In terms of gameplay, the most important two distances are 1) the shortest distance around a particular space (i.e., traveling along the spaces adjacent to that space), and 2) the distance to each adjacent space. The most intuitive, consistent conception of these distances begins with the idea that "around" something means, on average, "in a circle." In accordance with that idea, two key criteria arise: 1) that the shortest distance around a particular space should be consistent no matter which adjacent space you begin in, and 2) that the distance to each adjacent space should be 1.
Of all the systems listed above, only two of them meet both criteria. Those are the hex-based system and the 1-2-2-2 based system (although in the latter case, it's only so far as the "circle" encompasses only 1 square.)
Two of the systems fail to meet the first criterion. These are 1-1-1-1 and 1-2-1-2. In these three systems, it is harder to encircle an enemy if you start in a diagonally adjacent space than if you start in a non-diagonally adjacent space.
For example, in 1-1-1-1, it costs at least 5 squares to encircle an enemy when starting on a non-diagonally adjacent square and only 4 squares to encircle the enemy when starting on a diagonally adjacent square. Whatever people may think about some of the other 1-1-1-1 distortions, this is a distortion of the distance that matters the most out of all distances in gameplay. But it happens in 1-2-1-2 as well! It requires at least 7 squares when starting on a non-diagonally adjacent square, but only 6 squares when starting on a diagonally adjacent squares.
The 2-2-2-2 system does meet the first criterion, because it always takes 8 squares to move around the enemy. However, this system fails to meet the second criterion. (Incidentally, with 1-1-1-1 the problem is reversed. 1-1-1-1 meets the second criterion, but not the first.)
This leaves the hex-based system and the 1-2-2-2 system.
The 1-2-2-2 system, however, fails the first criteria if it is extended to beyond just the most immediate squares. Even if we let that slide, it is fair to bring other criteria into play now that there are only two systems left. The 1-2-2-2 system, for instance, requires inconsistent counting, which is certainly a heavy mark against it.
The hex-based system is somewhat unintuitive when it comes to drawing rectangular rooms (though such rooms can be consistently drawn, in addition to many other shapes of rooms, by putting intersections at the midpoints instead of the vertices). However, this sort of evaluation does not affect gameplay in any case. Just as one can imagine a firecube to be a fireball, one can imagine a hexagon to be a square.
Without practice using a hex-based system, it could be unintuitive for use with creatures larger than Large. However, by the time such creatures come into play, the DM will have had practice with the system. But even more importantly, this is where consistency in my two prime criteria becomes very beneficial: the DM can simply move the creature where it looks like the right position/distance, and there is a good chance that will be correct, or at least within an error less than that of other systems.
Thus, it seems that the hex-based system may actually be the most intuitive of all the systems when it comes to what matters most across all of them, on average.
Of all the systems listed above, only two of them meet both criteria. Those are the hex-based system and the 1-2-2-2 based system (although in the latter case, it's only so far as the "circle" encompasses only 1 square.)
Two of the systems fail to meet the first criterion. These are 1-1-1-1 and 1-2-1-2. In these three systems, it is harder to encircle an enemy if you start in a diagonally adjacent space than if you start in a non-diagonally adjacent space.
For example, in 1-1-1-1, it costs at least 5 squares to encircle an enemy when starting on a non-diagonally adjacent square and only 4 squares to encircle the enemy when starting on a diagonally adjacent square. Whatever people may think about some of the other 1-1-1-1 distortions, this is a distortion of the distance that matters the most out of all distances in gameplay. But it happens in 1-2-1-2 as well! It requires at least 7 squares when starting on a non-diagonally adjacent square, but only 6 squares when starting on a diagonally adjacent squares.
The 2-2-2-2 system does meet the first criterion, because it always takes 8 squares to move around the enemy. However, this system fails to meet the second criterion. (Incidentally, with 1-1-1-1 the problem is reversed. 1-1-1-1 meets the second criterion, but not the first.)
This leaves the hex-based system and the 1-2-2-2 system.
The 1-2-2-2 system, however, fails the first criteria if it is extended to beyond just the most immediate squares. Even if we let that slide, it is fair to bring other criteria into play now that there are only two systems left. The 1-2-2-2 system, for instance, requires inconsistent counting, which is certainly a heavy mark against it.
The hex-based system is somewhat unintuitive when it comes to drawing rectangular rooms (though such rooms can be consistently drawn, in addition to many other shapes of rooms, by putting intersections at the midpoints instead of the vertices). However, this sort of evaluation does not affect gameplay in any case. Just as one can imagine a firecube to be a fireball, one can imagine a hexagon to be a square.
Without practice using a hex-based system, it could be unintuitive for use with creatures larger than Large. However, by the time such creatures come into play, the DM will have had practice with the system. But even more importantly, this is where consistency in my two prime criteria becomes very beneficial: the DM can simply move the creature where it looks like the right position/distance, and there is a good chance that will be correct, or at least within an error less than that of other systems.
Thus, it seems that the hex-based system may actually be the most intuitive of all the systems when it comes to what matters most across all of them, on average.
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