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### Adding All Nine

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Age 11 to 14

Challenge Level

In a league of 5 football teams which play in a round robin tournament (i.e. two teams play each other once only with no ties/draws allowed) show that it is possible for all five teams to be league leaders at the end of the tournament.

In a league of 6 hockey teams which play in a round robin tournament (i.e. two teams play each other once only with no ties/draws allowed) show that it is impossible for all six teams to be league leaders.

Make a set of numbers that use all the digits from 1 to 9, once and once only. Add them up. The result is divisible by 9. Add each of the digits in the new number. What is their sum? Now try some other possibilities for yourself!

15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?