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Topic: square to pentomino (Read 869 times) 

JocK
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square to pentomino
« on: Feb 5^{th}, 2005, 2:39pm » 
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Pentominos are the connected planar rectangular shapes that originate when arranging five squares of equal size such that each square has at least one side coincident with another. Can you cut a square into pieces such that the pieces can be arranged* into any pentomino (of the same area as the square)? What is the minimum number of pieces you need? * translations, rotations and flipping of the pieces are all allowed

« Last Edit: Feb 6^{th}, 2005, 2:04am by JocK » 
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x^{y}  y = x^{5}  y^{4}  y^{3} = 20; x>0, y>0.



rmsgrey
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Re: square to pentomino
« Reply #1 on: Feb 5^{th}, 2005, 6:20pm » 
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on Feb 5^{th}, 2005, 2:39pm, JocK wrote:Pentominos are the rectangular shapes that originate when arranging five squares of equal size such that each square has at least one side coincident with another. 
 There's only one rectangular pentomino... Without the rectangular requirement, your definition includes all the shapes you can make by randomly placing a domino and a tromino in the plane,as well as various nonplanar shapes. An alternate way of defining them is: "the shapes you can make by picking 5 orthogonally connected squares on a chessboard"


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SWF
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Re: square to pentomino
« Reply #2 on: Feb 5^{th}, 2005, 7:17pm » 
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The pentomino in the shape of a "+" cannot be made from a domino and a triomino. One easy way, but perhaps not with the minimum piece count: Cut out two square pieces and a domino. The remainder easily be can be cut into pieces that form a rectangle. The rectangle is pretty simple to cut so it can form a square. The 3 squares and a domino can make all the required shapes..


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rmsgrey
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Re: square to pentomino
« Reply #3 on: Feb 5^{th}, 2005, 8:04pm » 
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on Feb 5^{th}, 2005, 7:17pm, SWF wrote:The pentomino in the shape of a "+" cannot be made from a domino and a triomino. 
 on Feb 5^{th}, 2005, 6:20pm, rmsgrey wrote: Without the rectangular requirement, your definition includes all the shapes you can make by randomly placing a domino and a tromino in the plane, as well as various nonplanar shapes. 
 So how many planar shapes which are not pentominoes fit JocK's definition but cannot be made up of a domino and a tromino?


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JocK
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Re: square to pentomino
« Reply #4 on: Feb 6^{th}, 2005, 1:53am » 
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on Feb 5^{th}, 2005, 6:20pm, rmsgrey wrote: There's only one rectangular pentomino... Without the rectangular requirement, your definition includes all the shapes you can make by randomly placing a domino and a tromino in the plane,as well as various nonplanar shapes. 
 OK, OK, ... Will update the 'definition'... on Feb 5^{th}, 2005, 6:20pm, rmsgrey wrote: An alternate way of defining them is: "the shapes you can make by picking 5 orthogonally connected squares on a chessboard" 
 Are you sure that if I use this definition here I will not be asked the question "Define chessboard" ... ?

« Last Edit: Feb 6^{th}, 2005, 1:57am by JocK » 
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solving abstract problems is like sex: it may occasionally have some practical use, but that is not why we do it.
x^{y}  y = x^{5}  y^{4}  y^{3} = 20; x>0, y>0.



Grimbal
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Re: square to pentomino
« Reply #5 on: Feb 7^{th}, 2005, 2:38am » 
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Whatever the definition, here is a list of all possible pentominoes. http://www.theory.csc.uvic.ca/~cos/inf/misc/PentInfo.html It is already an interesting question how to cut a square in a minimum number of pieces to make each of the pentominoes individually. But here, it seems we have to come up with a single set that will recombine into any one of the pentominoes. I have a solution in 5 pieces. :: ##################### #__####____________## #______####_______#_# #__________####__#__# #______________##___# ###____________#____# #__####_______#_____# #__#___####__#______# #_#________####_____# ##_____________####_# ##################### ::

« Last Edit: Feb 7^{th}, 2005, 2:39am by Grimbal » 
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JocK
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Re: square to pentomino
« Reply #6 on: Feb 7^{th}, 2005, 3:24pm » 
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If this works it beats my own solution.... But... are you sure you can create all 12 pentominos from these five pieces? How would you rearrange the five pieces into a 5x1 bar?


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solving abstract problems is like sex: it may occasionally have some practical use, but that is not why we do it.
x^{y}  y = x^{5}  y^{4}  y^{3} = 20; x>0, y>0.



Grimbal
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Re: square to pentomino
« Reply #7 on: Feb 8^{th}, 2005, 1:07am » 
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Oops.... Don't you want me to do the W instead? Actually the + would also be a problem.


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SWF
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Excuse me, rmsgrey, I thought you were suggesting a clue to the solution rather than pointing out an error in the phrasing of the question. The following uses 7 pieces (pieces of the same color pair up to form either a domino or a square, 3 squares and a domino can make any pentomino):


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JocK
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Re: square to pentomino
« Reply #9 on: Feb 9^{th}, 2005, 3:03pm » 
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Yes, cutting two of Grimbal's pieces does the job. As long as Barukh doesn't post a solution with fewer pieces this is the optimum. Well done SWF.


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solving abstract problems is like sex: it may occasionally have some practical use, but that is not why we do it.
x^{y}  y = x^{5}  y^{4}  y^{3} = 20; x>0, y>0.



