Revisions to Chess Knight minimum moves to destination on an infinite board

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edit approved Jan 27, 2019 at 21:54

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There is a closed form solution for finding the minimum number of moves the chess knight needs to move a specified displacement on the infinite chess board. Let $g$ be the requisite displacement expressed as a Gaussian integer; the real part of $g$ will be the horizontal displacement, and the imaginary part of $g$ will be the vertical displacement. Then we may write

$$g = ((1-i)g+(2-i)d)(2+i) - (g+(2+i)d)(2-i). $$

Here, $d$ can be any Gaussian integer; the value of $d$ which minimizes the number of knight moves is $d=Cint((2i-5)g/10)$, where $Cint$ is the closest Gaussian integer of the argument. The first term yields the requisite counterclockwise moves, and the second term yields the requisite clockwise moves. The real and imaginary parts of the counterclockwise and clockwise coefficients together yield the total minimum requisite moves of the chess knight, while simultaneously specifying all minimal paths.

N.B. Reply to amass.jack

I am an originator of this formula. I do not yet know of any prior publication. However, I have generated lecture notes for my talk at Acacia Creek to be given on the third Wednesday in February, as well as additional notes with numerous worked examples. These notes set forth the fundamental theory providing a foundation for the formula. Robert Word, Ph.D.

I have encountered difficulties in activating the present interface to post a reply to you, for unclear reasons. Hence, I copied the same reply into a number of related Facebook groups dealing with Mathematics, in case you might happen across them.

There is a closed form solution for finding the minimum number of moves the chess knight needs to move a specified displacement on the infinite chess board. Let $g$ be the requisite displacement expressed as a Gaussian integer; the real part of $g$ will be the horizontal displacement, and the imaginary part of $g$ will be the vertical displacement. Then we may write

$$g = ((1-i)g+(2-i)d)(2+i) - (g+(2+i)d)(2-i). $$

Here, $d$ can be any Gaussian integer; the value of $d$ which minimizes the number of knight moves is $d=Cint((2i-5)g/10)$, where $Cint$ is the closest Gaussian integer of the argument. The first term yields the requisite counterclockwise moves, and the second term yields the requisite clockwise moves. The real and imaginary parts of the counterclockwise and clockwise coefficients together yield the total minimum requisite moves of the chess knight, while simultaneously specifying all minimal paths.

There is a closed form solution for finding the minimum number of moves the chess knight needs to move a specified displacement on the infinite chess board. Let $g$ be the requisite displacement expressed as a Gaussian integer; the real part of $g$ will be the horizontal displacement, and the imaginary part of $g$ will be the vertical displacement. Then we may write

$$g = ((1-i)g+(2-i)d)(2+i) - (g+(2+i)d)(2-i). $$

Here, $d$ can be any Gaussian integer; the value of $d$ which minimizes the number of knight moves is $d=Cint((2i-5)g/10)$, where $Cint$ is the closest Gaussian integer of the argument. The first term yields the requisite counterclockwise moves, and the second term yields the requisite clockwise moves. The real and imaginary parts of the counterclockwise and clockwise coefficients together yield the total minimum requisite moves of the chess knight, while simultaneously specifying all minimal paths.

N.B. Reply to amass.jack

I am an originator of this formula. I do not yet know of any prior publication. However, I have generated lecture notes for my talk at Acacia Creek to be given on the third Wednesday in February, as well as additional notes with numerous worked examples. These notes set forth the fundamental theory providing a foundation for the formula. Robert Word, Ph.D.

I have encountered difficulties in activating the present interface to post a reply to you, for unclear reasons. Hence, I copied the same reply into a number of related Facebook groups dealing with Mathematics, in case you might happen across them.

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edited Jan 27, 2019 at 5:01

Yuval Filmus

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There is a closed form solution for finding the minimum number of moves the chess knight needs to move a specified displacement on the infinite chess board. Let g$g$ be the requisite displacement expressed as a Gaussian integer; the real part of g$g$ will be the horizontal displacement, and the imaginary part of g$g$ will be the vertical displacement. Then we may write

g = ((1-i)g+(2-i)d)(2+i) - (g+(2+i)d)(2-i).$$g = ((1-i)g+(2-i)d)(2+i) - (g+(2+i)d)(2-i). $$

Here, d$d$ can be any Gaussian integer; the value of d$d$ which minimizes the number of knight moves is d=Cint((2i-5)g/10)$d=Cint((2i-5)g/10)$, where Cint$Cint$ is the closest Gaussian integer of the argument. The first term yields the requisite counterclockwise moves, and the second term yields the requisite clockwise moves. The real and imaginary parts of the counterclockwise and clockwise coefficients together yield the total minimum requisite moves of the chess knight, while simultaneously specifying all minimal paths.

There is a closed form solution for finding the minimum number of moves the chess knight needs to move a specified displacement on the infinite chess board. Let g be the requisite displacement expressed as a Gaussian integer; the real part of g will be the horizontal displacement, and the imaginary part of g will be the vertical displacement. Then we may write

g = ((1-i)g+(2-i)d)(2+i) - (g+(2+i)d)(2-i).

Here, d can be any Gaussian integer; the value of d which minimizes the number of knight moves is d=Cint((2i-5)g/10), where Cint is the closest Gaussian integer of the argument. The first term yields the requisite counterclockwise moves, and the second term yields the requisite clockwise moves. The real and imaginary parts of the counterclockwise and clockwise coefficients together yield the total minimum requisite moves of the chess knight, while simultaneously specifying all minimal paths.

There is a closed form solution for finding the minimum number of moves the chess knight needs to move a specified displacement on the infinite chess board. Let $g$ be the requisite displacement expressed as a Gaussian integer; the real part of $g$ will be the horizontal displacement, and the imaginary part of $g$ will be the vertical displacement. Then we may write

$$g = ((1-i)g+(2-i)d)(2+i) - (g+(2+i)d)(2-i). $$

Here, $d$ can be any Gaussian integer; the value of $d$ which minimizes the number of knight moves is $d=Cint((2i-5)g/10)$, where $Cint$ is the closest Gaussian integer of the argument. The first term yields the requisite counterclockwise moves, and the second term yields the requisite clockwise moves. The real and imaginary parts of the counterclockwise and clockwise coefficients together yield the total minimum requisite moves of the chess knight, while simultaneously specifying all minimal paths.

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created Jan 26, 2019 at 19:13

Robert Wordar

53
1
2

There is a closed form solution for finding the minimum number of moves the chess knight needs to move a specified displacement on the infinite chess board. Let g be the requisite displacement expressed as a Gaussian integer; the real part of g will be the horizontal displacement, and the imaginary part of g will be the vertical displacement. Then we may write

g = ((1-i)g+(2-i)d)(2+i) - (g+(2+i)d)(2-i).

Here, d can be any Gaussian integer; the value of d which minimizes the number of knight moves is d=Cint((2i-5)g/10), where Cint is the closest Gaussian integer of the argument. The first term yields the requisite counterclockwise moves, and the second term yields the requisite clockwise moves. The real and imaginary parts of the counterclockwise and clockwise coefficients together yield the total minimum requisite moves of the chess knight, while simultaneously specifying all minimal paths.

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