Expert: Josh - 4/8/2010

Question
Find all the different boxes you can build that have a volume of 24 cubic units.  
In order for boxes to be different, their dimensions must differ.  I have
identified 4, my teacher says their are at least 8.

Answer
Hi Sheridan,

There are at least two ways of looking at this problem.

An informal approach may use paper and scissors, handcrafts etc. to model different combination. By trial-and-error, you hope to find new combination which you haven't encountered previously.

Then, again, we can look at this analytically using mathematical equations.

I think you have already considered the first approach, so I will elaborate on the second approach.

The volume of a rectangular prism (see http://www.math.com/tables/geometry/surfareas.htm#rec) is given by V = w*l*h, where w is the width, l is its length and h is its height.

========================================================
Since the volume is 24 cubic unit, we need to find three numbers (w, l and h, representing width, length and height, respectively) such that their product (w*l*h) equals 24.

IMPORTANT:
If we are allowed to pick any number, that is, set "w" to any value, we can always find a number for l (and h) to ensure that we end up with 24 [cubic units] when the three numbers (w,l and h) are multiplied together.

This means, there are an infinite number of combination (w, l and h dimensions) which will give the same volume if w, l and h can be any positive number.

HOWEVER, if we restrict our attention to integers only, then, we have only a finite number of possibilities. The number of combination will be countable and NOT infinite. In this situation, finding the number of different cuts becomes a prime number factorization problem.

Observe that 24 = 2*12 = 2*2*6 = 2*2*2*3, .......[#1]
i.e., the number "24" can be written as a product of 4 prime numbers. Why did I bring this up?

Recall that the problem of finding different cuts with the same volume is same as finding three numbers w, l and h, whose product equals 24.

If we fix one dimension to one (say, w=1), we have to come up with "l" and "h" such that w*l*h=24  .......[#2]

But from [#1] above, we already know that 24 = 2*2*2*3. When we set w=1 in [#2], the equation becomes l*h = 2*2*2*3  [note: we have simply replaced the right hand side of [#2], viz., 24, with its prime factorization 2*2*2*3].

So, when we fix w=1, the number of feasible combination is determined by the number of ways we can select "l" and "h" (length and height of the prism) from 2*2*2*3.

| Example: Here, we consider the case where width w=1.
| a) we may select length l=2, this would force height h=2*2*3=12.
|    we can verify that w=1, l=2, h=12 gives a total volume of V = w*l*h = 1*2*12 = 24.
| b) we may select length l=2*2, this would force height h=2*3.
|    we can verify that w=1, l=4, h=6 gives a total volume of V = w*l*h = 1*4*6 = 24.
| c) we may select length l=2*2*2, this would force height h=3.
|    we can verify that w=1, l=8, h=3 gives a total volume of V = w*l*h = 1*8*3 = 24.
| d) we may select length l=2*2*3, this would force height h=2.
|    we can verify that w=1, l=12, h=2 gives a total volume of V = w*l*h = 1*12*2 = 24.
| e) we may select length l=2*2*2*3, this would force height h=1.
|    we can verify that w=1, l=24, h=1 gives a total volume of V = w*l*h = 1*24*1 = 24.
| Note:
| (d) is really just a permutation of (a), we could simply label its length as "h" (instead
| of "l") and label its height as "l" (instead of "h"). 1*2*12 and 1*12*2 are indistinguishable
| because given a solid with identical shape, we have no sense of orientation (some people will
| refer to the "length" as "height", others vice versa) unless we mark one of the corners and
| let this point be the origin in a coordinate system. Only then will the meaning of "length"
| be well-defined and consistently referred as "l" instead of being mixed up as "h".
| When a corner is marked, we may consider (d) w=1,l=12,h=2 as a unique combination different
| from (a) w=1,l=2,h=12.
|
| So far, we have found 4 distinct possibilities (with (d) being redundant).
| They are {1,2,12}, {1,4,6}, {1,8,3}, {1,24,1}.

Now, consider how we can choose the three numbers (x,l and h) when none of them is allowed to be one. This is equivalent to forming a product of three numbers from "2*2*2*3".
| f) we may select w=2 and l=2. This would force h=2*3.
|    we can verify that w=2, l=2, h=6 gives a total volume of V = w*l*h = 2*2*6 = 24.
| g) we may select w=2 and l=3. This would force h=2*2.
|    we can verify that w=2, l=3, h=4 gives a total volume of V = w*l*h = 2*3*4 = 24.
| Again, I will argue that there are no further unique combination.
| That is, I do not consider w=4, l=3, h=2 to be different from (g), because we can
| call the width "h" and the height "w", simply by using a different coordinate system.

Conclusion:
When people are free to call any side the "length" and any one of the two remaining dimensions the "width" (i.e., have no way of distinguishing (d) and (a) after both boxes have been subject to arbitrary rotation):
- There are 6 unique cuts which yield a volume of 24 cubic units when "w", "l" and "h" can only take on integer values.
- These are {1,2,12}, {1,4,6}, {1,8,3}, {1,24,1}, {2,2,6} and {2,3,4}. .....[#3]

When the origin of a box is marked, and people have a sense of orientation
e.g., using a fixed coordinate system like the one below, where the meaning of "length", "width" and "height" are not ambiguous, we can make a distinction between w=1,l=2,h=12 and w=2,l=12,h=1. In fact, each of the dimensions given in [#3] can be reordered in 6 different ways. So, the number of possibilities (different cuts) expand by a factor of 6, from 6 to 36 permutation.

h
|
|   l
|  /
| /
|/
O-------------w

- These are (1,2,12), (1,12,2), (2,1,12), (2,12,1), (12,1,2), (12,2,1),
           (1,4,6), (1,6,4), (4,1,6), (4,6,1), (6,1,4), (6,4,1),
           (1,8,3),... and so forth.