Can you work out how to win this game of Nim? Does it matter if you go first or second?
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
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
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.
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?
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.
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.
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.
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
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?
Watch this film carefully. Can you find a general rule for
explaining when the dot will be this same distance from the
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?
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?
The NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
Find out what a "fault-free" rectangle is and try to make some of
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
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.
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?
For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?
A collection of games on the NIM theme
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?
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?
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
Are these statements relating to odd and even numbers always true, sometimes true or never true?
Find a route from the outside to the inside of this square, stepping on as many tiles as possible.
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
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
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
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?
Here are two kinds of spirals for you to explore. What do you notice?
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.
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.
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 challenge asks you to imagine a snake coiling on itself.
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?
How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?
It's easy to work out the areas of most squares that we meet, but
what if they were tilted?
These squares have been made from Cuisenaire rods. Can you describe
the pattern? What would the next square look like?
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.
Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?
Can you tangle yourself up and reach any fraction?
It would be nice to have a strategy for disentangling any tangled
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
This activity involves rounding four-digit numbers to the nearest thousand.
In this game for two players, the idea is to take it in turns to choose 1, 3, 5 or 7. The winner is the first to make the total 37.
Can you describe this route to infinity? Where will the arrows take you next?
It starts quite simple but great opportunities for number discoveries and patterns!