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?
Investigate the different distances of these car journeys and find
out how long they take.
Mr. Sunshine tells the children they will have 2 hours of homework.
After several calculations, Harry says he hasn't got time to do
this homework. Can you see where his reasoning is wrong?
This article for teachers suggests ideas for activities built around 10 and 2010.
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?
If you have only four weights, where could you place them in order
to balance this equaliser?
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
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?
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?
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.
How have the numbers been placed in this Carroll diagram? Which
labels would you put on each row and column?
How could you put eight beanbags in the hoops so that there are
four in the blue hoop, five in the red and six in the yellow? Can
you find all the ways of doing this?
Lolla bought a balloon at the circus. She gave the clown six coins
to pay for it. What could Lolla have paid for the balloon?
This article gives you a few ideas for understanding the Got It! game and how you might find a winning strategy.
Using the statements, can you work out how many of each type of
rabbit there are in these pens?
These two group activities use mathematical reasoning - one is
numerical, one geometric.
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10
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 Sudoku, based on differences. Using the one clue number can you find the solution?
Place the numbers 1 to 10 in the circles so that each number is the
difference between the two numbers just below it.
Write the numbers up to 64 in an interesting way so that the shape they make at the end is interesting, different, more exciting ... than just a square.
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?
Can you put the numbers 1 to 8 into the circles so that the four
calculations are correct?
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
Crosses can be drawn on number grids of various sizes. What do you
notice when you add opposite ends?
Three children are going to buy some plants for their birthdays. They will plant them within circular paths. How could they do this?
Exactly 195 digits have been used to number the pages in a book.
How many pages does the book have?
Look carefully at the numbers. What do you notice? Can you make
another square using the numbers 1 to 16, that displays the same
Bernard Bagnall recommends some primary school problems which use
numbers from the environment around us, from clocks to house
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?
Replace each letter with a digit to make this addition correct.
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 make a cycle of pairs that add to make a square number
using all the numbers in the box below, once and once only?
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?
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?
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?
Arrange eight of the numbers between 1 and 9 in the Polo Square
below so that each side adds to the same total.
Different combinations of the weights available allow you to make different totals. Which totals can you make?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
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?
Start by putting one million (1 000 000) into the display of your
calculator. Can you reduce this to 7 using just the 7 key and add,
subtract, multiply, divide and equals as many times as you like?
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?
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.
Use your logical-thinking skills to deduce how much Dan's crisps and ice-cream cost altogether.
Can you substitute numbers for the letters in these sums?
Place the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. How many different ways can you do this?