All in the Mind
Imagine you are suspending a cube from one vertex and allowing it to hang freely. What shape does the surface of the water make around the cube?
Age
11 to 14
Challenge level
Problem
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?
Image
This problem has been adapted from the book "Sums for Smart Kids" by Laurie Buxton, published by BEAM Education. This book is out of print but can still be found on Amazon.
Getting Started
This is quite a difficult problem to imagine in your head. If you've tried very hard without success, then actually dangle a cube into some water.
Student Solutions
Many of you correctly guessed that the
shape outlined by the water would be a hexagon. You provided us
with several different explanations.
Jimmy, from Lady Mary Primary School,
made a cube out of blue-tac and - with the help of his teacher! -
cut it in half:
I held the cube from one vertex, and
stood it on the opposite vertex. Then I measured the height of the
cube standing like that. It was 2.4cm. Half of 2.4cm is 1.2cm, so
half the cube was above that height and the other half was below.
Then I cut it in a horizontal line at 1.2cm from the bottom of the
cube. The cross-section I made was a hexagon.
Tilly thought about what happens when you
submerge the cube so that more than half or less than half is in
the water.
If you just submerge the very tip of the cube in water, you
get a triangle shape with the point of the triangle pointing up. If
you submerge more and more you get bigger triangles, still with the
point pointing up. If you start with all but the tip of the cube in
the water, you get a triangle shape with the point of the triangle
pointing down. If you submerge more you get bigger triangles, also
with the tip pointing down.
When they meet at the middle there's a
triangle with the tip pointing up and a triangle with the tip
pointing down, so it must be a six pointed star or a hexagon. It is
not a six pointed star so it must be a hexagon.
And James had yet another
reason!
The water touches all six sides and it
does the same thing to all of them, because if you rotate the cube
the same thing still happens. So the water must make a regular
six-sided shape: a hexagon!
Thank you, Jimmy, Tilly and
James!