Define an “m-n-knight” as a piece which moves m squares vertically, then n squares horizontally; or n squares vertically, then m squares horizontally; and can jump over all intervening pieces in doing so. m and n are each whole numbers in the range 0 through 7.
Observations:
- An m-n knight is the same thing as an n-m knight (commutativity).
- A 2-1-knight is just a conventional chess knight.
- Any m-n knight has at most 8 possible moves at any given time.
- An m-n knight may have as few as 0 possible moves, even on an empty board, but how did it get there?
Note 1: A 5-0 knight can move directly from b1 to g1, but unlike a rook, can jump over its own or enemy pieces in doing so.
Note 2: A 0-0 knight moves by simply having the player thump it, but it does not actually move to a different square. This can be useful in positions where the player wishes to lose a tempo (e.g. to gain the opposition or something similar).
Rule Variation 1: When promoting to an m-n knight, the player must state what m and n are. OR:
Rule Variation 2: When promoting to an m-n knight, the player must state what m and n are only when it makes its first move after promotion.
The second variation is interesting, because any threat to promote is also a mate threat. Just promote to an (undesignated) m-n knight, then on the next move, declare m and n to be the number of squares, vertically and horizontally respectively, that the m-n knight must move in order to land on top of the enemy king.
Hmm. Let’s go with Rule Variation 1.
Now define a root-RR knight as a piece that functions as an m-n knight for some m and n where m-squared plus n-squared equals RR. RR can be any whole number from 0 through 98 for which appropriate m and n can be found. (For example, a 7-7-knight is a root-98 knight.)
Note the Pythagorean formula here. A 2-1 knight is a root-5 knight because 2 squared plus 1 squared equals 5. To look at it another way, the distance from the center of the g1 square to the center of the f3 square is the square root of 5 (where 1 represents the distance between the centers of two horizontally or vertically adjacent squares).
Can a root-RR knight be an m-n knight for more than one pair m-n? As it turns out, there are only two possibilities. A root-25 knight can be either a 5-0 knight or a 4-3 knight. (Remember the famous 3-4-5 right triangle from 10th grade geometry?) And a root-50 knight can be either a 7-1 knight or a 5-5 knight. No other such “duals” exist.
Several years ago, one of the players in Helen Warren’s U.S. Masters tournament proposed a puzzle involving root-50 knights. (These could move either 7 and 1, or 5 and 5.) It was a mate in three or something like that. This conversation came up in the skittles room before the tournament started. I had showed up to soak up some master atmosphere. (Obviously, I was not qualified to enter the tournament.)
The following year, I again visited the skittles room of the U.S. Masters, and while milling around, I recognized a familiar voice. “Hey, I remember you, you’re the square root of 50 guy!” “Yes, I am the square root of 50 guy.” And he proceeded to show the same problem again, to anybody would listen.
I never did catch the name of that player. He had an Eastern European accent, and was probably an IM or something like that.
Would anybody out there have an inkling as to who this player might have been?
Bill Smythe