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Puzzles

21 December

There are ten ways to make a list of four As and Bs that don't contain an even* number of consecutive As:
How many ways are there to make a list of eleven As and Bs that don't contain an even number of consecutive As?
* We don't count 0 consecutive As as being an even number of consecutive As.

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2 December

Holly adds up the first six even numbers, then adds on half of the next even number. Her total is 49.
Next, Holly adds up the first \(n\) even numbers then adds on half of the next even number. This time, her total is 465124. What is \(n\)?

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15 December

There are 3 even numbers between 3 and 9.
What is the only odd number \(n\) such that there are \(n\) even numbers between \(n\) and 729?

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