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Counting Factors

Is there an efficient way to work out how many factors a large number has?

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Summing Consecutive Numbers

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

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Helen's Conjecture

Helen made the conjecture that "every multiple of six has more factors than the two numbers either side of it". Is this conjecture true?

Chameleons

Age 11 to 14 Challenge Level:

On a certain island there are 12 green, 15 brown and 18 yellow chameleons. Whenever two chameleons of different colours meet they always change colour to the third colour (e.g. a brown and a yellow would both change to green when they met). This is the only time they change colour. Describe the shortest sequence of meetings in which all the chameleons change to green.

Now suppose there are 13 green, 15 brown and 17 yellow chameleons and they change colour in exactly the same circumstances. Is it possible now for all the chameleons eventually to be the same colour?