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

Delight your friends with this cunning trick! Can you explain how it works?

Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.

Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten. Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it. . . .

Consider all two digit numbers (10, 11, . . . ,99). In writing down all these numbers, which digits occur least often, and which occur most often ? What about three digit numbers, four digit numbers. . . .

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

Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?

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

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 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 happens when you round these numbers to the nearest whole number?

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?

What happens when you round these three-digit numbers to the nearest 100?

Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?

This activity involves rounding four-digit numbers to the nearest thousand.

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?

Take any two digit number, for example 58. What do you have to do to reverse the order of the digits? Can you find a rule for reversing the order of digits for any two digit number?

Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?

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.

Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?

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

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?

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.

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.

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

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?

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.

Ben’s class were making cutting up number tracks. First they cut them into twos and added up the numbers on each piece. What patterns could they see?

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

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?

How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?

How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?

Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.

What can you say about these shapes? This problem challenges you to create shapes with different areas and perimeters.

Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?

Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.

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

This challenge encourages you to explore dividing a three-digit number by a single-digit number.

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 focuses on finding the sum and difference of pairs of two-digit numbers.

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.

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

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

How many centimetres of rope will I need to make another mat just like the one I have here?

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?