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...

Squares, Squares and More Squares

Can you dissect a square into: 4, 7, 10, 13... other squares? 6, 9, 12, 15... other squares? 8, 11, 14... other squares?

More Number Pyramids

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

Maxagon

Age 11 to 14 Challenge Level:

For this problem you may wish to download and print some dotty paper.
Alternatively, you could explore the problem using this interactivity.

Draw some polygons by joining the dots on a $3$ by $3$ grid.

What is the greatest number of sides that your polygon could have?

What about on a $3$ by $4$ grid, or a $3$ by $5$ grid?

What about on a $3$ by $n$ grid?

Can you explain the pattern by which the 'number of sides' increases?

Explore some polygons on grids that are $4$ dots high.

What is the maximum number of sides a polygon could have on a $4$ by $n$ grid?

Can you explain how you know?

What is the maximum number of sides a polygon could have on a $6$ by $6$ grid?
And on a $6$ by $n$ grid?

With thanks to Don Steward, whose ideas formed the basis of this problem.