Cheese cutting
In this problem we see how many pieces we can cut a cube of cheese
into using a limited number of slices. How many pieces will you be
able to make?
Age
16 to 18
Challenge level
Problem
Visualisation of 3D objects is a very useful real-life skill used in a range of everyday situations, from packing suitcases and boxes into the back of a car to travel to university through to parking the car in a tight space upon arrival. We can practise and develop our 3D skills and intuition through the study of the following sorts of mathematical problems. Spatial problems are mathematically quite different to algebraic problems and you may find one type of problem easier or more difficult depending on how your brain works!
I have some cubes of cheese and cut them into pieces using straight cuts from a very sharp cheese wire. In between cuts I do not move the pieces from the original cube shape.
For example, with just one cut I will obviously get two smaller pieces of cheese, with two cuts I can get up to 4 pieces of cheese and with three cuts I can get up to $8$ pieces of cheese, as shown in the picture:
Image
Suppose I now make a fourth cut. How many individual pieces of
cheese can I make?
Suppose now that I am allowed more generally to cut the block
$N$ times. Can you say anything about the maximum or minimum number
of pieces of chesse that you will be able to create?
Although you will not be able to determine the theoretical
maximum number of pieces of cheese for $N$ cuts, you can always
create a systematic cutting system which will generate a
pre-detemined number of pieces (for example, making $N$ parallel
cuts will always result in $N+1$ pieces of cheese). Investigate
developing better cutting algorithms which will provide larger
numbers of pieces. Using your algorithm what is the largest number
of pieces of cheese you can make for $10$, $50$ and $100$
cuts?
Getting Started
Simplify the problem by making the initial cuts all at right angles so that the large cube is sliced into$8$ smaller cubes.
Align the cube so that the $8$ corners lie at the points $(\pm 1, \pm 1, \pm 1).$
Now consider the plane $x+y+z=0$.
How many of the $8$ smaller cubes does this plane pass
through?
What can you say about any cubes that this plane does not pass
through? How can you alter this plane to try to get more
intersections?
For the second part of the problem don't attempt to work out
the theoretical maximum number of pieces as this is exceedingly
difficult even for $5$ cuts! You can, however, consider how many
pieces each small piece of cheese can be cut into per slice. You
can also experiment with various pre-determined 'cutting schemes'
to try to find a guaranteed minimum number of possible
pieces.
Student Solutions
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 pattern.
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 summary:
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
- 3-DIM 1 2 4 8 15 26 42 64 93 ...
Pippa and Mike from South Hunsley provided this solution
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. Cuts 0 1 2 3 4 5 6
- No. Regions 1 2 4 7 11 16 22
this gives the quadratic formula
- 1/2x^2 + 1/2x + 1
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.
This gives
- No. Cuts 0 1 2 3 4 5 6
- No. Pieces 1 2 4 8 15 26 42
This gives the cubic formula
- 1/6x^3 + 5/6x + 1
So 10 cuts gives 176
50 cuts gives 20876
100 cuts gives 166751
Teachers' Resources
This is a difficult visualisation problem to tackle, not least because is it very difficult to create a physical model with which it is possible to experiment with the slicing. Visualisation is thus crucially required.
The problem can be attempted at various levels, from having an educated guess at the answer to providing a proof of the maximum possible number of pieces.
If solvers are unable to prove the maximum number of pieces they should certainly be encouraged to describe their best effort at a slicing as clearly and convincingly as possible. Try to establish the existence of an upper bound on the number of pieces.
In practice, the solver may be able to spot the answer quite quickly, but providing an explanation of the answer may be very hard. Solvers should be encouraged to try to explain their answer as clearly as possible in words if a sound mathematical argument cannot intially be provided.