Patterns that lead to algebra and proof - October 2010, All Stages

Mathematicians are fascinated by pattern - one writer has even described 'pattern sniffing' as a mathematical behaviour. In this month's activities and problems we invite you to discover patterns, describe them in different ways and begin to think about how you can convince yourself, and others that you have sniffed out all there is to find!

Problems

Cube Bricks and Daisy Chains

Stage: 1 Challenge Level:

Daisy and Akram were making number patterns. Daisy was using beads that looked like flowers and Akram was using cube bricks. First they were counting in twos.

How Odd

Stage: 1 Challenge Level:

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?

Sitting Round the Party Tables

Stage: 1 and 2 Challenge Level:

Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.

Carrying Cards

Stage: 2 Challenge Level:

These sixteen children are standing in four lines of four, one behind the other. They are each holding a card with a number on it. Can you work out the missing numbers?

Three Dice

Stage: 2 Challenge Level:

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

Which Is Quicker?

Stage: 2 Challenge Level:

Which is quicker, counting up to 30 in ones or counting up to 300 in tens? Why?

Shapes in a Grid

Stage: 2 Challenge Level:

Can you find which shapes you need to put into the grid to make the totals at the end of each row and the bottom of each column?

Stage: 3 Challenge Level:

Think of a number... follow the machine's instructions. I know what your number is! Can you explain how I know?

Stage: 3 Challenge Level:

Think of a number and follow my instructions. Tell me your answer, and I'll tell you what you started with! Can you explain how I know?

Special Numbers

Stage: 3 Challenge Level:

My two digit number is special because adding the sum of its digits to the product of its digits gives me my original number. What could my number be?

Stage: 4 Challenge Level:

Explore the two quadratic functions and find out how their graphs are related.

Stage: 4 Challenge Level:

Robert noticed some interesting patterns when he highlighted square numbers in a spreadsheet. Can you prove that the patterns will continue?

Stage: 4 Challenge Level:

Here are some more quadratic functions to explore. How are their graphs related?

Weekly Challenges

Stage: 4 and 5 Challenge Level:

The NRICH Stage 5 weekly challenges are shorter problems aimed at Post-16 students or enthusiastic younger students. There are 52 of them.

Patterns of Inflection

Stage: 5 Challenge Level:

Find the relationship between the locations of points of inflection, maxima and minima of functions.

Calculus Analogies

Stage: 5 Challenge Level:

Consider these analogies for helping to understand key concepts in calculus.

Agile Algebra

Stage: 5 Challenge Level:

Observe symmetries and engage the power of substitution to solve complicated equations.

Interactive Workout - Mathmo

Stage: 5 Short Challenge Level:

Mathmo is a revision tool for post-16 mathematics. It's great installed as a smartphone app, but it works well in pads and desktops and notebooks too. Give yourself a mathematical workout!

Tens

Stage: 5 Challenge Level:

When is $7^n + 3^n$ a multiple of 10? Can you prove the result by two different methods?