Problem-Solving Narrative

I will be narrating the problem titled 'products of sums'. We worked on it in class on Wednesday April the 3rd and I gained knowledge and insight about the problem in class that day and also when I went home and thought about it more.The problem states briefly, " If n is a positive integer, what is the maximum product that can be formed of a list of positive integers that sum to n?". Many approaches exist when tackling this problem, some simple and some complex.
1. First off you must understand the problem.
Many things have to be considered when looking at a solution to this problem. Only positive integers are being used so you can immediately eliminate the negative integers which helps us. The fact that we are also only working with integers also helps us because when working with real numbers we would have to consider different techniques to solve this problem. There are various ways to represent a solution to this problem, none of them which are wrong in any sense; the solution can represented by a sentence in English, by an equation or by a python program. All of which would have their benefits, disadvantages and difficulty in interpreting.

2. Devising the plan to tackle the problem.
First off, like always, I try to solve the problem using examples, or in other terms brute force. Sometimes if the problem is easy enough then this will work, but for any problem worth solving this method usually doesn't hold up. The 'best case' I would have from this problem would be that the examples from brute force technique showed a pattern in which I could derive a simple math equation from and then the problem would be solved. Maybe if the solution takes many steps to complete, the answer could be transferred to a python program. And then from there we could even translate it to English for a wider audience. After using brute force for a little while, in this problem particularly, it seems to be useless and we need to start thinking conceptually.

3. Carrying out a plan and working through concepts.
Some large idea that I thought of and that were worked on individually were:
- Splitting up your number into just 1's, having enough 1's to add up to the particular number you are dealing with. Then from there, continue adding them up from the left to the right into different combinations or somehow organize these lists. This seemed to fail though and wasn't very practical.
- One solution I found was just breaking the number in half and then the largest product would come from those two numbers. But I soon realized that it didn't work for small numbers, or odd ones at that, and your list of sums could be bigger two numbers so that idea quickly dissolved.
- Another strain of thought came form the idea that the highest product of the sum will never include the factor of itself, examples like 199 + 1 = 200, but this idea does not work for small numbers such as three and two.

After trying these techniques I thought of dividing these numbers into two sets, odd and even numbers, which came to a moving conclusion that ceiling or floors had to be used in the solution to this problem. After then thinking about this, I still could not come up with a solution for the problem unfortunately.

When it comes to other SLOGS I thought George Abu Salehs SLOG was the best one I read. He made the posts fun and enjoyable to read as they also gave insight to understanding concepts.


Alec out