Number Patterns and Infinity

Mathematicians seek and find patterns in a variety of situations.

The Number Patterns and Infinity pathway on wild.maths.org invites students to explain patterns, explore their underlying structure, and make sense of infinity...

The collection of related NRICH tasks below are ideal for teachers who want to promote creativity in the classroom. They are designed for classroom use, with accompanying Teachers' Notes and Resources.

problem

Summing consecutive numbers

Age

11 to 14

Challenge level

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?

problem

Frogs

Age

11 to 14

Challenge level

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

problem

Number pyramids

Age

11 to 14

Challenge level

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

problem

Crossed ends

Age

11 to 14

Challenge level

Crosses can be drawn on number grids of various sizes. What do you notice when you add opposite ends?

problem

Odds, evens and more evens

Age

11 to 14

Challenge level

Alison, Bernard and Charlie have been exploring sequences of odd and even numbers, which raise some intriguing questions...

problem

Magic letters

Age

11 to 14

Challenge level

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

problem

Where can we visit?

Age

11 to 14

Challenge level

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?

problem

Marbles in a box

Age

11 to 16

Challenge level

How many winning lines can you make in a three-dimensional version of noughts and crosses?

problem

Route to infinity

Age

11 to 14

Challenge level

Can you describe this route to infinity? Where will the arrows take you next?

problem

Triangle numbers

Age

11 to 14

Challenge level

Take a look at the multiplication square. The first eleven triangle numbers have been identified. Can you see a pattern? Does the pattern continue?

problem

Tower of Hanoi

Age

11 to 14

Challenge level

The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.

problem

Always a multiple?

Age

11 to 14

Challenge level

Think of a two digit number, reverse the digits, and add the numbers together. Something special happens...

problem

Differences

Age

11 to 14

Challenge level

Can you guarantee that, for any three numbers you choose, the product of their differences will always be an even number?

problem

Pair products

Age

14 to 16

Challenge level

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

problem

Steel cables

Age

14 to 16

Challenge level

Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?

problem

Attractive tablecloths

Age

14 to 16

Challenge level

Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?

problem

Painted cube

Age

14 to 16

Challenge level

Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?

problem

Mystic rose

Age

14 to 16

Challenge level

Use the animation to help you work out how many lines are needed to draw mystic roses of different sizes.

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For younger learners

Number patterns and infinity

Summing consecutive numbers

Frogs

Number pyramids

Crossed ends

Odds, evens and more evens

Magic letters

Where can we visit?

Marbles in a box

Route to infinity

Triangle numbers

Tower of Hanoi

Always a multiple?

Differences

Pair products

Steel cables

Attractive tablecloths

Painted cube

Mystic rose