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?
A triangle PQR, right angled at P, slides on a horizontal floor
with Q and R in contact with perpendicular walls. What is the locus
A box of size a cm by b cm by c cm is to be wrapped with a square piece of wrapping paper. Without cutting the paper what is the smallest square this can be?
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:
One can swap the columns 1234 to create 4X3X2X1=24 different
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