Question

I have a number n, and I want to find three numbers whose product is n but are as close to each other as possible. That is, if n = 12 then I’d like to get 3, 2, 2 as a result, as opposed to 6, 1, 2.

Another way to think of it is that if n is the volume of a cuboid then I want to find the lengths of the sides so as to make the cuboid as much like a cube as possible (that is, the lengths as similar as possible). These numbers must be integers.

I know there is unlikely to be a perfect solution to this, and I’m happy to use something which gives a good answer most of the time, but I just can’t think where to go with coming up with this algorithm. Any ideas?

Answer 1

Here’s my first algorithm sketch, granted that n is relatively small:

• Compute the prime factors of n.
• Pick out the three largest and assign them to f1, f2, f3. If there are less than three factors, assign 1.
• Loop over remaining factors in decreasing order, multiply them into the currently smallest partition.

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Edit

Let’s take n=60.

Its prime factors are 5 3 2 2.

Set f1=5, f2=3 and f3=2.

The remaining 2 is multiplied to f3, because it is the smallest.

We end up with 5 * 4 * 3 = 60.

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Edit

This algorithm will not find optimum, notice btillys comment:

Consider 17550 = 2 * 3 * 3 * 3 * 5 * 5
* 13. Your algorithm would give 15, 30, 39 when the best is 25, 26, 27.

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Edit

Ok, here’s my second algorithm sketch with a slightly better heuristic:

• Set the list L to the prime factors of n.
• Set r to the cube root of n.
• Create the set of three factors F, initially set to 1.
• Iterate over the prime factors in descending order:
  • Try to multiply the current factor L[i] with each of the factors in descending order.
    • If the result is less than r, perform the multiplication and move on to the next
      prime factor.
    • If not, try the next F. If out of Fs, multiply with the smallest one.

This will work for the case of 17550:

n=17550
L=13,5,5,3,3,3,2
r=25.98

F = { 1, 1, 1 }

Iteration 1:

• F[0] * 13 is less than r, set F to {13,1,1}.

Iteration 2:

• F[0] * 5 = 65 is greated than r.
• F[1] * 5 = 5 is less than r, set F to {13,5,1}.

Iteration 3:

• F[0] * 5 = 65 is greated than r.
• F[1] * 5 = 25 is less than r, set F to {13,25,1}.

Iteration 4:

• F[0] * 3 = 39 is greated than r.
• F[1] * 3 = 75 is greated than r.
• F[2] * 3 = 3 is less than r, set F to {13,25,3}.

Iteration 5:

• F[0] * 3 = 39 is greated than r.
• F[1] * 3 = 75 is greated than r.
• F[2] * 3 = 9 is less than r, set F to {13,25,9}.

Iteration 6:

• F[0] * 3 = 39 is greated than r.
• F[1] * 3 = 75 is greated than r.
• F[2] * 3 = 27 is greater than r, but it is the smallest F we can get. Set F to {13,25,27}.

Iteration 7:

• F[0] * 2 = 26 is greated than r, but it is the smallest F we can get. Set F to {26,25,27}.