We had lots of solutions suggesting that it is possible to make 16 pieces
of cheese. This isn't quite right because you cannot cut all of your 8 pieces in half with one straight
cut, although it is a good start because we know that we cannot get more pieces than this. How many do
you think will have to be left out from the cut?
Duncan from QEGS Horncastlemade a brave attempt at the problem using
extensions of Euler's formula for the edges and vertices of a polyhedra, but, unfortunately, this
didn't quite work out correctly -- however, we were still very impressed by this analysis which indicates
great mathematical promise and using topology was a clever idea
Jason correctly realised that you couldn't cut through each piece each
time and suggested that 12 was the maximum.However, we think that more pieces are possible.
Well, we had to wait a while, but two excellent solutions were finally given to this tricky problem. Very well done!
ChanWC from Hong Kong suggested reducing the problem to lower dimension as follows
To reduce the 3D case into lower dimension so that we can see the
for 0-Dimension, no matter how it is cut, the point can not be divided.
for 1-Dimension, the line can only be cut one segment each time, so each cut increases the number of pieces by 1.
for 2-Dimensions, a plane is cut by line. For each non-parallel line cutting the
plane, it can cut the previous lines at most one time
cut time 1 2 3 4 5 6 7 8 9 ... n
- 0-DIM 1 1 1 1 1 1 1 1 1 ... 1
- 1-DIM 1 2 3 4 5 6 7 8 9 ... n
- 2-DIM 1 2 4 7 11 16 22 29 37 ... n(n-1)/2 + 1
from the above pattern, i found that the
(x-1) DIM of n cut + x DIM of n cut = x DIM of n+1 cut
by using this observation, i get this
Pippa and Mike from South Hunsley provided this solution
- 3-DIM 1 2 4 8 15 26 42 64 93 ...
Firstly we looked at the 2-D version of this problem.
We discovered that the greatest number of regions that could be created
happened when each existing line was cut by the new line. This new line
could not cut at the intersection of existing lines.
This means that the number of regions increases by the number of existing
lines +1. This is because if a new line cuts through n lines it cuts
through n+1 regions. This will create n+1 new regions.
The sequence we generated is shown below
- No. Regions 1 2 4 7 11 16 22
this gives the quadratic formula
Moving on to the 3-D case, we assumed that the greatest number of pieces
were formed by cutting through all the existing planes without cutting
through ti intersection of three existing planes.
We then thought that the cross section formed by this latest cut would be
the same as the 2-d model with the same number of cuts. So if we were
putting in the 4th cut it would pass through 7 pieces of cheese, (the
number of regions for 3 cuts on the 2-D model) so our new sequence is
generated by increasing each term by the terms in the 2-D sequence.
- No. Pieces 1 2 4 8 15 26 42
This gives the cubic formula
So 10 cuts gives 176
50 cuts gives 20876
100 cuts gives 166751