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### Number and algebra

### Geometry and measure

### Probability and statistics

### Working mathematically

### Advanced mathematics

### For younger learners

# Odd Differences

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Age 14 to 16

Challenge Level

The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n ² Use the diagram to show that any odd number is the difference of two squares. Note that 15 = 8 ² - 7 ² as well as 4 ² - 1 ². Write the number 105 as the difference of two squares in as many different ways as you can? The number 1155 can be written as the difference of two squares in eight different ways, can you find them? |

Show that 8778, 10296 and 13530 are three triangular numbers and that they form a Pythagorean triple.

Take a triangular number, multiply it by 8 and add 1. What is special about your answer? Can you prove it?

Mr Smith and Mr Jones are two maths teachers. By asking questions, the answers to which may be right or wrong, Mr Jones is able to find the number of the house Mr Smith lives in... Or not!