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mrmitchell
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Zero got it right--a moderator.
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I think a remember a combinatoric solution to this
I have to go to sleep now though.
Might post tomorrow if I have time.
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One_Brow
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quote:
Originally posted by Zero-Tau The following solitaire game is played on an m by n rectangular board:
First, a rook is placed on some square. At each move, the rook can be moved any number of squares horizontally or vertically, with the extra condition that each move has to be made in the 90 degree clockwise direction compared to the previous one (e.g. after a move to the left, the next one has to be done upwards, the next one to the right etc). For which values of m and n is it possible that the rook visits every square of the board exactly once and returns to the first square? (The rook is considered to visit only those squares it stops on, and not the ones it steps over.) |
Outside of 1x1, neither m nor n can be odd. Every time a row is entered from a column, exactly two squares of the row are marked off, and the same applies to columns.
2x2 is trivial.
2xn, where n is even and greater than 2:
Move to second column of first row, then up one. continue in pattern left one, down one, left one, up one until you reach the end of the row, than move to the first column and down one.
If 2xn is solvable, than you can solve mxn (m even) by moving from the first column to the second column, moving up to the third row, filling in the third and fourth rows using the pattern of the 2xn, with the roles of first and second column reversed. This will end in the second column of the fourth row. Move up to fifth row, repeat as needed. Finally, move down to the second column of the second row after you run our of higher rows, and complete the 2xn pattern.
The answer is: m and n are both 1 or both even.
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One_Brow
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I think we've already seen teh problem where you have a 3 and 5 gallon jog, but need exactly 4 gallons of water.
Let's say you have jugs of size x and y gallons, with no internal markings. You can get any number of gallons between 1 and x+y using these two jugs, by filling and pouring out water as needed.
What is the key property of x and y?
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Zero-Tau
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Elsewhere
Aug 2002
time: 06:23
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quote:
Originally posted by One_Brow
2xn, where n is even and greater than 2:
Move to second column of first row, then up one. continue in pattern left one, down one, left one, up one until you reach the end of the row, than move to the first column and down one.
If 2xn is solvable, than you can solve mxn (m even) by moving from the first column to the second column, moving up to the third row, filling in the third and fourth rows using the pattern of the 2xn, with the roles of first and second column reversed. This will end in the second column of the fourth row. Move up to fifth row, repeat as needed. Finally, move down to the second column of the second row after you run our of higher rows, and complete the 2xn pattern.
The answer is: m and n are both 1 or both even. |
The answer is correct, but the method is not. You have to turn clockwise each time. You change between clockwise and counter-clockwise.
Oh yeah, and bonus question: does anyone know where I got this?
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Zero-Tau
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Elsewhere
Aug 2002
time: 06:23
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quote:
Originally posted by One_Brow
I think we've already seen teh problem where you have a 3 and 5 gallon jog, but need exactly 4 gallons of water.
Let's say you have jugs of size x and y gallons, with no internal markings. You can get any number of gallons between 1 and x+y using these two jugs, by filling and pouring out water as needed.
What is the key property of x and y? |
x and y are coprime (they have no common factors). If x and y had a common factor, then you could only get multiples of that factor by filling and pouring.
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One_Brow
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quote:
Originally posted by Zero-Tau
The answer is correct, but the method is not. You have to turn clockwise each time. You change between clockwise and counter-clockwise.
Oh yeah, and bonus question: does anyone know where I got this? |
My fault for not reading carefully. Still, that's a small change.
2xn: Start at bottom left. First moves are right to the last column, up one, left two, down one, right one, up one. Pattern of three left, one down, one right, one up until you reach the top of the third-from-left column. Then two left (and one down to return to staring square).
mxn: Start at bottom left. Follow the 2xn pattern using exclusively row 1 (the bottom row) and row m/2 + 1. This finishes with a leftt move into row m/2 +1, column 1. Move down to row 2, repeat in rows 2 and m/2 + 2. Continue repeating through rows m/2 and m, then down from row m to row 1 in column 1.
However, Zero-Tau has already answered my puzzle. so it's on him again.
I have no idea regarding your source.
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Zero-Tau
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Elsewhere
Aug 2002
time: 06:23
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A period symbol ( . ).
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mrmitchell
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I mean, I'm not insulting you with the head of cabbage reference, I'm insulting the man in your brain teaser. You, on the other hand, were able to provide an riddle that could stump me.
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mrmitchell
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Okay, I think this is one, already well known, I'm just seeing if 'Poly-ers know it:
When Bob leaves his 15th-floor apartment, he uses the elevator all the way to the 1st floor. However, when Bob goes back up, sometimes he uses the elevator all the way to the 15th floor and sometimes he rides the elevator only up to the 7th floor and walks the stairs the rest of the way up. Why?
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30-11-2002 21:21
)
(Basically you'd probably just have to be online to get my riddles right 
A nice twist on the problem. I didn't expect that. 
That's correct. 
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