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An arithmagon is a shape with numbers and a rule attached to it. What are the rules? Aim Our aim is to investigate arithmagon solutions. We will look for patterns and relationships in the numbers. This will help us to answer two questions. What are those questions? Using the square arithmagons blank sheet, find a solution for: E = 12 F = 9 G = 4 H = 1 What is your solution? Can you find a different one? How many solutions are there altogether? Record your results in a table: Given E = 12 Found: A F=9 B G=4 C D H=1 Now try these: 1. E = 1 2. E = 7 3. E = 6 4. E = 17 5. E = 8 F=7 F=3 F=8 F = 15 F=6 G=4 G = 13 G = 11 G = 13 G=9 H = 10 H=9 H = 13 H = 11 H=5 In each case, try to make sure that you have all the possible solutions Key questions Have you got a system for working out your solution? Are there any rules about how many solutions there will be? What other patterns or rules have you discovered? Are there any sets of numbers (E, F, G, H) that do not work. If so, why do you think this is? Tip 1 Look at the number of solutions for any one square arithmagon. Compare it with the smallest number in the squares. How does it compare? Is this always true? Tip 2 Add up the numbers opposite one another in the squares. Eg E + H and F +G What do you notice? Tip 3 Add up E + F + G + H Add up A + B + C + D What do you notice? Tip 4 Why does problem 6 not work? Can you find another combination which does not work? Analysis of Results • How can you predict the number of solutions there will be? • How do you know if an arithmagon is solvable? • Is there a relationship between the numbers in squares and circles? • Consider patterns of odd and even numbers, opposite numbers, diagonally opposite numbers.