Reasoning, convincing and proving

problem
Find the fake

Age

14 to 16

Challenge level

There are 12 identical looking coins, one of which is a fake. The counterfeit coin is of a different weight to the rest. What is the minimum number of weighings needed to locate the fake coin?
problem
Legs eleven

Age

11 to 14

Challenge level

Take any four digit number. Move the first digit to the end and move the rest along. Now add your two numbers. Did you get a multiple of 11?
problem
Marbles

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?
problem
Compare areas

Age

14 to 16

Challenge level

Which has the greatest area, a circle or a square, inscribed in an isosceles right angle triangle?
problem
Always the same

Age

11 to 14

Challenge level

Arrange the numbers 1 to 16 into a 4 by 4 array. Choose a number. Cross out the numbers on the same row and column. Repeat this process. Add up you four numbers. Why do they always add up to 34?
problem
Rule of three

Age

11 to 14

Challenge level

If it takes four men one day to build a wall, how long does it take 60,000 men to build a similar wall?
problem
No right angle here

Age

14 to 16

Challenge level

Prove that the internal angle bisectors of a triangle will never be perpendicular to each other.
problem
Always two

Age

14 to 18

Challenge level

Find all the triples of numbers a, b, c such that each one of them plus the product of the other two is always 2.
problem
Number sandwiches

Age

7 to 14

Challenge level

Can you arrange the digits 1, 1, 2, 2, 3 and 3 to make a Number Sandwich?
problem
2-digit square

Age

14 to 16

Challenge level

A 2-Digit number is squared. When this 2-digit number is reversed and squared, the difference between the squares is also a square. What is the 2-digit number?