I actually did give that idea some thought during the analysis of this puzzle.

I figured that if I could find the worst possible cluster – the one that took the most colour changes to absorb it – by using a similar technique to the one used to calculate the matrix, D in the typical diff algorithm. I think each of the three stages – calculating minimum distance for all nodes, finding the most distant node, and backtracking the colour steps could be done in O(n), and they are done sequentially, so the whole solution is O(n). </handwaving>

I was patting myself on the back, prematurely, thinking I could use that technique to solve the puzzle completely! But then I realised that just because I had coloured the hardest possible cell, that didn’t necessarily mean I had solved every single cell.

I could use it to find an underestimate of the minimum possible solution time, which might be useful as a heuristic for an A*-like solution, but I didn’t want to go that direction.

Yours is an interesting idea: repeat the process after every move – it won’t necessarily find the best possible solution, but it is still a fruitful area to consider.

The thinking here is that in the process of going to the furthest away hex (taking into account cluster topology) you will make changes that include lots of other hexes, and you may not have to worry about any other line. It’s not an analytic solution, but if some of your movers make big changes, the recalculation of minimum distance will sort it out.

The calculations themselves are O(n^3) I think – minimum distance for a single cluster is O(n), and there are O(n) clusters to calculate, and you need O(n) steps… or possibly less, would that be O(logn)?

This algorithm would tend to grow the cluster towards the center early on, in the manner of Andrew D.

I think I did.

Firstly, I abstracted up from hex cells to clusters. Two hex cells of the same colour can be treated as one cluster, and reasoned about at that level. If your aim is to absorb the largest number of hex cells to improve your reach into untouched cells, then you should first realise touching two adjacent cells of the same colour doesn’t extent your coverage – what matters is the largest number of clusters.

Once that abstraction was made, then the Most Neighbours sub-metric looks for the move that covers the biggest number of clusters – which I think maps to the same basic conceptual approach as “biggest number of hexes”.

I think it is supposed to be reminiscent of how a virus grows, but I suspect the authors are confusing virus with amoebae.

Maybe it’s trying to be reminiscent of how a virus spreads, with each hex being a larger organism which is susceptible to infection from a certain strain of the virus, signified by its color. You get to control how the virus mutates, and your goal is to infect the entire population of larger organisms in as few mutations as possible. (Maybe the name of the game should be “Contagion” rather than “Virus”?)