Filter by: Content type: ALL Problems Articles Games Stage: All Stage 1&2 Stage 2&3 Stage 3&4 Stage 4&5 Challenge level:
A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
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.
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.
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?
Find the sum of all three-digit numbers each of whose digits is odd.
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?
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?
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.
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?
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?
Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?
For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?
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?
What happens if you join every second point on this circle? How about every third point? Try with different steps and see if you can predict what will happen.
Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?
How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.
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.
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?
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. . . .
Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.
Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?
Charlie and Lynne put a counter on 42. They wondered if they could visit all the other numbers on their 1-100 board, moving the counter using just these two operations: x2 and -5. What do you think?
In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.
Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?
How many moves does it take to swap over some red and blue frogs? Do you have a method?
Can you find sets of sloping lines that enclose a square?
How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?
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.
Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.
A package contains a set of resources designed to develop pupils’ mathematical thinking. This package places a particular emphasis on “generalising” and is designed to meet the. . . .
Find out what a "fault-free" rectangle is and try to make some of your own.
An investigation that gives you the opportunity to make and justify predictions.
Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.
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. . . .
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 find an efficient method to work out how many handshakes there would be if hundreds of people met?
A collection of games on the NIM theme
Can you find the values at the vertices when you know the values on the edges?
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?
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
Can you work out how to win this game of Nim? Does it matter if you go first or second?
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?
Try adding together the dates of all the days in one week. Now multiply the first date by 7 and add 21. Can you explain what happens?
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.
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?
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. . . .