Challenge

Task 56 ... Years 2 - 8

Summary

Blocks are placed on the board so that the difference between the blocks on either end of any line is not 1.

Can you find one solution?
Can you find more than one solution?
What happens if...?

Materials

Blocks numbered from 1 to 8
Playing board

Content

difference between two numbers
spatial perception
reasoning skills

Iceberg

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

There are many answers to this problem, so it's surprising how time consuming it can be to find the first. It's very common to think a solution has been found only to discover that one of the lines is a difference of 1. This can be quite frustrating when it happens more than once. When and if to intervene has to be considered.

One approach might be to ask students what they are learning about the problem from their trials. They might notice for instance that 1 and 8 each only have to be 'scared of' one other number. All the other numbers have to be 'scared of' two others. They might notice that some spaces are connected by 2 lines and some by 4. The more connections there are, the more 'dangerous' the spaces are. Ask them to record such discoveries in their journals and keep trying.

Another approach might be to suggest 'Break the problem into smaller parts' and combine that with 'Try every possible case'. One way to do this is to suggest something like:

Have you thought of placing the smallest number in the left box and then working out the possibilities?

Doing so immediately limits what can be placed in the top and bottom of the left column of three and could lead to a solution such as:

	5	3
1	7	8	6
	4	2

So, what happens if we try each of the numbers in turn in the left box? Can a solution be found in each case?

	6	8
2	4	3	5
	7	1

	1	4
3	7	8	6
	5	2

	7	5
4	1	3	2
	6	8

	3	1
5	7	8	6
	2	4

	4	7
6	1	2	3
	8	5

	4	6
7	1	8	2
	3	5

	3	1
8	6	5	7
	2	4

Could there be more solutions in each case? Studying the solutions so far indicates there can be. For example, consider the solutions for 2 and 4 above. Reflecting the solution for 4 in an imaginary line down the middle gives:

	5	7
2	3	1	4
	8	6

which is not the same as the previous solution for 2.

So, now the problem has been broken into smaller parts by choosing the left box number, and in each case these questions can be asked:

How many solutions are there?
How do we know when we have found them all?

Examining the data above could also lead to students realising that whatever number is placed in the left box, it's immediate neighbours are (or in the cases of 1 and 8, it's only neighbour is) in the right half of the puzzle. Therefore a way of proceeding might be to first put a number in the left box, then work through all the possible ways its neighbours can be placed in the opposite half of the puzzle.

Trouble is that in the reflected solution of 4 above, the new solution for 2 does not have both its neighbours in the right half. The hypothesis needs refining to 'at least one of its neighbours in the right half of the puzzle'.

This approach is likely to produce more solutions, but even if all the possible ones from this approach were found, that would not mean all solutions had been found. It would be necessary to prove that once a number had been decided for the left box, at least one of its neighbours must be in the right half of the puzzle.

Finding all the solutions is a legitimate challenge, but it isn't the aim of the task. The aim is that students continue to develop their reasoning, their ability to express it and their journalising of it.

We still don't know how many solutions there are, but you are invited to have fun finding as many as you can. We would be happy to publish any found by your students. Here's one we found to start that collection.

	7	1
3	5	4	6
	8	2

and its reflection:

	1	7
6	4	5	3
	2	8

which is not the same as the solution for 6 above.

Recognising that a mathematician is never finished with a problem could also lead to these additional challenges.

What happens if we change the set of numbers?
What happens if we change the difference that is not allowed?
What happens if we change the design of the playing board?

You might also invite students to explore what happens if the rule is changed to the difference must be one and explain what they discover.

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.
To convert this task to a whole class lesson you will need copies of the playing board in the photograph, which is easy to make with a drawing package, or the drawing tools within a word processor. Instead of the blocks you can make a set of numbers to match the squares on your board using the table tools of the word processor. Students can quickly cut up the sheet you give them into the numbers they need. Alternatively, folding and tearing scrap paper will produce number tiles more quickly. However, it should be said that if you can use blocks, their three dimensionality makes it easier to pick them up and move them around.
The focus for the lesson would be on finding one solution; then another and another; gradually shifting to discussing strategies for finding more and applying them to build data. There are so many solutions it is likely in a whole class situation that most, perhaps all pairs could find one. Certainly that challenge would be implied if, when solutions were recorded on the board they were labelled with the initials of the pair who discovered them.
Journal recording would be built in and modelled by the teacher as an essential part of the work of a mathematician. The search for all solutions would be introduced as a higher level challenge but not necessarily followed through.
The problem could be introduced in several ways:

With a printed page, as if from a textbook, and an expectation to use pencil and paper because nothing else is provided.

Similar approach but save the printing by making one slide to display.

Either of those approaches but supplying or creating sets of numbers 1 - 8 for each pair as described above.

Tucking the task version of the problem (card and materials as in the photo above) under your arm and inviting the class to join you at a central table where a particular pair of students help you begin its investigation in a 'fish bowl', before adjourning to their own table to continue with materials or pencil and paper as they choose.

Preparing A4 sheets, each with a number from 1 - 8, and preparing an enlargement of the puzzle board on the floor with, for example, sheets of newspaper and masking tape, or chalked on the basketball court outside. Number sheets are handed out to 8 students who sit randomly on the pieces of newspaper with their number obvious. Others gather around; differences are explored; the requirement of the puzzle is introduced; 'sitters' are moved around to seek a solution as advised by the crowd; the first solution is recorded on the board and journalised; pairs begin a search for more with concrete materials or pen and paper as they choose.

Each of these is a legitimate pedagogical choice. As teachers it is our responsibility to choose one likely to fascinate, captivate and absorb the most learners. Which would be your choice. Why?
At this stage, Challenge does not have a matching lesson on Maths300.

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 Challenge task is an integral part of:

MWA Number & Computation Years 3 & 4

This task is also included in the Task Centre Kit for Aboriginal Students.

Follow this link to Task Centre Home page.