Reasoning, convincing and proving

problem
Repetitiously

Age

14 to 16

Challenge level

Can you express every recurring decimal as a fraction?
problem
Power quady

Age

16 to 18

Challenge level

Find all real solutions of the equation (x^2-7x+11)^(x^2-11x+30) = 1.
problem
Postage

Age

14 to 16

Challenge level

The country Sixtania prints postage stamps with only three values 6 lucres, 10 lucres and 15 lucres (where the currency is in lucres).Which values cannot be made up with combinations of these postage stamps? Prove that all other values can be made up.
problem
Magic W wrap up

Age

16 to 18

Challenge level

Prove that you cannot form a Magic W with a total of 12 or less or with a with a total of 18 or more.
problem
Halving

Age

5 to 7

Challenge level

These pictures show squares split into halves. Can you find other ways?
problem
Remainders

Age

7 to 14

Challenge level

I'm thinking of a number. My number is both a multiple of 5 and a multiple of 6. What could my number be?
problem
Classifying solids using angle deficiency

Age

11 to 16

Challenge level

Toni Beardon has chosen this article introducing a rich area for practical exploration and discovery in 3D geometry
problem
Got it

Age

7 to 14

Challenge level

A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.
problem
Sealed solution

Age

7 to 11

Challenge level

Ten cards are put into five envelopes so that there are two cards in each envelope. The sum of the numbers inside it is written on each envelope. What numbers could be inside the envelopes?
problem
Think of two numbers

Age

11 to 14

Challenge level

Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?