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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.

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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.

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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. . . .

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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.

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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.

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

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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.

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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?”

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

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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?

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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?

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

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

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

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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. . . .

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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.

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.

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

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

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

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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.

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.

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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. . . .

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

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

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

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

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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?

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

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

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

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What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =

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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. . . .

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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

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

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

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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. . . .

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

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

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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. . . .

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

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

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

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

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

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It would be nice to have a strategy for disentangling any tangled ropes...

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