List any 3 numbers. It is always possible to find a subset of adjacent numbers that add up to a multiple of 3. Can you explain why and prove it?
Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?
Can you explain the strategy for winning this game with any target?
Find some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48.
Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it?
Delight your friends with this cunning trick! Can you explain how it works?
A game for 2 players. Set out 16 counters in rows of 1,3,5 and 7. Players take turns to remove any number of counters from a row. The player left with the last counter looses.
The NRICH team are always looking for new ways to engage teachers and pupils in problem solving. Here we explain the thinking behind maths trails.
Four bags contain a large number of 1s, 3s, 5s and 7s. Pick any ten numbers from the bags above so that their total is 37.
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.
A game for 2 players with similaritlies to NIM. Place one counter on each spot on the games board. Players take it is turns to remove 1 or 2 adjacent counters. The winner picks up the last counter.
The triangle OMN has vertices on the axes with whole number co-ordinates. How many points with whole number coordinates are there on the hypotenuse MN?
Can you explain how this card trick works?
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
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?
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
The aim of the game is to slide the green square from the top right hand corner to the bottom left hand corner in the least number of moves.
A game for 2 players
A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.
Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?
This article for teachers describes several games, found on the site, all of which have a related structure that can be used to develop the skills of strategic planning.
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
Build gnomons that are related to the Fibonacci sequence and try to explain why this is possible.
An account of some magic squares and their properties and and how to construct them for yourself.
Problem solving is at the heart of the NRICH site. All the problems give learners opportunities to learn, develop or use mathematical concepts and skills. Read here for more information.
You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . .
An article which gives an account of some properties of magic squares.
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
Jo has three numbers which she adds together in pairs. When she does this she has three different totals: 11, 17 and 22 What are the three numbers Jo had to start with?”
What is the volume of the solid formed by rotating this right angled triangle about the hypotenuse?
Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?
These gnomons appear to have more than a passing connection with the Fibonacci sequence. This problem ask you to investigate some of these connections.
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable.
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
Can you find the values at the vertices when you know the values on the edges?
Consider all two digit numbers (10, 11, . . . ,99). In writing down all these numbers, which digits occur least often, and which occur most often ? What about three digit numbers, four digit numbers. . . .
We can show that (x + 1)² = x² + 2x + 1 by considering the area of an (x + 1) by (x + 1) square. Show in a similar way that (x + 2)² = x² + 4x + 4
How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?
Can you use the diagram to prove the AM-GM inequality?
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?
What would be the smallest number of moves needed to move a Knight from a chess set from one corner to the opposite corner of a 99 by 99 square board?
Can you explain the surprising results Jo found when she calculated the difference between square numbers?