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.
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.
How many different journeys could you make if you were going to visit four stations in this network? How about if there were five stations? Can you predict the number of journeys for seven stations?
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.
Take a look at the multiplication square. The first eleven triangle
numbers have been identified. Can you see a pattern? Does the
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.
How many pairs of numbers can you find that add up to a multiple of
11? Do you notice anything interesting about your results?
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
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?
Imagine an infinitely large sheet of square dotty paper on which you can draw triangles of any size you wish (providing each vertex is on a dot). What areas is it/is it not possible to draw?
How many moves does it take to swap over some red and blue frogs? Do you have a method?
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
In how many ways can you arrange three dice side by side on a
surface so that the sum of the numbers on each of the four faces
(top, bottom, front and back) is equal?
Imagine a large cube made from small red cubes being dropped into a
pot of yellow paint. How many of the small cubes will have yellow
paint on their faces?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
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?
Three circles have a maximum of six intersections with each other.
What is the maximum number of intersections that a hundred circles
Draw a square. A second square of the same size slides around the
first always maintaining contact and keeping the same orientation.
How far does the dot travel?
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?
Can you tangle yourself up and reach any fraction?
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?
It would be nice to have a strategy for disentangling any tangled
Can you find the values at the vertices when you know the values on
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
Pick a square within a multiplication square and add the numbers on
each diagonal. What do you notice?
What happens when you round these numbers to the nearest whole number?
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?
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?
A collection of games on the NIM theme
This challenge focuses on finding the sum and difference of pairs of two-digit numbers.
Find the sum and difference between a pair of two-digit numbers. Now find the sum and difference between the sum and difference! What happens?
This challenge encourages you to explore dividing a three-digit number by a single-digit number.
Nim-7 game for an adult and child. Who will be the one to take the last counter?
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.
Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?
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 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.
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
What happens when you round these three-digit numbers to the nearest 100?
Charlie has moved between countries and the average income of both
has increased. How can this be so?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
It starts quite simple but great opportunities for number discoveries and patterns!
Can you find an efficient method to work out how many handshakes
there would be if hundreds of people met?
This challenge asks you to imagine a snake coiling on itself.
A country has decided to have just two different coins, 3z and 5z
coins. Which totals can be made? Is there a largest total that
cannot be made? How do you know?
Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?
Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.