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Calendar Capers

Choose any three by three square of dates on a calendar page...

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Adding All Nine

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!

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Rotating Triangle

What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?

Tower of Hanoi

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

Look at the sequence below:
$1, 2, 4, 8, 16...$

Can you describe how to get from one term to the next?

Can you describe the $n^{th}$ term of the sequence?

Now try adding together terms from the sequence:
$1 + 2$
$1 + 2 + 4$
$1 + 2 + 4 + 8$
Do you notice anything interesting?

Can you predict what $1 + 2 + 4 + ... + 64 + 128$ would be? Check to see if you are right.

How could you write the answer to $1 + 2 + 4 + ... + 2^n$?
Justify why your formula works.

Hint for building block B: Think about moving all the discs except the largest one onto the middle peg first.

Hint for Extension question: Current theory and observations suggest that between 13.5 and 14 billion years have elapsed since the Big Bang.