Challenge Level

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. . . .

Challenge Level

Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.

Challenge Level

A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?

Challenge Level

Investigate the different ways that fifteen schools could have given money in a charity fundraiser.

Challenge Level

Is there an efficient way to work out how many factors a large number has?

Challenge Level

You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . .

Challenge Level

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

Challenge Level

Imagine we have four bags containing numbers from a sequence. What numbers can we make now?

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

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. . . .

Challenge Level

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

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

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

Challenge Level

First of all, pick the number of times a week that you would like to eat chocolate. Multiply this number by 2...

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

Think of a number, add one, double it, take away 3, add the number you first thought of, add 7, divide by 3 and take away the number you first thought of. You should now be left with 2. How do I. . . .

Challenge Level

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

Challenge Level

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

Challenge Level

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

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

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 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.

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

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

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

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

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

Can you explain the surprising results Jo found when she calculated the difference between square numbers?

Challenge Level

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?

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

Great Granddad is very proud of his telegram from the Queen congratulating him on his hundredth birthday and he has friends who are even older than he is... When was he born?

Challenge Level

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

Challenge Level

Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?

Challenge Level

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

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

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

Challenge Level

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

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

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

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

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

Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?

Challenge Level

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

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?