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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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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.

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

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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. . . .

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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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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. . . .

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

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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.

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

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

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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?

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

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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.

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

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

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

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

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

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

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Charlie and Abi 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?

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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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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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Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?

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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?

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I added together some of my neighbours' house numbers. Can you explain the patterns I noticed?

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

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

Dave Hewitt suggests that there might be more to mathematics than looking at numerical results, finding patterns and generalising.

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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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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?

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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.

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.

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

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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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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.

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

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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.

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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?