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?
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.
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 we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
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.
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
How many pairs of numbers can you find that add up to a multiple of
11? Do you notice anything interesting about your results?
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. . . .
Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?
Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?
Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?
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?
For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?
Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
How many moves does it take to swap over some red and blue frogs? Do you have a method?
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
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?
Think of a number, add one, double it, take away 3, add the number
you first thought of, add 7, divide by 3 and take away the number
you first thought of. You should now be left with 2. How do I. . . .
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.
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. . . .
Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.
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.
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
Start with any number of counters in any number of piles. 2 players
take it in turns to remove any number of counters from a single
pile. The winner is the player to take the last counter.
A package contains a set of resources designed to develop
pupils’ mathematical thinking. This package places a
particular emphasis on “generalising” and is designed
to meet the. . . .
Charlie has moved between countries and the average income of both
has increased. How can this be so?
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
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
It starts quite simple but great opportunities for number discoveries and patterns!
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
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
Can you find the values at the vertices when you know the values on
A collection of games on the NIM theme
Start with any number of counters in any number of piles. 2 players take it in turns to remove any number of counters from a single pile. The loser is the player who takes the last counter.
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
Can you tangle yourself up and reach any fraction?
It would be nice to have a strategy for disentangling any tangled
A game for 2 players
Can you find sets of sloping lines that enclose a square?
The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
Take a look at the multiplication square. The first eleven triangle
numbers have been identified. Can you see a pattern? Does the
To avoid losing think of another very well known game where the
patterns of play are similar.
A red square and a blue square overlap so that the corner of the red square rests on the centre of the blue square. Show that, whatever the orientation of the red square, it covers a quarter of the. . . .
Take any two digit number, for example 58. What do you have to do to reverse the order of the digits? Can you find a rule for reversing the order of digits for any two digit number?
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 NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
Build gnomons that are related to the Fibonacci sequence and try to
explain why this is possible.
Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten.
Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it. . . .