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?

A circle is inscribed in an equilateral triangle. Smaller circles touch it and the sides of the triangle, the process continuing indefinitely. What is the sum of the areas of all the circles?

Just Rolling Round

P is a point on the circumference of a circle radius r which rolls, without slipping, inside a circle of radius 2r. What is the locus of P?

Instant Insanity

Stage: 3, 4 and 5 Challenge Level:

This solution is from Shu Cao of the Oxford High School for Girls. Well done Shu!

Let B=blue, W=white, Y=yellow and R=red.

The colours of the walls of the tower listed from top to bottom in columns 1-4 are:

 Column 1 2 3 4 Wall 1 B W Y R Wall 2 W R B Y Wall 3 R B W Y Wall 4 B Y R W

One can swap the columns 1234 to create 4X3X2X1=24 different towers.

Method:

 Column 1 2 3 4 Line 1 B Y B Y Line 2 YWY RWR RWW WWB Line 3 R R R R Line 4 B B Y Y

The tower has 4 walls so each colour only appears 4 times on the walls. There are 7 red, 6 yellow, 6 white and 5 blue so 3 red, 2 yellow, 2 white and 1 blue will be either within the tower or directly on the top and bottom of the tower.

Incidentally, there are 3 red flaps, 2 yellow flaps, 2 white flaps and 1 blue flap in line 2. Assuming that they can be folded over into the tower or onto the top and bottom, the faces left in columns 1234 will make up the walls of the tower.

All that one has to do now is to move the columns up and down so as to line different colours together. Line up 1 and 2 first, when there is a protruding flap, move it onto the top or bottom of the column depending on circumstances. Proceed to columns 3 and 4 in the same way.

Another method is to use graphs. The edges in the graphs below join the colours that appear on opposite faces of the cubes.

To solve the problem next combine all 4 graphs, writing 1, 2, 3, and 4 on the edges denoting which of the 4 cubes they represent. Then look for 2 subgraphs, in this case two square circuits chosen from combined graph, such that each contains all 4 colours and precisely one edge of each numbered cube. One subgraph will represent the colours on the front and back walls of the tower and the other subgraph will represent the colours on the left and right hand walls of the tower.

If you can draw the combined graph and the two subgraphs giving the solutions then do send them in for publishing with this solution.