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Pebbles

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

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Happy Numbers

Take any whole number between 1 and 999, add the squares of the digits to get a new number. Make some conjectures about what happens in general.

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Intersecting Circles

Three circles have a maximum of six intersections with each other. What is the maximum number of intersections that a hundred circles could have?

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.