A Rook on the square a8 of an empty 8x8 board could move to a1, a distance of seven squares. According to the rule, "half, rounded up", a Halfling Rook could move four squares in that direction, to a4; and next turn it could move only two squares in that direction, to a2; and so it would need three moves to reach a1. This example demonstrates a special weakness of Halfling pieces -- they have trouble reaching the edge of the board, and in fact cannot move to the edge of the board except by taking a one step move. Thus the new Chess proverb will be "When hounded by halflings, hide on the edge."
There are two possible kinds of Halfling pieces, relative halflings and absolute halflings.
A relative halfling White Rook on a8 could not capture a Black Pawn on a6, because a normal Rook would only be able to move two squares Southwards (and thereby capture the Pawn), and the Halfling can only move half that far. Although it would be futile to chase the Pawn, the Halfling Rook can intercept it if the rest of the board is empty, by moving to b8, b4, b2, and a2. Alas for the poor Halfling, the Pawn can capture it on b4; and so, the Pawn wins against the Halfling Rook in this miniscule endgame study.
An absolute Halfling Rook on a8 can simply capture the Pawn. Its range is half of what a normal Rook could move on an empty board, and so it could move as many as four squares to the South. Of course, even the absolute Halfling Rook cannot force mate in the K+R versus K endgame -- but at least the Halfling Q can do so.
All my examples and discussions from here on will be about absolute Halflings. Relative Halflings have a confusing move, that is, it's very easy to make mistakes and try to make illegal moves with them; and relative Halflings also are very weak, as they can capture only at short range.
All my examples and discussions will be about Halflings, as opposed
to Thirdlings and Quarterlings and Octolings and Duodecalings --
however, such pieces are possible, and can have interesting uses in
It is obvious that the rule of Halfling movement immediately defines
a game that must be named Halfling Chess. The Pawns cannot make a
two step advance, Knights still move as they always did (one step
Knightwise), the Rooks, Bishops, and Queens are Halflings of the
same type. Castling is evidently illegal. One could argue that
Castling is a special move of no distance, of course, but the judge
has ruled that Castling is illegal. Halfling Chess is obviously a
playable and enjoyable game, and one can safely predict that people
who like to play Shatranj or DemiChess will also like to play
Halfling Chess. It is equally obvious that Progressive
Chess and Avalanche Halfling and so on are playable games,
suit players of different tastes.
The limitation of distance may have particularly piquant effects in games such as Halfling Dynamo Chess, or Halfling Conversion Chess.
The Halfling Chess army is presumably decisively stronger than the DemiChess army, even though both armies try to be "half strength", and one reason is that the Halfling Knight is exactly the same as the normal Knight. Things can be somewhat evened up by using a CrabRider instead of the normal Demichess Crab, but the additive nature of piece power probably still leaves the Halflings decisively stronger.
The Halfling Nutty Knights will be stronger than the Halfling Fabulous FIDEs, because the short range pieces are not handicapped by their Halfling halving. The Halfling Colorbound Clobberers and the Halfling Remarkable Rookies also gain in strength relative to the FIDEs, though they do not gain as much as the Nutty Knights do. Finally, the Halfling Forward FIDEs are probably still a good match for the Halfling Fab FIDEs. (Knights versus Clobberers is probably also a good game.)
On a cylindrical board, the Rook can move an infinite distance East or West; and so, therefore, can the Halfling Rook.
On an 8x8 board, the empty board mobility of the Halfling Rook is precisely 8.0 because when it can move 4 spaces North, it cannot move South at all, and when 3 North is possible, the greatest Southing is always 1; and so on.
Allowing for the fact that the board is not always empty, as per http://www.chessvariants.com/d.betza/pieceval/betterway.html , we get a value of about 6.0 for the average real mobility of the Halfling Rook; and interpolating 0.77 (the ratio of 6.0 to 7.88) into the chart given in http://www.chessvariants.com/d.betza/chessvar/pieces/short rook.html we find that the Halfling Rook's mobility gives it a likely value that is midway between the value of a R3 and the value of a R4.
In other words, the numbers predict that a Halfling Rook has very nearly the same value as a FIDE Knight, which is worth something between 60 percent and two thirds of a FIDE Rook.
Where the Halfling Rook has 0.77 times as much empty board mobility as the FIDE Rook, the Halfling Bishop has a mere 0.6 times the empty board mobility of the FIDE B. However, a high percentage of its moves are short, and its mobility allowing for the presence of other pieces is proportional -- and so it seems safe to take a shortcut and say that a Halfling piece is generally worth 0.6 to 0.66 of the value of the corresponding non-Halfling piece.
Thus, the Halfling Queen should be worth a bit more than a Rook. These are theoretical values which have not been play-tested, but are likely to be reasonably accurate.
So far, I have presented a new and unprecedented rule of movement, a chess variant that uses it, several likely candidates for good matchups in different-armies games, and a good try at estimating the value of pieces using the new rule. That should be enough for one day.