, 36, 38
• 40, 42, 46, 48
• 50, 52, 54, 56, 58
• 60, 62, 64, 68
• 70, 72, 74, 76, 78
• 80, 82, 84, 86
• 90, 92, 94, 96, 98
Including the third box means the first digit must be divisible by 1, the first two digits by 2 and the whole three digit number by 3.
Students may well know one test for divisibility by 3 is that the sum of the digits must add to a multiple of 3. As the problem proceeds, knowledge of divisibility tests for other numbers will be useful. They are listed below, but this need to know could be a good reason to visit the Web. Entering 'divisibility tests' into a search engine produces many useful results.

For the three box case, one strategy might be to start with the 41 solutions above and add a third digit so that the whole 3-digit number is divisible by '3'. For example:

• Starting with 10 the numbers would be 102, 105 and 108. (three solutions)
• Starting with 12, the numbers would be 120, 123, 126 and 129 (four solutions)
• Starting with 14, the numbers are 147 (only one solution)
• Starting with 16, the numbers are 162, 165, and 168 (three solutions and a similarity to the '10' group above)
• Starting with 18, the numbers are 180, 183, 186 and 189 (four solutions and a similarity to the '12' group above.
Hence when the first digit is '1' there are 3 + 4 + 1 + 3 + 4 = 15 solutions.

 This could be developed in a sequence of tree diagrams. 1st level ... is each of the 9 possible digits for the first box (taken one at a time). 2nd level ... has 5 branches to produce 2 digit numbers divisible by 2. 3rd level ... has different numbers of branches appropriate to the divisibility by 3 condition. A tree diagram is a systematic visual representation of the strategy of testing every possible combination, and may be more appropriate for learners with a visual intelligence preference. Continue, with 2 at the top of the tree:

• Starting with 20 the numbers are 201,204, 207.
• Starting with 24, the numbers are 240, 243, 246 and 249.
• Starting with 26, the numbers are 261, 264 and 267.
• Starting with 28, the only number is 285.
And hence a total of 11 solutions for the group starting with a 2. The group starting with 3 produces 13 solutions:
• 306, 309, 321, 324, 327, 342, 345, 348, 360, 369, 381, 384, and 387
This method can clearly be continued right through to the 10 digits. It does seem that the 'tree' might become impossibly large, but indeed after the fourth box the possibilities start to significantly reduce.

A major benefit for students of this systematic search is firstly seeing a viable problem solving method, but also realising the need to be very systematic and rigorous in checking each possibility.

#### Divisibility Tests

• 2: If the last digit is even, then number is divisible by 2.
• 3: If the sum of the digits is divisible by 3, then the number is also divisible by 3.
• 4: If the last two digits form a number divisible by 4, then the number is also divisible by 4.
• 5: If the last digit is a 5 or a 0, then the number is divisible by 5.
• 6: If the number is divisible by both 3 and 2, it is also divisible by 6.
• 7: Double the last digit and subtract it from the rest of the number, eg: for 301, double 1 and subtract it from 30. If the answer is divisible by 7 (including 0), then the number is too. You can keep on applying this rule until you do recognise a result which is, or is not, divisible by 7.
• 8: If the last three digits form a number divisible by 8, then the number is also divisible by 8.
• 9: If the sum of the digits is divisible by 9, then the number is also divisible by 9.
Many of the web sources also explain why these tests work. Seeking and reporting on this information would be a worthy project at this level. Also, Ask Dr. Math at Math Forum has an excellent alternative approach to solving Division Boxes.

#### 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.

Begin the class by asking pairs to tear one piece of paper in half. Then each person tears their half into five pieces. The ten pieces are then numbered from 1 - 10. You will also need 10 large cards numbered the same so you can start the investigation with a fish bowl demonstration on the floor or at a central table. Use the large cards to make 1, 2, 3 4, ... digit numbers as appropriate.

Perhaps starting by making 40 and asking Is this number divisible by 10? introduce a discussion about the meaning of divisibility and what the students already know about it. Record key points on the board and set a few exercises for students to create and record numbers divisible by 3.

Return to the fish bowl and introduce the challenge on the card. Write the rules on the board and invite students to record any solutions underneath them. After a while ask:

• How many solutions are there?
• How do we know when we have found them all?
Explore these questions, then introduce the additional challenges of:
• What happens if we use four boxes and divisibility tests up to 4?
• What happens if we use four boxes and divisibility tests up to 4?
• ...
• What happens if we use four boxes and divisibility tests up to 4?
• What happens if we allow digits to be repeated?
The unique solution for the 10 digit problem with no repeats is 3,816,547,290. This might start a further discussion about how to say and read large numbers.

For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 146, Division Boxes, which includes companion software that helps to explore the problem.

#### 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 Division Boxes task is an integral part of:

• MWA Number & Computation Years 9 & 10

The Division Boxes lesson is not included in any kit, but can be used to enrich Number & Computation kits in Years 5 & 6 and Years 9 & 10. 