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Topic: Tetraminoes On A Checkerboard (Read 5066 times) |
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william wu
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Tetraminoes On A Checkerboard
« on: Nov 17th, 2002, 8:39pm » |
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A sequel to the tiling problem "Dominoes On A Checkerboard".
Shown immediately below is a checkerboard mutilated by the removal of two squares from each of two opposite corners.
Shown immediately below are two T-shaped tiles each of which can cover four squares of the checkerboard exactly.
If tiles of both kinds are abundant, and if tiles may be rotated, can the mutilated checkerboard be covered exactly with non-overlapping tiles that match the colors of covered squares? Why?
P.S. If there's a more official name for the "T" shapes than tetramino, feel free to let me know. The word tetramino is used by Tetris fans to describe the shapes in the game of Tetris (J-tetramino, T-tetramino, L-tetramino, etc). I don't know if the mathematics community agrees.
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| « Last Edit: Nov 17th, 2002, 9:04pm by william wu » |
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towr
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Re: Tetraminoes On A Checkerboard
« Reply #1 on: Nov 18th, 2002, 1:04am » |
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there's an equal number of blue and white square, so there must be equally many of both kinds of 'tetramino'
but there are only 15 * 4 squares left, so there can't be equal numbers of each kind (since they are 4 squares)
It'd been harder to find if you didn't explicitly state there are two kinds..
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| « Last Edit: Nov 18th, 2002, 3:40am by william wu » |
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Chronos
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Re: Tetraminoes On A Checkerboard
« Reply #2 on: Nov 24th, 2002, 7:13pm » |
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Of course, you could remove all reference to color in the original puzzle. The same solution still works, although the solver must put the colors back himself.
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titan
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Re: Tetraminoes On A Checkerboard
« Reply #3 on: Oct 13th, 2013, 2:53am » |
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Can someone pls explain what's going on in the ques.? It's not clear to me. What is the ques. asking for?
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towr
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Re: Tetraminoes On A Checkerboard
« Reply #4 on: Oct 13th, 2013, 8:06am » |
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Try to cut the shape on the top into ones the size of the other two without anything left over.
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titan
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Re: Tetraminoes On A Checkerboard
« Reply #5 on: Oct 13th, 2013, 8:28am » |
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We need to cut the bigger figure into 2 pieces which look like the other two. r8?
And theargument is that since 60 boxes are left and halves mean 30 each. But, for a figure to look the other two Ts, it shud have no. of boxes in a multiple of 4. So, not possible!
Correct?
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titan
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Re: Tetraminoes On A Checkerboard
« Reply #6 on: Oct 13th, 2013, 8:48am » |
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Ok! ok! Got it!!
Since 60 squares (not boxes) have equal no. of blue n while squares, so, we need to use both kind of Ts. Since, if I use 1st T only it will add while and blue in a ratio which is not 1:1.
So, I use equal no. of both the Ts.
So, no. of blue squares=4x and similarly for while ones. But, we have 30 blue squares n 4x!=30. Hence, not possible!
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titan
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Re: Tetraminoes On A Checkerboard
« Reply #7 on: Oct 13th, 2013, 8:52am » |
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And what if the checkerboard was only in one color and Ts having that single color were to be used? Then was it possible to cover the board with the Ts?
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towr
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Re: Tetraminoes On A Checkerboard
« Reply #8 on: Oct 13th, 2013, 10:09am » |
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on Oct 13th, 2013, 8:52am, yoyoy wrote:| And what if the checkerboard was only in one color and Ts having that single color were to be used? Then was it possible to cover the board with the Ts? |
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Why would coloring them in a checkerboard pattern make any difference?
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titan
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Re: Tetraminoes On A Checkerboard
« Reply #9 on: Oct 13th, 2013, 10:16am » |
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Ah! By checkerboard, I meant a board with 8*8 squares i.e. replace all blues with white. So, all squares are in while color. Replace all blues of Ts too by white.
It wud make a difference because the logic we gave for the original problem was based on counting no. of blue and white squares.
Now since both the Ts are essentially same, so, basically now I have to cover 60 squares with 4x squares. And 4x is a multiple of 60. So, we need to use a different logic now.
Cool?
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towr
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Re: Tetraminoes On A Checkerboard
« Reply #10 on: Oct 13th, 2013, 10:48am » |
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An 8x8 board has 64 squares, so it might work there (probably does work).
The coloration doesn't matter, because if you can tile the board, you can always add the coloration afterwards. So if it works without color, it must also work with and vice versa.
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pex
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on Oct 13th, 2013, 10:48am, towr wrote:| An 8x8 board has 64 squares, so it might work there (probably does work). |
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Yep. (I hope no one's confused by the colors.)
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towr
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Re: Tetraminoes On A Checkerboard
« Reply #12 on: Oct 13th, 2013, 1:00pm » |
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Just the center would have been enough to prove it works 
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rmsgrey
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Re: Tetraminoes On A Checkerboard
« Reply #13 on: Oct 14th, 2013, 11:46am » |
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The official name for the shapes formed by four squares appears to be "tetromino[e]s" - the inclusion of 'e' in the plural appears to be optional.
As any serious Tetris player knows, only the 'S'/'Z' tetronimo can't be used to tile any 4n*4m block, and the 'I', 'L'/'J', and 'O' tetrominos can tile 2n*4m blocks.
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pex
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Re: Tetraminoes On A Checkerboard
« Reply #14 on: Oct 14th, 2013, 12:12pm » |
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on Oct 13th, 2013, 1:00pm, towr wrote:| Just the center would have been enough to prove it works |
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But that'd be too easy... 
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