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1 Generalising Gosford Hill May 2010

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2 Remainders of the Day (1) Write down a number which when you subtract 1 is divisible by 5 and another Write down one which you think no-one else here will write down. How can you check quickly whether someone’s example number is correct?

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3 Remainders of the Day (2) Write down a number which when you subtract 1 is divisible by 2 and when you subtract 1 from the quotient, the result is divisible by 3 and when you subtract 1 from that quotient the result is divisible by 4 Why must any such number be divisible by 3?

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4 Four Consecutives Write down four consecutive numbers and add them up and another Now be more extreme! What is the same, and what is different about your answers? + 1 + 2 + 3 + 6 4 Can 44 be the sum of four consecutive numbers?

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5 Up & Down Sums == 1 + 3 + 1 = 2x2 + 1= 2 2 + 1 2 == 1 + 3 + 5 + 3 + 1= 3x4 + 1= 3 2 + 2 2

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6 One Sum Diagrams 1 1 (1- ) 2 Anticipating, not waiting 1- 2

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7 Reading a Diagram x 3 + x(1–x) + (1-x) 3 x 2 + (1-x) 2 x 2 z + x(1-x) + (1-x) 2 (1-z)xz + (1-x)(1-z) xyz + (1-x)y + (1-x)(1-y)(1-z) yz + (1-x)(1-z)

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8 Which looks bigger … … Yellow Annulus or Light Green Circle?

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