### Days and Dates

Investigate how you can work out what day of the week your birthday will be on next year, and the year after...

### 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?

# Number Pyramids

##### Stage: 3 Challenge Level:

In a number pyramid, the numbers on the lower levels determine the numbers above them.

Choose three single-digit numbers and enter them in the bottom row of the interactive number pyramid.

Full Screen Version
If you can see this message Flash may not be working in your browser
Please see http://nrich.maths.org/techhelp/#flash to enable it.

Try entering some different numbers in the bottom row.
Can you work out how the numbers in the upper layers are generated?

Here are some questions to consider:

If I tell you the numbers on the bottom layer, can you work out the top number without working out the middle layer?

If you change the order of the numbers on the bottom layer, will the top number change?

Given any three numbers for the bottom, how can you work out the largest possible number that could go at the top?

If I give you a target for the top number, can you quickly find three possible numbers for the bottom?

Can you adapt what you learned about 3-layer pyramids to larger pyramids?

Here is an interactive number pyramid with four layers so you can test out your ideas:

Full Screen Version

If you can see this message Flash may not be working in your browser
Please see http://nrich.maths.org/techhelp/#flash to enable it.

You can create a number pyramid puzzle by erasing numbers from a completed number pyramid and challenging someone to find the missing numbers.
Here's a couple for you to try:

What is the minimum number of cells that need to be filled, in order for a puzzle to have a unique solution?
Does it matter which cells?

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