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Here is a solution with a constant product of which I think is the minimum possible: Some partial progress for a lower bound on the product: This leaves only a few possibilities for improvement:


I found a solution trying to minimize prime factors. And finding a balance between the minimum value and minimum number of factors. I noticed that some fields are connected, in that multiplying one of them results in the multiplication of fixed other fields. There are three such patterns: -C, D, H, L, and M (and any mirror image of that) -A, E, G, L, M (...


Edit: In an effort to find the minumum, here is a much smaller solution in which the mutual product is Solution As MKBakker pointed out we could further reduce this by dividing each of the entries 4,8,16,96 and 192 by 2 to get a mutual product of although they have subsequently improved on this.


There are a number of ways to do this. An easy strategy:


And here is the number you are probably thinking of: It works only for $ab$ where $a \le b$. I suppose that it is a mistake in the problem statement. Others have proven that as it is, the problem is unsolvable. And here is how I came to that number. PS: I have been playing with this problem. You can extend it to $ab$ with $a > b$ with the ...


I am going to prove that Indeed Let me dump here previous thoughts that turned out not to be useful but might be in the future. First, Second


Just a small expansion of OmegaKrypton answer: A Perfect Trio Word:


A Trio word: A Not-Trio word: A Perfect Trio Word:

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