Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
How could Penny, Tom and Matthew work out how many chocolates there
are in different sized boxes?
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.
Find out what a "fault-free" rectangle is and try to make some of
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 work out how to win this game of Nim? Does it matter if you go first or second?
What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
Square numbers can be represented as the sum of consecutive odd
numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
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?
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.
Use the interactivity to investigate what kinds of triangles can be
drawn on peg boards with different numbers of pegs.
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
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?
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?
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
Nim-7 game for an adult and child. Who will be the one to take the last counter?
Three circles have a maximum of six intersections with each other.
What is the maximum number of intersections that a hundred circles
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?
Do you notice anything about the solutions when you add and/or
subtract consecutive negative numbers?
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?
Think of a number, square it and subtract your starting number. Is
the number you’re left with odd or even? How do the images
help to explain this?
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?
What would you get if you continued this sequence of fraction sums?
1/2 + 2/1 =
2/3 + 3/2 =
3/4 + 4/3 =
Take any two positive numbers. Calculate the arithmetic and geometric means. Repeat the calculations to generate a sequence of arithmetic means and geometric means. Make a note of what happens to the. . . .
In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.
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 challenge asks you to imagine a snake coiling on itself.
Only one side of a two-slice toaster is working. What is the
quickest way to toast both sides of three slices of bread?
Imagine you have a large supply of 3kg and 8kg weights. How many of each weight would you need for the average (mean) of the weights to be 6kg? What other averages could you have?
This challenge focuses on finding the sum and difference of pairs of two-digit numbers.
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
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 winner is the player to take the last counter.
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10
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?
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?
Strike it Out game for an adult and child. Can you stop your partner from being able to go?
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.
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?
Got It game for an adult and child. How can you play so that you know you will always win?
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
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?
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.
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?
For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?
Find the sum of all three-digit numbers each of whose digits is
The NRICH team are always looking for new ways to engage teachers
and pupils in problem solving. Here we explain the thinking behind
Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.