Square numbers

problem
Seven square numbers

Age

7 to 11

Challenge level

Add the sum of the squares of four numbers between 10 and 20 to the sum of the squares of three numbers less than 6 to make the square of another, larger, number.
problem
Fitted

Age

7 to 11

Challenge level

Nine squares with side lengths 1, 4, 7, 8, 9, 10, 14, 15, and 18 cm can be fitted together to form a rectangle. What are the dimensions of the rectangle?
problem
Square subtraction

Age

7 to 11

Challenge level

Look at what happens when you take a number, square it and subtract your answer. What kind of number do you get? Can you prove it?
problem
Picturing square numbers

Age

11 to 14

Challenge level

Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
problem
Factors and multiples puzzle

Age

11 to 14

Challenge level

Using your knowledge of the properties of numbers, can you fill all the squares on the board?
problem
Sticky numbers

Age

11 to 14

Challenge level

Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?
problem
Four coloured lights

Age

11 to 14

Challenge level

Imagine a machine with four coloured lights which respond to different rules. Can you find the smallest possible number which will make all four colours light up?
problem
Generating triples

Age

14 to 16

Challenge level

Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?
problem
Odd differences

Age

14 to 16

Challenge level

The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.
problem
Summing squares

Age

14 to 16

Challenge level

Discover a way to sum square numbers by building cuboids from small cubes. Can you picture how the sequence will grow?