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Multilink Cubes

If you had 36 cubes, what different cuboids could you make?

The Big Cheese

Investigate the area of 'slices' cut off this cube of cheese. What would happen if you had different-sized block of cheese to start with?

Making Boxes

Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?

All Wrapped Up

Age 7 to 11
Challenge Level

This was a tricky problem, but many of you tackled it successfully. Your solutions depended on whether you assumed that you could cut the paper or not. Rachel from Charter Primary said:

I solved this problem by drawing a net.

The length of my cuboid was $38 \; \text{cm}$. The width of my cuboid was 14cm. The depth of my cuboid was $1\; \text{cm}$.

I tried to make the unused space as small as possible.

So, Rachel's solution kept the paper in one piece (as you would usually do when you wrap a present, for example).

This gives a cuboid with the maximum possible surface area ($1168 \;\text{cm}^2$).

I wonder how you arrived at this solution, Rachel? How many differently-sized cuboids did you try before you discovered this was the largest? Rohaan from Longbay Primary looked at it in a different way, assuming that you could cut the paper:

We decided to find a cuboid that that had a surface area that matched with the wrapping paper ($1200 \; \text{cm}^2$).

We made a starting estimate of a cuboid that was $1\times20\times20$. It had a surface area of $880 \; \text{cm}^2$.

Then we thought there could be a bigger cuboid that would fit. We wondered by how much the cuboid's surface area would go up if we changed its measurements from $1\times20\times20$ to $2\times20\times20$. It went up $80 \; \text{cm}^2$.

We thought if we changed it to $3\times20\times20$ it would go up by $80 \; \text{cm}^2$ again. It did. We went up until we reached $5\times20\times20$ which had a surface area of exactly $1200 \; \text{cm}^2$. It matched the surface area of the wrapping paper.

Just to make sure it fitted, we drew up the surface area ($1200 \; \text{cm}^2$) on a piece of A3 paper. It fitted!

Did anyone try to find the cuboid with the largest volume that could be wrapped up in this paper? That's another challenge for you!