Enumerate the ways of putting six armies of queens on a humongous chessboard
$begingroup$
This is a sort of a sub-problem of the open puzzle Peaceful Encampments, for high numbers of armies.
Consider a chessboard with an astronomically large number of vanishingly small squares, on which we place six contiguous armies of queens — a red army, a green army, a blue army, a magenta army, a yellow army, and a cyan army — such that no queen is ever threatened by a queen of any other color. Here are four possible ways those armies could appear on the board:
Consider the upper-left image here. Scan it from left to right; you encounter the armies in the order "G-R-B-M-C-Y". Scan it from top to bottom: "M-R-G-Y-C-B". Scan it slashwards from lower left to upper right: "B-G-R-C-Y-M". Scan it backslashwards from upper left to lower right: "G-R-M-B-C-Y".
I claim that these four scans uniquely identify this particular arrangement of armies. So the four arrangements in the picture above can be summarized as:
Upper left: GRBMCY MRGYCB BGRCYM GRMBCY.
Upper right: GRBMCY MRYGCB BGRCYM RGMBYC.
Lower left: RGBMYC RMGYBC BGRCMY RGMBYC.
Lower right: RGBMYC MRYGCB BGRCYM RMGYBC.
So my combinatorial puzzle is:
How many different ways are there to arrange six armies on the board?
It would be reasonable to rearrange the colors so that the encoding always begins with, say, RGBCMY. I was a bit too lazy to do that when making these diagrams. But if we do that, we get these four encodings:
Upper left: RGBCMY CGRYMB BRGMYC RGCBMY.
Upper right: RGBCMY CGYRMB BRGMYC GRCBYM.
Lower left: RGBCMY RCGMBY BGRYCM RGCBMY.
Lower right: RGBCMY CRMGYB BGRYMC RCGMBY.
geometry combinatorics chess
asked 7 mins ago
QuuxplusoneQuuxplusone
232110
$endgroup$
add a comment |
$begingroup$
This is a sort of a sub-problem of the open puzzle Peaceful Encampments, for high numbers of armies.
Consider a chessboard with an astronomically large number of vanishingly small squares, on which we place six contiguous armies of queens — a red army, a green army, a blue army, a magenta army, a yellow army, and a cyan army — such that no queen is ever threatened by a queen of any other color. Here are four possible ways those armies could appear on the board:
Consider the upper-left image here. Scan it from left to right; you encounter the armies in the order "G-R-B-M-C-Y". Scan it from top to bottom: "M-R-G-Y-C-B". Scan it slashwards from lower left to upper right: "B-G-R-C-Y-M". Scan it backslashwards from upper left to lower right: "G-R-M-B-C-Y".
I claim that these four scans uniquely identify this particular arrangement of armies. So the four arrangements in the picture above can be summarized as:
Upper left: GRBMCY MRGYCB BGRCYM GRMBCY.
Upper right: GRBMCY MRYGCB BGRCYM RGMBYC.
Lower left: RGBMYC RMGYBC BGRCMY RGMBYC.
Lower right: RGBMYC MRYGCB BGRCYM RMGYBC.
So my combinatorial puzzle is:
How many different ways are there to arrange six armies on the board?
It would be reasonable to rearrange the colors so that the encoding always begins with, say, RGBCMY. I was a bit too lazy to do that when making these diagrams. But if we do that, we get these four encodings:
Upper left: RGBCMY CGRYMB BRGMYC RGCBMY.
Upper right: RGBCMY CGYRMB BRGMYC GRCBYM.
Lower left: RGBCMY RCGMBY BGRYCM RGCBMY.
Lower right: RGBCMY CRMGYB BGRYMC RCGMBY.
geometry combinatorics chess
asked 7 mins ago
QuuxplusoneQuuxplusone
232110
$endgroup$
add a comment |
$begingroup$
This is a sort of a sub-problem of the open puzzle Peaceful Encampments, for high numbers of armies.
Consider a chessboard with an astronomically large number of vanishingly small squares, on which we place six contiguous armies of queens — a red army, a green army, a blue army, a magenta army, a yellow army, and a cyan army — such that no queen is ever threatened by a queen of any other color. Here are four possible ways those armies could appear on the board:
Consider the upper-left image here. Scan it from left to right; you encounter the armies in the order "G-R-B-M-C-Y". Scan it from top to bottom: "M-R-G-Y-C-B". Scan it slashwards from lower left to upper right: "B-G-R-C-Y-M". Scan it backslashwards from upper left to lower right: "G-R-M-B-C-Y".
I claim that these four scans uniquely identify this particular arrangement of armies. So the four arrangements in the picture above can be summarized as:
Upper left: GRBMCY MRGYCB BGRCYM GRMBCY.
Upper right: GRBMCY MRYGCB BGRCYM RGMBYC.
Lower left: RGBMYC RMGYBC BGRCMY RGMBYC.
Lower right: RGBMYC MRYGCB BGRCYM RMGYBC.
So my combinatorial puzzle is:
How many different ways are there to arrange six armies on the board?
It would be reasonable to rearrange the colors so that the encoding always begins with, say, RGBCMY. I was a bit too lazy to do that when making these diagrams. But if we do that, we get these four encodings:
Upper left: RGBCMY CGRYMB BRGMYC RGCBMY.
Upper right: RGBCMY CGYRMB BRGMYC GRCBYM.
Lower left: RGBCMY RCGMBY BGRYCM RGCBMY.
Lower right: RGBCMY CRMGYB BGRYMC RCGMBY.
geometry combinatorics chess
asked 7 mins ago
QuuxplusoneQuuxplusone
232110
$endgroup$
This is a sort of a sub-problem of the open puzzle Peaceful Encampments, for high numbers of armies.
Consider a chessboard with an astronomically large number of vanishingly small squares, on which we place six contiguous armies of queens — a red army, a green army, a blue army, a magenta army, a yellow army, and a cyan army — such that no queen is ever threatened by a queen of any other color. Here are four possible ways those armies could appear on the board:
Consider the upper-left image here. Scan it from left to right; you encounter the armies in the order "G-R-B-M-C-Y". Scan it from top to bottom: "M-R-G-Y-C-B". Scan it slashwards from lower left to upper right: "B-G-R-C-Y-M". Scan it backslashwards from upper left to lower right: "G-R-M-B-C-Y".
I claim that these four scans uniquely identify this particular arrangement of armies. So the four arrangements in the picture above can be summarized as:
Upper left: GRBMCY MRGYCB BGRCYM GRMBCY.
Upper right: GRBMCY MRYGCB BGRCYM RGMBYC.
Lower left: RGBMYC RMGYBC BGRCMY RGMBYC.
Lower right: RGBMYC MRYGCB BGRCYM RMGYBC.
So my combinatorial puzzle is:
How many different ways are there to arrange six armies on the board?
It would be reasonable to rearrange the colors so that the encoding always begins with, say, RGBCMY. I was a bit too lazy to do that when making these diagrams. But if we do that, we get these four encodings:
Upper left: RGBCMY CGRYMB BRGMYC RGCBMY.
Upper right: RGBCMY CGYRMB BRGMYC GRCBYM.
Lower left: RGBCMY RCGMBY BGRYCM RGCBMY.
Lower right: RGBCMY CRMGYB BGRYMC RCGMBY.
geometry combinatorics chess
geometry combinatorics chess
asked 7 mins ago
QuuxplusoneQuuxplusone
232110
asked 7 mins ago
QuuxplusoneQuuxplusone
232110
asked 7 mins ago
QuuxplusoneQuuxplusone
232110
asked 7 mins ago
QuuxplusoneQuuxplusone
232110
asked 7 mins ago
QuuxplusoneQuuxplusone
232110
232110
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