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Seven from Nine

Age 11 to 14 Short
Challenge Level

Answer: 12 ways

Choosing 7 numbers is the same as choosing which 2 numbers to leave out
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9
                      = (1 + 9) + (2 + 8) + (3 + 7) + (4 + 6) + 5 = 45

45 is a multiple of 3
$\therefore$ the two left out add up to a multiple of 3

3,6   3,9   6,9   are the ways of pairing two multiples of 3

1,2   1,5   1,8   are the ways of pairing 1 with a number 1 less than a multiple of 3

4,2   4,5   4,8
7,2   7,5   7,8   are the ways of pairing the other numbers which are 1 more than
                       a multiple of 3 with a number 1 less than a multiple of 3



This problem is taken from the UKMT Mathematical Challenges.
You can find more short problems, arranged by curriculum topic, in our short problems collection.