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?
Suppose we allow ourselves to use three numbers less than 10 and
multiply them together. How many different products can you find?
How do you know you've got them all?
Investigate the different shaped bracelets you could make from 18 different spherical beads. How do they compare if you use 24 beads?
Imagine a wheel with different coloured markings - red, green and blue painted on it at regular intervals.
As the wheel goes round, a trail is painted on the ground.
RGBRGBRGBRGBRGBRGB ... ...
Satisfy yourself that you can predict where the blues will appear.
How about the red and green marks?
Can you predict the colour of the 18th? 19th? 31st? 59th? 299th? 3311th? 96 312th?
How did you work it out?
Now consider the wheels that produce:
BBYGBBYGBBYG ... ...
BYBRBYBRBYBRBYBR ... ...
RRRBBYRRRBBYRRRBBYRRRBBYRRRBBY ... ...
RRBRRRRBRRRRBRRRRBRRRRBRR ... ...
You could continue this investigation by asking yourself some "what if ...?" questions:
A wheel has six markings. Where would red be painted on it so that the 100th mark made is red?
What other wheels (with more/fewer markings) would give you a red in the 100th position?