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Doplication
We can arrange dots in a similar way to the 5 on a dice and they usually sit quite well into a rectangular shape. How many altogether in this 3 by 5? What happens for other sizes?
Making Cuboids
Let's say you can only use two different lengths - 2 units and 4 units. Using just these 2 lengths as the edges how many different cuboids can you make?
Quadrilaterals
How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?
Age 7 to 11
Challenge Level
I started with a clock without hands or the minute divisions (except for those where there is a number). The $12$ was replaced by a $0$ and the numbers placed outside the face.
I ruled lines joining up the numbers.
I started by counting in ones and I got a $12$-gon (that is a $12$-sided polygon - if you like long words you can call it a dodecagon).
Then I ruled lines counting round in $2$s. And I got .....?
Perhaps you do not need to put the numbers round the circles.
I tried $5$s (wow!) and $6$s (well!).
Each time I go on drawing lines until I get to the point where I first started.
Then I tried $7$s, $8$s, $9$s, $10$s, and $11$s.
Something interesting was happening.
Why don't you try it? What patterns do you notice emerging?
And what about counting round in $12$s?
Which shapes are the same? Can you think of a reason why?
Can you see a connection between the number in which you are counting around the circle and the number of sides in the shape you are making?
NRICH is part of the family of activities in the Millennium Mathematics Project.