|Witchcraft and sorcery!|
Unlike just about every other game before, Civ V is now played on a hex grid, not square.
Yup. Sit down for a moment and let the ramifications sink in.
Actually, the change is wonderful. Perhaps most significantly, it eliminates that silliness where if you have a troop moving straight north into an unexplored plain, it's always more efficient to zig-zag rather than go straight. That's actually part of a larger problem: The distances get all screwed up in square grids. Going back to the movement thing I just mentioned, in a square grid it takes 1 movement point to go diagonally and one to move horizontally or vertically. However, the latter move covers 1 square, whereas the former move covers √12+12 =1.41 squares. Moving diagonally, you end up moving almost 3 squares for the price of 2. Same happens for things like ranged attacks (bombardment and airplanes in Civ IV and before) and city radii. A hex grid neatly fixes a lot of these issues.
Of course, there are still opportunities for strange micromanagement gains from funky movement patterns. Let's take a typical situation: You just started the game and are exploring with your warrior unit. It has 2 movement points, but moving over difficult terrain like forest and hills costs double. However, if you have moved 1 tile, and have only 1 point left, you will still be able to move to the tile which would normally take 2 points to reach.
It's not really a big deal in the grand scheme of things, I don't even know (or care) if the AI takes advantage of this. But it's just... weird. Fairly easy to solve, too. Why not just remember the move so that next turn the unit starts with 1 point spent? Eh, maybe it's hard for the AI to understand that way or something. Like I said, though, this probably doesn't even matter if you don't have OCD.
Second big benefit of hexes is aesthetics. Maybe it's because there's now 3 possible angles as opposed to 2, or maybe it's because you can't make a right angle with hexes, but the terrain looks much more organic and natural. Civs, for a long time, had been doing all sorts of clever tricks with tile graphics to make them look less square-y but what the hexes can do is just miles ahead. Plus, I'm sure the artists are glad for all the work they've been saved.
Hexes work so well for this game, I don't even know why they didn't just use them from the beginning with CivI. It's not like it makes things more complicated pathfinding-wise, there were lots of games from that era, and before, that used hexes successfully. Maybe it was an ill-conceived attempt to differentiate CivI from its competitors or something... But in any case, I'm glad we've left that behind now.