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I think I've seen something similar in a game named Rotary, Does anyone remember? Nice idea!
Greg, the Lancer must move before rotating; it cannot rotate as a move by itself. The reason is to prevent the lancer from being too powerful, so choosing the direction after a move is obviously very important (one must be sure there are safe squares it could move to if attacked - or if it needs to turn around). I also wanted to prevent someone from turning the lancer as a way out of zugzwang positions (or to avoid stalemate in a lame way).
Aurelian, I hadn't heard of Rotary before but that looks interesting too. That one appears to have mostly sliding pieces. For this variant, I mostly wanted to create an alternate "jumping" piece other than the knights we're used to. It would create some different approaches to all phases of the game. The lancers probably lose a little value in open positions just like the knight, but I'm sure they can be used for some great tactics even late in the game (such as putting a lancer behind passed pawns- maybe even more useful than a rook in that regard)
I believe a single lancer and king cannot force checkmate, but two lancers could.
I've altered the original setup slightly. Lancers should begin the game facing diagonally inward to provide more squares for development. Otherwise, having them face straight forward makes fianchetto-style openings a rote necessity, and it could detract from the usual option to castle early for those who prefer it.
This would mean that the English and Sicilian openings would not make much sense any more, as enemy lancers can immediately capture those pawns. Nevetheless, I believe other openings become a possibility where they otherwise don't work. Traditionally unsound openings like King's Gambit might be more appealing with a lancer. Even an opening like the infamous 1. f3 could work nicely.
I don't understand why you say the Queen can take the Pawn. This would leave the King in check, wouldn't it?
Note that 'Centennial Chess' features 'Rotating Spearmen', which are somewhat similar to your Lancers. Except that they are unidirectional sliders, rather than hoppers, have a non-capture retreat move, and can only be oriented in 3 directions.
I think your proposed notation sucks. (But I don't consider notation conventions really part of a game.) I don't like it that the direction indication is absolute, i.e. that looking 'forward' is N for white, and S for black. I also don't like it that the separator 'T' is a capital like the direction specification itself. Why not stick to normal SAN, and write the re-orietation as a promotion move (La1=NE)? Strictly speaking it is a promotion move (changing one piece type for another, as a Lancer moving in a different direction is really a different piece type). Or better yet, La1=fr (forward right), reserving capitals for indicating piece types.
H.G. I respect your opinion on the notation. You raise a good point about capital letters reserved for pieces, and I'd be open to adopting the promotion-type notation.
The spearmen do look similar to these lancers, but as we've both mentioned it's an exploration into a different type of "jumper" to try.
Regarding your question about the queen, it would be captured regardless, so Black would want to go ahead and take what it can: the b2 pawn. Remember that after Black takes that pawn, the lancer is directly attacking the queen (not the king yet, per the rules about the first enemy piece in one direction) Nevertheless, I actually need to fix my example to have the White king on f1. Otherwise, Bb6+ and then Bd4 would guarantee a win.
UPDATE ON OCTOBER 9th: Here is the original sample position, which had a demonstrative flaw, that I refer to in this comment:
Nicolino, good question, and sorry if the rules were unclear - the lancer can only jump as far as the first enemy piece that it's facing. Assuming you were talking about the very first move for white, after Le4TNW, the lancer is only attacking the b7 pawn. Then, Black could respond with d5 and win the lancer.
Giving some credit to H.G. Muller, perhaps we could notate that first move you mentioned Le4=nw
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Can a Lancer turn without moving?