Robot Welders

A company specialises in producing large rectangular metal sheets, starting from unit square metal plates. The welding is performed by a range of robots of increasing size. Unfortunately, the programming options of these robots are rather limited. Each one can only process up to $25$ identical rectangles of metal, which they can weld along either edge to produce a larger rectangle. The only programmable variables are the number of rectangles to be processed (up to and including $25$), and whether to weld the long or short edge.
For example, the first robot could be programmed to weld together $11$ raw unit square plates to make a $11 \times 1$ strip. The next could take $10$ of these $11 \times 1$ strips, and weld them either to make a longer $110 \times 1$ strip, or a $11 \times 10$ rectangle. Many, but not all, possible dimensions of metal sheets can be constructed in this way.
One regular customer has a particularly unusual order: The finished product should have an exact area, and the long side must not be more than $10\%$ larger than the short side. If these requirements can be met in more than one way, in terms of the exact dimensions of the two sides, then the customer will demand that all variants be produced. For example, if the order calls for a metal sheet of area $889200$, then there are three final dimensions that can be produced: $900 \times 988$, $912 \times 975$ and $936 \times 950$. The target area of $889200$ is the smallest area which can be manufactured in three different variants, within the limitations of the robot welders.
Let $M(n)$ be the minimal area that can be manufactured in exactly $n$ variants with the longer edge not greater than $10\%$ bigger than the shorter edge. Hence $M(3) = 889200$.
Find $\sum_{n=2}^{100} M(n)$.

This problem is a number theory problem involving integers and it involves programming the robot in a way to make $n$ rectangles with a given area constraint while maintaining a maximal aspect ratio.

The goal is to find $M(n)$ where $n$ happens to be the number of rectangles that can be manufactured with the same area and satisfying the condition that the longer edge may not be more than 10% larger than the shorter edge.

This problem can be solved using a computer algorithm to simulate all possible ways to program the robot and find the minimal area of the generated rectangles.

Let’s overview the algorithm. Suppose we have to find $M(2)$.

1. We start by setting the area as a high number to begin the iterative process of reducing it, once we get the number of rectangles to $2$.
2. We generate all possible pairs of dimensions whose product is equal to the set area, and also satisfy the given condition. (i.e. long side <= 1.1 * short side) 3. Next, we count the number of rectangles generated. 4. If the number of rectangles obtained is less than $2$, we decrement the area, generate new pairs of dimensions and count again. 5. We continue this process until we get exactly $2$ rectangles. This gives us the minimal area that can be manufactured in exactly $2$ ways. This process can be carried out for other $n$ as well from $3$ to $100$ to find $M(n)$ for each. Finally, $\sum_{n=2}^{100} M(n)$ can be calculated by iterating through $M(n)$ for $n$ from $2$ to $100$ and adding all $M(n)$. However, due to the iterations involved at each step and also for all $n$ from due $2$ to $100$, this program would require a lot of computational power and may take a lot of time. One possible optimization we could use is to iteratively decrement the area and, for each area, find the pair of dimensions. At the same time, we could save the number of rectangles made for that area. For the next decrement in area, we only count the pairs lost as a result of the reduction. In that way, we would not need to count all pairs for each decrement in area which might make the program run significantly faster. Additionally, we can parallelize the algorithm to process different values of $n$ simultaneously. However, this does require multiple processors, and depending on the available resources and the programming environment, this might not always be feasible. Please note, exact calculations are not provided as they depend on computational methods and not manual calculations.

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