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Can you describe this route to infinity? Where will the arrows take you next?
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?
A game for 2 players
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.
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
To avoid losing think of another very well known game where the patterns of play are similar.
Build gnomons that are related to the Fibonacci sequence and try to explain why this is possible.
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
Try entering different sets of numbers in the number pyramids. How does the total at the top change?
A game for 2 players with similaritlies 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.
A collection of games on the NIM theme
An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
Problem solving is at the heart of the NRICH site. All the problems give learners opportunities to learn, develop or use mathematical concepts and skills. Read here for more information.
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.
Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
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.
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?
Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?
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.
Take a look at the multiplication square. The first eleven triangle numbers have been identified. Can you see a pattern? Does the pattern continue?
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 the vertices of a pentagon given the midpoints of its sides.
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
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?
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
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.
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?
Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?
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. . . .
Take any whole number between 1 and 999, add the squares of the digits to get a new number. Make some conjectures about what happens in general.
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.
Can you find the values at the vertices when you know the values on the edges?
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?
Can you explain the surprising results Jo found when she calculated the difference between square numbers?
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?
How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?
A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you. . . .
What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
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 × 1 [1/3]. What other numbers have the sum equal to the product and can this be so for. . . .
How many moves does it take to swap over some red and blue frogs? Do you have a method?
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. . . .
Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.
Can you find sets of sloping lines that enclose a square?
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.
The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
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. . . .
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?