Henry Baker wrote:

Are there any examples of an obviously discrete game -- e.g., tic-tac-toe, checkers, chess, go, etc. -- that can be converted into a continuous game ?

It seems fairly obvious roughly how you'd make go continuous. 1. Playing field is some compact 2-manifold with boundary. (A square region in the plane would do.) 2. Players take turns placing one piece of their colour on the playing field. (For our purposes, a "piece" occupies a single point.) A player also has the option of passing. 3. Each piece must be at distance >= 1 from all others. 4a. Pieces at distance exactly 1 and of the same colour are "directly connected". A "group" is a maximal set of pieces any two of which can be linked by a chain of direct connections. OR, probably better, at least for noneuclidean playing fields, 4b. Pieces at distance <= 1+h (for some suitably chosen h, maybe somewhere around 1/5 or maybe much smaller) and of the same colour are "directly connected". A "group" is a maximal set of pieces any two of which can be linked by a chain of direct connections. OR, perhaps, 4c. Pieces at distance <= 1+h (for some suitably chosen h, maybe somewhere around 1/5) and of the same colour are "directly connected". A "group" is a maximal set of pieces any two of which can be linked by a chain of direct connections whose total length is at most the number of links plus some extra (small, slower-growing) term. (The idea here is that basically your pieces should be 1 unit apart, but you get a small amount of slack and you should keep your use of it small in case the group grows larger.) 5. A group for which no point exists that's legally playable (under rule 3, ignoring this rule and rule 7) and directly connected to a piece in the group is "dead".If you play a move that renders one or more of your opponents' groups dead, the pieces of those groups are immediately removed from the board. You may not play a move that, after removing opponents' dead groups, leaves you with anything dead. 6. There is no rule 6. 7. You may not play a move that recreates a previous position in the following sense: (a) the same player is to move, (b) you can pair up pieces between old and new positions such that each piece's new place is within 1/2 unit of its old place and the belongs-to-same-group relation is the same for new as for old. 8. Play stops when two consecutive passes occur. 9. At the end of the game, the winner is the player with more stones on the field. In real games of go, one counts "territory plus prisoners" but then one has to have rules like "at the end, remove any dead stones". It turns out that (discrete versions of) the nice clear-cut rules above are approximately equivalent to this for an ordinary discrete square board. You could easily make territory-based continuous rules; for actual equivalence with the above, the value of a territory would have to be something like the number of stones one could cram into it (which would be roughly proportional to its area for territories with no thin bits) but there would be complications involving "sacrificial" stone placement by the opponent. It turns out, unsurprisingly, that this has been considered before; see, e.g., http://www.di.fc.ul.pt/~jpn/gv/boards.htm . -- g