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Numbers and the Number System > Number theory

##### Age 11 to 14 Challenge Level:

I start with a red, a green and a blue marble. I can trade any of my marbles for two others, one of each colour. Can I end up with five more blue marbles than red after a number of such trades?

##### Age 11 to 14 Challenge Level:

I start with a red, a blue, a green and a yellow marble. I can
trade any of my marbles for three others, one of each colour. Can I
end up with exactly two marbles of each colour?

##### Age 11 to 14 Challenge Level:

A country has decided to have just two different coins, 3z and 5z
coins. Which totals can be made? Is there a largest total that
cannot be made? How do you know?

##### Age 11 to 14 Challenge Level:

What is the remainder when 2^{164}is divided by 7?

##### Age 11 to 14 Challenge Level:

Can you guarantee that, for any three numbers you choose, the
product of their differences will always be an even number?

##### Age 11 to 14 Challenge Level:

All strange numbers are prime. Every one digit prime number is
strange and a number of two or more digits is strange if and only
if so are the two numbers obtained from it by omitting either. . . .

##### Age 11 to 14 Challenge Level:

Charlie and Abi put a counter on 42. They wondered if they could visit all the other numbers on their 1-100 board, moving the counter using just these two operations: x2 and -5. What do you think?

##### Age 11 to 14 Challenge Level:

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