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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?


Investigate the different shaped bracelets you could make from 18 different spherical beads. How do they compare if you use 24 beads?

Sweets in a Box

How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction?

Cubes Within Cubes

Age 7 to 14
Challenge Level

We had interlocking cubes (all the same size) in ten different colours, up to 1000 of each colour. We started with one yellow cube. This was covered all over with a single layer of red cubes:

diagram of one yellow cube being covered with a layer of red cubes with a pile of blue cubes to the side, not yet used

This was then covered with a layer of blue cubes.
Then came a layer of green, followed by black, brown, white, orange, pink and purple for as long as there were enough cubes of that colour to cover the layer that came before.

piles of different coloured cubes

The unused cubes were put away.
The many-layered cube was then broken up and each colour made into cubes. These were just of the one colour and the largest cubes possible made.
For example, the red layer made three 2x2x2 cubes with two 1x1x1 cubes left over, whereas the larger layers made much larger cubes as well as smaller ones.

What colour was the largest cube that was made?

Which colour made into cubes had no 1x1x1 cubes?

Which colour was made into the most cubes including the 1x1x1 cubes?