Arithmagons 2

Task 194 ... Years 4 - 8

Summary

Using only the digits 0 to 20, numbers are placed at the corners of a square and the lines joining them are assigned a value that is the sum of the connected corners. If you know the line totals can you work out the numbers at each corner?

How many solutions are there?
How do you know when you have found them all?
What happens if you don't know all four line totals?

Arithmagons 2 is a partner to Arithmagons 1 which involves similar reasoning using a triangle as the shape. It also describes a significant reasoning contribution from one Year 7 class.

Materials

Counters numbered 0 - 20

Content

addition
algebra, generalisation in words & symbols
arithmetic, addition / subtraction
concept of proof
flow charts
mental arithmetic
reasoning
recording mathematics

Iceberg

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

Tasks exist to support a Working Mathematically curriculum - a curriculum with the core focus of students learning to work like a mathematician. Therefore, although this task involves mental arithmetic practise and the opportunity to generalise, its main focus is on the questioning, reasoning, justifying and communication skills that are the work of a mathematician. In fact, this task can be viewed as an extension of Task 188, Arithmagons 1, through the question What happens if we change the starting shape?.

Students won't have too much trouble finding solutions to questions 1 (a) - (d) since guess, check and improve can be used. So the implied challenge in these first questions is to find a process that will always produce the answers and be able to explain that process to someone else. There are several ways to think through this process. One is explained here, but it should be compared to how 'real' Year 7 students reasoned in the triangle shape case as explained in From The Classroom in Task 188.

Solving Arithmagons

Start with the biggest line total.
Make all the pairs of counters that equal this total and arrange them in order.
Start with the largest number in the first pair. Place it in the corner of the chosen line so that it is connected to the second largest line total.
Put the other counter from the pair in the circle at the other end of the largest line total.
Ask: Can this pair work?
If it does you will be able to find a counter for the third corner and all the line totals will be correct.
Record the solution if there is one.
Whether or not it works, remove the extra counters and swap the pair of counters to opposite ends of the highest line total.
Ask: Can this pair work?
If it does you will be able to find a counter for the third corner and all the line totals will be correct.
Record the solution if there is one.
Repeat this process with the next pair of numbers and continue until all pairs have been checked.

This system will work for the Challenge questions too. Try it for yourself with the puzzles on the card. You should find, for example, these solutions:

1(a) 11 1 1(b) 7 1 6 2 Challenge 1 11 0 10 1

20 12 3 13 4 12 2 7 3 6

1(a) has at least one more solution and 1(b) has several more than the two shown. In Challenge 1, seven pairs of numbers can make 13, the largest line total - (13, 0), (12, 1), (11, 2), (10, 3), (9, 4), (8, 5), (7, 6). The first two can be shown to give no solutions. The missing line total for (11, 2) is 9 and the missing line total for (10, 3) is also 9. There are other solutions for Challenge 1. Will the missing line total also be 9 for those solutions?

In these, and all other cases, checking for all solutions, requires explaining why other possibilities don't work. This is just as important to a mathematician as knowing why things do work.

If your students do create their own 'rules' - and them doing so is certainly the hope of the task - you might still introduce this process and ask them to compare it to their own. Then for either or both you have the opportunity to introduce the concept of a flow chart to explain this process in diagram form. The option is also there, especially if a flow chart works, to write a computer program to solve Arithmagon puzzles.

However, observant students might see another way of solving for the missing side in the Challenges. The clue is in examining differences between line totals in 1(a) to 1(b). In particular, examine the line totals on adjacent (not opposite) pairs of sides.

Extensions

The general picture for a square arithmagon is:

Which means that:

... A + B = t_ab
... B + D = t_bd
... C + A = t_ac
... D + C = t_cd

Subtracting Equation 2 from Equation 1, for example, gives:
(A + B) - (B + D) = t_ab - t_bd
which implies that:
A - D = t_ab - t_bd

So, the difference between the two opposite corner numbers is the same as the difference between the two totals that lead from those corners to the corner eliminated from the equation.

This understanding can provide another way to solve an arithmagon puzzle.
In 1(a), t_ab = 12 and t_bd = 13.
The difference between these values is 1, which means the difference between A and D is also 1.

So, to find A and D we need two numbers that have a difference of 1 and can join with B to equal both line totals. The first possibility is 13 and 12 which will only work if D = 13 and B = 0. Then, C = 19 and we have a new solution to 1(a), which is:

1(a) 12 0

19 13

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.
If you already have sets of tiles or counters numbered from 0 to 20, then all you need to do is design a square frame to fit them and make copies. If not, then each person folds a piece of A4 paper into horizontal quarters and tears it into strips. Then each strips is folded into thirds and torn. Now each person has twelve pieces, so partners can number them 0 - 20 and have a few spares. One set between two is all you need.
The lesson could start by one person in each pair secretly placing four corner numbers, then calculating the line totals and giving just those totals to their partner with the challenge: Can you work out my corner numbers?. This will lead you into a statement like:
A mathematician might be interested in finding a way to always be able to work out the corner numbers if you know the line totals. Our challenge today is to try to find a way to do that. So, keep exploring Arithmagon puzzles and when you discover something, let me know so we can share it.
The notes above will guide the lesson further. For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 63, Arithmagons, which also includes an Investigation Guide with answers and discussion. You might also find the Calculating Changes threaded activity Number Shapes to be a useful companion to Arithmagons.

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 Arithmagons 2 task is an integral part of:

MWA Number & Computation Years 7 & 8

The Arithmagons lesson is an integral part of:

MWA Pattern & Algebra Years 5 & 6

MWA Number & Computation Years 7 & 8

Follow this link to Task Centre Home page.