In the third stage of the problem-solving process, learners are aiming for generalisation and possibly proof. (See the article Mastering Mathematics: The Challenge of Generalising and Proof.) Being able to generalise a situation involves identifying its underlying mathematical structure. Having transferred our thinking from one example to another, to
another, to another ..., the emerging similarities and differences offer us insight into what will always be true in that situation, which in turn can be explained by its underlying mathematical structure.

The tasks below provide opportunities for learners to get better at generalising.

The tasks below provide opportunities for learners to get better at generalising.

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### Largest Even

How would you create the largest possible two-digit even number from the digit I've given you and one of your choice?

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### Nim-7

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

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### Ring a Ring of Numbers

Choose four of the numbers from 1 to 9 to put in the squares so that the differences between joined squares are odd.

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### How Odd

This problem challenges you to find out how many odd numbers there are between pairs of numbers. Can you find a pair of numbers that has four odds between them?

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### Lots of Lollies

Frances and Rishi were given a bag of lollies. They shared them out evenly and had one left over. How many lollies could there have been in the bag?

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### Stop the Clock

This is a game for two players. Can you find out how to be the first to get to 12 o'clock?

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### Magic Vs

Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?

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### Three Dice

Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?

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### Got It

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.

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### Button-Up Some More

How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?

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### Six Numbered Cubes

This task combines spatial awareness with addition and multiplication.