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        riddles >> medium >> Tetraminoes On A Checkerboard
        
(Message started by: william wu on Nov 17th, 2002, 8:39pm)
Title: Tetraminoes On A Checkerboard
Post by william wu on Nov 17th, 2002, 8:39pm
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.

http://www.ocf.berkeley.edu/~wwu/images/riddles/chessboard_mutilated.gif


Shown immediately below are two T-shaped tiles each of which can cover four squares of the checkerboard exactly.

http://www.ocf.berkeley.edu/~wwu/images/riddles/chessboard_mutilated_piece1.gif

http://www.ocf.berkeley.edu/~wwu/images/riddles/chessboard_mutilated_piece2.gif



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.
Title: Re: Tetraminoes On A Checkerboard
Post by towr on Nov 18th, 2002, 1:04am
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..
Title: Re: Tetraminoes On A Checkerboard
Post by Chronos on Nov 24th, 2002, 7:13pm
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.
Title: Re: Tetraminoes On A Checkerboard
Post by yoyoy on Oct 13th, 2013, 2:53am
Can someone pls explain what's going on in the ques.? It's not clear to me. What is the ques. asking for?
Title: Re: Tetraminoes On A Checkerboard
Post by towr on Oct 13th, 2013, 8:06am
Try to cut the shape on the top into ones the size of the other two without anything left over.
Title: Re: Tetraminoes On A Checkerboard
Post by yoyoy on Oct 13th, 2013, 8:28am
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?
Title: Re: Tetraminoes On A Checkerboard
Post by yoyoy on Oct 13th, 2013, 8:48am
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!
Title: Re: Tetraminoes On A Checkerboard
Post by yoyoy on Oct 13th, 2013, 8:52am
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?
Title: Re: Tetraminoes On A Checkerboard
Post by towr on Oct 13th, 2013, 10:09am

on 10/13/13 at 08:52:51, 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?
Why would coloring them in a checkerboard pattern make any difference?
Title: Re: Tetraminoes On A Checkerboard
Post by yoyoy on Oct 13th, 2013, 10:16am
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?
Title: Re: Tetraminoes On A Checkerboard
Post by towr on Oct 13th, 2013, 10:48am
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.
Title: Re: Tetraminoes On A Checkerboard
Post by pex on Oct 13th, 2013, 11:35am

on 10/13/13 at 10:48:01, towr wrote:
An 8x8 board has 64 squares, so it might work there (probably does work).

Yep. (I hope no one's confused by the colors.)
Title: Re: Tetraminoes On A Checkerboard
Post by towr on Oct 13th, 2013, 1:00pm
Just the center would have been enough to prove it works ;)
Title: Re: Tetraminoes On A Checkerboard
Post by rmsgrey on Oct 14th, 2013, 11:46am
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.
Title: Re: Tetraminoes On A Checkerboard
Post by pex on Oct 14th, 2013, 12:12pm

on 10/13/13 at 13:00:15, towr wrote:
Just the center would have been enough to prove it works ;)

But that'd be too easy... :-[


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