Tic-tac-toe with one mark type

Solution 1:

I dont have a general-case answer rather some insight regarding the c=3 case leading to a general conclusion: (In the c=3 case:)Given some tic already placed 'p', you lose if you place a tic 'q' in a place allowing the opponent to complete a 3-line with p & q. One notices quickly that all such tics lie within the Manhattan-distance circle of radius 2 centered around p. This means that if when you start playing one can place a tic everywhere, each tic placed limits the non-losing space with the area of a square (a Manhattan-distance circle is a square rotated). This means each player 'survives' their turn by placing another square inside the remaining area, so this is a classic Packing problem. Many variants exist such as circle packing (https://en.wikipedia.org/wiki/Circle_packing#:~:text=In%20geometry%2C%20circle%20packing%20is,enlarged%20without%20creating%20an%20overlap.) or square packing (https://en.wikipedia.org/wiki/Square_packing_in_a_square#:~:text=Square%20packing%20in%20a%20square%20is%20a%20packing%20problem%20where,wasted%20space%20for%20non%2Dinteger), both of which dont have a closed form for their packing capacity, this leads me to believe that, other than some specific cases, this generalization your searching for does not exist!

I think in cases c>3 one can express the area taken by a move similarly (much more complex but still an area taken-up) and thus some weird packing-problem. This leads me to believe no generalization exists at all for any c.

The specific cases are cool though. Cheers!

(Manhattan-distance AKA Taxicab distance : https://en.wikipedia.org/wiki/Taxicab_geometry)