Puzzle of the Week #153 - Multiply and Subtract

Here is an interesting mathematical game to play. The idea is to reach a total by multiplying and/or subtracting positive integers, whilst minimising the sum of those integers. Rather than explain I’ll demonstrate:

For example if the target is 99 you could have:

11 x 3 x 3 = 99, giving a score of 17

Alternatively you could save a couple of points thus:

5 x 5 x 4 – 1 = 99, giving 15 points

But even this can be improved upon:

3 x 4 – 1 x 3 x 3 = 99, scoring 14 points.

Note that each calculation, whether multiplying or subtracting, is performed stepwise, rather than using the BODMAS order of operations.


So, the target I have chosen for the puzzle is 12345. What is the lowest score you can achieve?

I should point out that I am not confident that the answer that I have is the best achievable (twice before I thought I had the best answer and then found a better one). If your score is higher I’ll be able to tell you that a lower score is achievable, but I am fully prepared to discover that my score is not the best either!

Additionally, if anyone can find an algorithm that finds the best solution in the general case I’d love to hear of it. At present all I have been able to establish is a loose set of rules of thumb.