# Tag Info

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Another one of those puzzles you can brute force, and therefore would have a coding answer. The answer is: Try online Brute force code: function permut(string) { if (string.length < 2) return string; // This is our break condition var permutations = []; // This array will hold our permutations for (var i = 0; i < string.length; i++) { ...

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If we set x=9 then we have x*(x-4)*(x-8)=45 (x-1)(x-5)(x-7)=64 (x-2)(x-3)(x-6)=126 so I obtain 951*842*763=610966146 My opinion is that if we obtain minimum results from the multiplications of monomials, then we will obtain the maximum product of the three 3-digit numbers.

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The answer is as @avi already mentioned. But here is my justification: The conclusion is that Now we know that the 9 digits are splitted in 3 groups It now remains to form the three factors by picking one digit in each group. Using a similar reasoning As a result, the three factors formed for the three groups are such that

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We can calculate: $(a*100 + b*10 + c)*(d*100 + e*10 + f)*(g*100 + h*10 + i)= \\ = adg1000000 + (adh+aeg+bdg)100000 + (adi+aeh+bdh+afg+beg+cdg)10000 + (aei+bdi+afh+beh+cdh+bfg+ceg)1000 + (afi+bei+cdi+bfh+ceh+cfg)100 + (bfi+cei+cfh)10 + cfi$ If we now maximise coefficients in order: $max(adg)$ a = 9 d = 8 g = 7 $max(adh+aeg+bdg) = max(72h+63e+56b)$ h ...

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By simply trying out a couple of combinations, I found: Some of the other combinations I tried are: My methodology: Edit: FYI, I did the calculations by hand, rather than with a program, so I didn't try all options.

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The answer is: Here's the intuition: Finally, here's

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Building on the answers by Elias and CiaPan... Elias gave this "binary search"-based answer. But it's interesting to notice that we can also do it with this "unbalanced binary search": Or even like these: Or the two solutions with the red piece on the outside, which I'm too lazy to draw out. I don't know if there are other solutions. I actually asked ...

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I used a nonlinear optimization solver, with variables $x_i$, $y_i$, $w$, $h$. The problem is to minimize $w\cdot h$ subject to: \begin{align} i \le x_i &\le w - i &&\text{for $i\in\{1,\dots,12\}$}\\ i \le y_i &\le h - i &&\text{for $i\in\{1,\dots,12\}$}\\ (x_i - x_j)^2 + (y_i - y_j)^2 &\ge (i + j)^2 &&\text{for \$1\le i&...

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Work in progress (may not be optimal) Picture: Diagram of contacts: Coordinates (unconstrained circles at reduced precision):

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Improved answer This is similar to another answer but with a smaller area. I worked it out completely independently, then noticed its similarity. The 3" dish is in a different place, and it is not an adjustment based on that answer. It was generated by a C program I wrote for this purpose. It gave my previous answers and has been spitting out smaller ...

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