Challenge Level

Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?

Challenge Level

Can you find the values at the vertices when you know the values on the edges?

Challenge Level

Can you work out how to win this game of Nim? Does it matter if you go first or second?

Challenge Level

To avoid losing think of another very well known game where the patterns of play are similar.

The NRICH team are always looking for new ways to engage teachers and pupils in problem solving. Here we explain the thinking behind maths trails.

Challenge Level

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.

Challenge Level

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.

Challenge Level

Janine noticed, while studying some cube numbers, that if you take three consecutive whole numbers and multiply them together and then add the middle number of the three, you get the middle number. . . .

Challenge Level

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.

Challenge Level

This task encourages you to investigate the number of edging pieces and panes in different sized windows.

An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.

Challenge Level

Investigate sequences given by $a_n = \frac{1+a_{n-1}}{a_{n-2}}$ for different choices of the first two terms. Make a conjecture about the behaviour of these sequences. Can you prove your conjecture?

Challenge Level

Make some loops out of regular hexagons. What rules can you discover?

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.

Challenge Level

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. . . .

Challenge Level

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?

Challenge Level

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.

Challenge Level

A game for 2 players with similarities 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.

Challenge Level

Can you explain the strategy for winning this game with any target?

Challenge Level

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 moves.

Challenge Level

The Egyptians expressed all fractions as the sum of different unit fractions. Here is a chance to explore how they could have written different fractions.

Challenge Level

Nim-7 game for an adult and child. Who will be the one to take the last counter?

Challenge Level

Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?

Challenge Level

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?

Challenge Level

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?

Challenge Level

Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?

Challenge Level

Can all unit fractions be written as the sum of two unit fractions?

Challenge Level

Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?

Challenge Level

Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?

Challenge Level

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.

Challenge Level

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

Challenge Level

Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?

Challenge Level

The sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 ï¿½ 1 [1/3]. What other numbers have the sum equal to the product and can this be. . . .

Challenge Level

It starts quite simple but great opportunities for number discoveries and patterns!

Challenge Level

Got It game for an adult and child. How can you play so that you know you will always win?

Challenge Level

Can you find sets of sloping lines that enclose a square?

Challenge Level

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?

Challenge Level

Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?

Challenge Level

Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?

Challenge Level

How many moves does it take to swap over some red and blue frogs? Do you have a method?

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.

Challenge Level

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.

Challenge Level

Try entering different sets of numbers in the number pyramids. How does the total at the top change?

Challenge Level

How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?

Challenge Level

Jo has three numbers which she adds together in pairs. When she does this she has three different totals: 11, 17 and 22 What are the three numbers Jo had to start with?”

Challenge Level

Can you see how to build a harmonic triangle? Can you work out the next two rows?

Challenge Level

Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?