### Summing Consecutive Numbers

Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?

### Always the Same

Arrange the numbers 1 to 16 into a 4 by 4 array. Choose a number. Cross out the numbers on the same row and column. Repeat this process. Add up you four numbers. Why do they always add up to 34?

### Fibs

The well known Fibonacci sequence is 1 ,1, 2, 3, 5, 8, 13, 21.... How many Fibonacci sequences can you find containing the number 196 as one of the terms?

# Number Pyramids

##### Stage: 3 Challenge Level:

Choose three single-digit numbers and write them in the bottom row of the pyramid.
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Try some different numbers.
Can you work out how the numbers in the upper layers are generated?

Here are some questions to consider:

• Given the numbers on the bottom layer in order, can you find a quick way to work out the number at the top?
• If you change the order of the numbers on the bottom layer, will the top number change?
• If you can rearrange the numbers on the bottom layer, can you find a quick way to work out the largest possible number that could go at the top?
• Given the number at the top, how can you come up with possible numbers to go at the bottom?
Test out your observations and insights. You could use big numbers, small numbers, negative numbers, decimals...

Can you explain what is happening?
Can you explain why it is happening?
Can you explain it algebraically

Can you adapt your insights so that they apply to pyramids with more than three layers?

Here is a number pyramid with four layers so that you can test some of your ideas:

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To experiment with 5 and 6 layer pyramids, you may find this spreadsheet useful.
You could adapt it to work on even larger pyramids!

For other problems that use this idea go to More Number Pyramids.