### 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?

# Pick's Theorem

##### Age 11 to 14 Challenge Level:

* How many different figures can be described as $(4, 0)$?

* What do you notice about $(4,0)$ figures?

* Choose another particular value for $(p,i)$ and explore different shapes.

* Have you tried drawing shapes with the same area?

* What do you notice about those figures whose areas are the same?

* What ways are there of increasing the area by $1$ unit?

* Draw more figures; tabulate the information about their perimeter points ($p$), interior points ($i$) and their areas ($A$).