1 Generalising Gosford Hill May 2010

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?

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?

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? Can 44 be the sum of four consecutive numbers?

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

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

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)

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