An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
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.
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.
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
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.
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 collection of games on the NIM theme
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.
Here are two kinds of spirals for you to explore. What do you notice?
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
Are these statements relating to odd and even numbers always true, sometimes true or never true?
Can you work out how to win this game of Nim? Does it matter if you go first or second?
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
Watch this video to see how to roll the dice. Now it's your turn! What do you notice about the dice numbers you have recorded?
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?
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.
Nim-7 game for an adult and child. Who will be the one to take the last counter?
Delight your friends with this cunning trick! Can you explain how
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
Find a route from the outside to the inside of this square, stepping on as many tiles as possible.
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
Can you explain how this card trick works?
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?
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.
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10
Use the interactivity to investigate what kinds of triangles can be
drawn on peg boards with different numbers of pegs.
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
In this problem we are looking at sets of parallel sticks that
cross each other. What is the least number of crossings you can
make? And the greatest?
Here are some arrangements of circles. How many circles would I need to make the next size up for each? Can you create your own arrangement and investigate the number of circles it needs?
What would you get if you continued this sequence of fraction sums?
1/2 + 2/1 =
2/3 + 3/2 =
3/4 + 4/3 =
Find out what a "fault-free" rectangle is and try to make some of
Can you explain the strategy for winning this game with any target?
One block is needed to make an up-and-down staircase, with one step up and one step down. How many blocks would be needed to build an up-and-down staircase with 5 steps up and 5 steps down?
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?
A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you. . . .
This challenge asks you to imagine a snake coiling on itself.
This activity involves rounding four-digit numbers to the nearest thousand.
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
Can you dissect an equilateral triangle into 6 smaller ones? What
number of smaller equilateral triangles is it NOT possible to
dissect a larger equilateral triangle into?
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?
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.
The NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
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?
Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?
Watch this film carefully. Can you find a general rule for
explaining when the dot will be this same distance from the
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?
Are these statements always true, sometimes true or never true?
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.