Reasoning, convincing and proving

problem
Growing garlic

Age

5 to 11

Challenge level

Ben and his mum are planting garlic. Can you find out how many cloves of garlic they might have had?
problem
Triangle numbers

Age

11 to 14

Challenge level

Take a look at the multiplication square. The first eleven triangle numbers have been identified. Can you see a pattern? Does the pattern continue?
problem
Rarity

Age

16 to 18

Challenge level

Show that it is rare for a ratio of ratios to be rational.
problem
Proof of Pick's theorem

Age

16 to 18

Challenge level

Follow the hints and prove Pick's Theorem.
problem
So many sums

Age

11 to 14

Challenge level

In this addition each letter stands for a different digit, with S standing for 3. What is the value of YxO?
problem
Old order

Age

11 to 14

Challenge level

Who is the youngest in this family?
problem
Birthday party

Age

11 to 14

Challenge level

The 30 students in a class have 25 different birthdays between them. What is the largest number that can share any birthday?
problem
Wallpaper

Age

5 to 7

Challenge level

These pieces of wallpaper need to be ordered from smallest to largest. Can you find a way to do it?
problem
Sizing them up

Age

5 to 7

Challenge level

Can you put these shapes in order of size? Start with the smallest.
problem
Trapezium four

Age

14 to 16

Challenge level

The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?