Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

I realized chess pieces can be redesigned to be geometric attack directions

twitter.com

Post reply on HN
Print view View original

121–130 of 269 posts

Re: I realized chess pieces can be redesigned to be geometric attack directions

#121

Post by mananaysiempre »

godelski wrote: post #115
ComplexSystems wrote: post #104

Earlier quoted context omitted.

You mean a 2-square Chebyshev distance, I think (the Linf norm). A 2-square Manhattan distance (the L1 norm) wouldn't include any of the squares that a knight can get to.

For those that want a visual description there's a blog I serendipitous recently ran into[0]. If you're too lazy to open: - n-square Chebyshev distance forms concentric squares around your starting point - n-square Manhattan Distance forms diamonds (meaning up/down/left/right = 1 space away but all diagonals are 2 spaces away) We could also say the Knight can go everywhere a queen can't in the 3-square Manhattan dist…

That's very much a picture that you learn on the usual mathematics route of learning about these things, yes. More generally, there's a fair amount to be said about how you can derive shapes of unit balls from distance functions and vice versa (in functional analysis in particular).

Re: I realized chess pieces can be redesigned to be geometric attack directions

#124

Post by godelski »

mananaysiempre wrote: post #121
godelski wrote: post #115

Earlier quoted context omitted.

For those that want a visual description there's a blog I serendipitous recently ran into[0]. If you're too lazy to open: - n-square Chebyshev distance forms concentric squares around your starting point - n-square Manhattan Distance forms diamonds (meaning up/down/left/right = 1 space away but all diagonals are 2 spaces away) We could also say the Knight can go everywhere a queen can't in the 3-square Manhattan dist…

That's very much a picture that you learn on the usual mathematics route of learning about these things, yes. More generally, there's a fair amount to be said about how you can derive shapes of unit balls from distance functions and vice versa (in functional analysis in particular).

I think, unfortunately, not many get visual descriptions when learning about these topics. To be fair, these visualizations can harm one's intuitions about how the metrics work in higher dimensions, so I can understand arguments against visualization. But in another example, I think even many people can come out of a Linear Algebra class and not understand that a matrix operating on a vector always performs an affine transformation on that vector (yes, I know we can abuse and get other forms but let's keep general due to context).

I think this is something math education can improve on since we are highly visual creatures. But it is always a tough balance since that same tool to help learning can decrease generalization if internalized improperly. But that can also be said about anything and is why I dislike the common test focused paradigm of education. As I think we've been doing it long enough that students have found studying for the test -- as opposed to study for learning -- is optimal. Goodhart always wins lol

Re: I realized chess pieces can be redesigned to be geometric attack directions

#125

Post by basscomm »

paulddraper wrote: post #107
zelphirkalt wrote: post #94

Earlier quoted context omitted.

Most people think of the knight move as one straight and then one diagonal. However, this is still ambiguous. Imagine one up, then one down left for example, resulting in the square left to the original square. A better description is moving 2 squares straight, not diagonal, into one direction, then moving 1 square into a 90° turned direction.

Most people think of the knight moving 2 and then 1.

I was taught that a knight moves by going to the opposite corner of a 3x2 rectangle

Re: I realized chess pieces can be redesigned to be geometric attack directions

#126

Post by th4tg41 »

paulddraper wrote: post #109
catapart wrote: post #8

That is very clever! But I hate the way the pawns face. Makes sense! But it still reads, to me, like one of those "color, but the text is in a different color" things. Feels like the pawns should be facing the opposite direction. =/

Yeah, I wonder if it would be a slight improvement to combine move+capture. So pawns would be /|\ /|\ \|/ \|/ which looks less anti-directional.

Also visualizes en-passant in a better way.

Re: I realized chess pieces can be redesigned to be geometric attack directions

#128

Post by dheera »

basscomm wrote: post #125
paulddraper wrote: post #107

Earlier quoted context omitted.

Most people think of the knight moving 2 and then 1.

I was taught that a knight moves by going to the opposite corner of a 3x2 rectangle

Both western and Chinese chess knights move to the opposite corner of a 3x2 rectangle.

However in western chess, it doesn't matter what pieces are in the other 4 locations of the 3x2 rectangle.

In Chinese chess, you can only move from A to B if X is unoccupied. The locations marked "-" don't matter if they are occupied or not.

    A X -
    - - B

    A -
    X -
    - B

    - - B
    A X -

    - A
    - X
    B -
etc.

Of course, all the other pieces have their own rules as well, so it's a significantly different game. There is no queen, the elephant is sort of like a bishop but must move 2 squares at a time, the cannon moves horizontally and vertically but cannot kill without having exactly one (not zero) piece in-between, the "guards" and king all cannot leave the king's palace which is a 3x3 box, the pawns can move sideways after they cross the river in the middle of the board, etc. Come to think of it the only piece that has the exact same rules are the 2 "chariots" which are positioned in the same place as western rooks and function exactly like western rooks.

Post reply on HN