Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?
Can you find six numbers to go in the Daisy from which you can make all the numbers from 1 to a number bigger than 25?
How many solutions can you find to this sum? Each of the different letters stands for a different number.
Different combinations of the weights available allow you to make different totals. Which totals can you make?
A game for 2 people. Take turns placing a counter on the star. You win when you have completed a line of 3 in your colour.
Starting with four different triangles, imagine you have an
unlimited number of each type. How many different tetrahedra can
you make? Convince us you have found them all.
This cube has ink on each face which leaves marks on paper as it is rolled. Can you work out what is on each face and the route it has taken?
Five numbers added together in pairs produce: 0, 2, 4, 4, 6, 8, 9, 11, 13, 15 What are the five numbers?
Have a go at this well-known challenge. Can you swap the frogs and toads in as few slides and jumps as possible?
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?
Can you coach your rowing eight to win?
Can you put the numbers 1 to 8 into the circles so that the four
calculations are correct?
Place six toy ladybirds into the box so that there are two ladybirds in every column and every row.
Mr McGregor has a magic potting shed. Overnight, the number of
plants in it doubles. He'd like to put the same number of plants in
each of three gardens, planting one garden each day. Can he do it?
How many different symmetrical shapes can you make by shading triangles or squares?
A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle
contains 20 squares. What size rectangle(s) contain(s) exactly 100
squares? Can you find them all?
Place the numbers 1 to 10 in the circles so that each number is the
difference between the two numbers just below it.
If you are given the mean, median and mode of five positive whole numbers, can you find the numbers?
Make your own double-sided magic square. But can you complete both
sides once you've made the pieces?
Place the numbers 1 to 8 in the circles so that no consecutive
numbers are joined by a line.
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?
Four numbers on an intersection that need to be placed in the
surrounding cells. That is all you need to know to solve this
The letters of the word ABACUS have been arranged in the shape of a
triangle. How many different ways can you find to read the word
ABACUS from this triangular pattern?
This magic square has operations written in it, to make it into a
maze. Start wherever you like, go through every cell and go out a
total of 15!
There were chews for 2p, mini eggs for 3p, Chocko bars for 5p and
lollypops for 7p in the sweet shop. What could each of the children
buy with their money?
Suppose we allow ourselves to use three numbers less than 10 and
multiply them together. How many different products can you find?
How do you know you've got them all?
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?
An extra constraint means this Sudoku requires you to think in
diagonals as well as horizontal and vertical lines and boxes of
You have two egg timers. One takes 4 minutes exactly to empty and
the other takes 7 minutes. What times in whole minutes can you
measure and how?
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?
This problem is based on the story of the Pied Piper of Hamelin. Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!
What do you notice about the date 03.06.09? Or 08.01.09? This
challenge invites you to investigate some interesting dates
Ben has five coins in his pocket. How much money might he have?
This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?
Choose four different digits from 1-9 and put one in each box so that the resulting four two-digit numbers add to a total of 100.
Can you find all the different triangles on these peg boards, and
find their angles?
Charlie and Abi put a counter on 42. They wondered if they could visit all the other numbers on their 1-100 board, moving the counter using just these two operations: x2 and -5. What do you think?
A dog is looking for a good place to bury his bone. Can you work
out where he started and ended in each case? What possible routes
could he have taken?
Find the product of the numbers on the routes from A to B. Which
route has the smallest product? Which the largest?
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.
Two sudokus in one. Challenge yourself to make the necessary
A few extra challenges set by some young NRICH members.
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
Bellringers have a special way to write down the patterns they
ring. Learn about these patterns and draw some of your own.
Find out what a "fault-free" rectangle is and try to make some of
This second Sudoku article discusses "Corresponding Sudokus" which are pairs of Sudokus with terms that can be matched using a substitution rule.
Can you put the numbers from 1 to 15 on the circles so that no
consecutive numbers lie anywhere along a continuous straight line?
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?
Find a cuboid (with edges of integer values) that has a surface
area of exactly 100 square units. Is there more than one? Can you
find them all?
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