All in the Mind

Imagine you are suspending a cube from one vertex (corner) and allowing it to hang freely. Now imagine you are lowering it into water until it is exactly half submerged. What shape does the surface of the water make around the cube?

Rotating Triangle

What happens to the perimeter of triangle ABC as the two smaller circles change size and roll around inside the bigger circle?

Instant Insanity

Given the nets of 4 cubes with the faces coloured in 4 colours, build a tower so that on each vertical wall no colour is repeated, that is all 4 colours appear.

Christmas Boxes

Stage: 3 Challenge Level:

There were 11 possibilites in total, found by George and Jannis. Well Done for cracking ths Toughnut!

George (Rosebank Primary School, Leeds) recognised this type of problem. He described his answers in groups as well as drawing them, finding 7 of the possibilities.

"We did this sum at school last year and this is what I think the answer is. First I did this kind of boxes. They were 4 squares down, 1 square at the top left and one at the bottom right. I looked for more and I found that there was only 1:

Then I did this kind of boxes. They were 4 squares down, then one square

at the left and opposite it the right one. There were 2 of these:

After that I did this kind of boxes. They were 4 squares down, 1 square at the left and then a right square which will be opposite then 1 square up or down. There were 2 of these:

Fourthly I did this kind of boxes. They were 4 squares down, 1 square at the left and a right one which will be opposite then 2 squares up or down. There was only 1 of them: At last I did this kind of boxes. They were 3 squares down and 3 squares which the last of their square is touching the last of the 1st 3 squares. There was only one of them:

The answer is you can make 7 boxes".

Jannis (Long Bay Primary) then correctly found the other 4:

"I started by looking at the two already given box-nets. Using these box-nets I shifted the small squares around to get another few:"

Finally, Matthew (Eastwood Primary) noticed that:

"You can turn them around or reflect them to make a lot more."

So there are other possibilites, but they would just be rotations or reflections of the above 11.