During the third hour after midnight the hands on a clock point in
the same direction (so one hand is over the top of the other). At
what time, to the nearest second, does this happen?
Is it possible to rearrange the numbers 1,2......12 around a clock
face in such a way that every two numbers in adjacent positions
differ by any of 3, 4 or 5 hours?
Find the sum of all three-digit numbers each of whose digits is
A lady has a steel rod and a wooden pole and she knows the length
of each. How can she measure out an 8 unit piece of pole?
Which times on a digital clock have a line of symmetry? Which look
the same upside-down? You might like to try this investigation and
What happens when you add three numbers together? Will your answer be odd or even? How do you know?
Investigate the different distances of these car journeys and find
out how long they take.
These two group activities use mathematical reasoning - one is
numerical, one geometric.
What do you notice about the date 03.06.09? Or 08.01.09? This
challenge invites you to investigate some interesting dates
Crosses can be drawn on number grids of various sizes. What do you notice when you add opposite ends?
Arrange eight of the numbers between 1 and 9 in the Polo Square
below so that each side adds to the same total.
Put the numbers 1, 2, 3, 4, 5, 6 into the squares so that the
numbers on each circle add up to the same amount. Can you find the
rule for giving another set of six numbers?
Place the numbers 1 to 10 in the circles so that each number is the
difference between the two numbers just below it.
There are 44 people coming to a dinner party. There are 15 square
tables that seat 4 people. Find a way to seat the 44 people using
all 15 tables, with no empty places.
Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?
Katie had a pack of 20 cards numbered from 1 to 20. She arranged
the cards into 6 unequal piles where each pile added to the same
total. What was the total and how could this be done?
Bernard Bagnall recommends some primary school problems which use
numbers from the environment around us, from clocks to house
Can you put the numbers 1 to 8 into the circles so that the four
calculations are correct?
Place this "worm" on the 100 square and find the total of the four
squares it covers. Keeping its head in the same place, what other
totals can you make?
This addition sum uses all ten digits 0, 1, 2...9 exactly once.
Find the sum and show that the one you give is the only
In the following sum the letters A, B, C, D, E and F stand for six
distinct digits. Find all the ways of replacing the letters with
digits so that the arithmetic is correct.
What do the digits in the number fifteen add up to? How many other
numbers have digits with the same total but no zeros?
Here is a chance to play a fractions version of the classic
Three children are going to buy some plants for their birthdays. They will plant them within circular paths. How could they do this?
Find out why these matrices are magic. Can you work out how they were made? Can you make your own Magic Matrix?
Three dice are placed in a row. Find a way to turn each one so that the three numbers on top of the dice total the same as the three numbers on the front of the dice. Can you find all the ways to do. . . .
There are 78 prisoners in a square cell block of twelve cells. The
clever prison warder arranged them so there were 25 along each wall
of the prison block. How did he do it?
Place six toy ladybirds into the box so that there are two ladybirds in every column and every row.
Exactly 195 digits have been used to number the pages in a book.
How many pages does the book have?
Replace each letter with a digit to make this addition correct.
Using the statements, can you work out how many of each type of
rabbit there are in these pens?
We start with one yellow cube and build around it to make a 3x3x3 cube with red cubes. Then we build around that red cube with blue cubes and so on. How many cubes of each colour have we used?
Can you make square numbers by adding two prime numbers together?
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.
Ben has five coins in his pocket. How much money might he have?
This task, written for the National Young Mathematicians' Award 2016, invites you to explore the different combinations of scores that you might get on these dart boards.
This problem is based on a code using two different prime numbers
less than 10. You'll need to multiply them together and shift the
alphabet forwards by the result. Can you decipher the code?
There are 4 jugs which hold 9 litres, 7 litres, 4 litres and 2
litres. Find a way to pour 9 litres of drink from one jug to
another until you are left with exactly 3 litres in three of the
An environment which simulates working with Cuisenaire rods.
A group of children are using measuring cylinders but they lose the labels. Can you help relabel them?
This task, written for the National Young Mathematicians' Award 2016, focuses on 'open squares'. What would the next five open squares look like?
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?
Ram divided 15 pennies among four small bags. He could then pay any sum of money from 1p to 15p without opening any bag. How many pennies did Ram put in each bag?
Use your logical reasoning to work out how many cows and how many
sheep there are in each field.
Tim had nine cards each with a different number from 1 to 9 on it. How could he have put them into three piles so that the total in each pile was 15?
Who said that adding couldn't be fun?
Number problems at primary level that require careful consideration.
Try adding together the dates of all the days in one week. Now
multiply the first date by 7 and add 21. Can you explain what
Can you substitute numbers for the letters in these sums?
This Sudoku, based on differences. Using the one clue number can you find the solution?