Puzzle: Playing chess with dominoes

Puzzle: Playing chess with dominoes
by Ari Siletz
27-Jul-2010
 

A chess board can be completely tiled with 32 dominoes in several different ways (a pretty example is given at the bottom of the above image). A harder challenge is to remove the two squares shown and try to tile the chessboard with 31 dominoes. Can you do it? Or prove it is impossible. 

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Ari Siletz

Divaneh

by Ari Siletz on Tue Jul 27, 2010 10:54 PM PDT

An odd/even way to see this is to put coordinates on the board from 0 to 7 along the x axis and from 0 to 7 along the y axis. Each square is then labeled by a coordinate pair. For example the top left square is labeled (0,7). When you place a dominoe on the board, it covers a square whose x and y coordinates sum to an odd number and a square whose x and y coordinates sum to an even number. So any dominoe tiling requires the same number of odd sum squares as even sum squares. But we have removed two odd sum squares (7+0=7and 0+7=7) without removing two even sum squares to compensate. So the tiling is not possible. This is basically the same as hamsade's proof except with odd/even replacing black/white.

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divaneh

What a teaser

by divaneh on Tue Jul 27, 2010 07:14 PM PDT

There is an odd number of boxes in the first row. In any way that you place the dominos in the first row , you will end up with an odd number of squares in the next row. Carried through it leaves an even number of boxes in the last row, but not in the right order.

 I think Hamsade's theory is the best one so far and makes complete sense.


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Ari Siletz

MPD

by Ari Siletz on Tue Jul 27, 2010 06:14 PM PDT

Your attention to the arts as a means of education is already doing something to remove those squares. To quote Moliere's Monsieur Jourdain
 "Well, what do you know about that! These forty years now, I've been speaking
in prose without knowing it! How grateful am I to you for teaching me that!"

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Multiple Personality Disorder

I could do things only when I know what I am supposed to do,

by Multiple Personality Disorder on Tue Jul 27, 2010 05:55 PM PDT

...including removing those two blocks at the corners.


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Ari Siletz

Khar

by Ari Siletz on Tue Jul 27, 2010 05:49 PM PDT

Yes, as hamsade ghadimi has pointed out, removing the right two squares leaves a solvable problem. For example if you remove the squares under the pink domino on the upper left of the tiling example, the remaining board is solvable. Estebdad is one of the squares to take out, the other is the cultural willingess to invite it. Even the secular mindset of the Iranian masses is still Mahdi oriented in that we still follow individuals  or specific ideals rather than build and manintain civic institutions where lots of people with different ideals are able to participate in leadership. 

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Khar

Ari Jaan

by Khar on Tue Jul 27, 2010 05:35 PM PDT

Since the impossibility of this new game "Domiess" (forgive my abbreviation) has been proven, will it be possible to set battling sides not as Amaameh vs. Taaj, but battle between Democracy vs. Estebdaad? in which anything maybe possible perhaps even this new game ;-)


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Red Wine

:=)

by Red Wine on Tue Jul 27, 2010 03:52 PM PDT

آقا این فَرَنگی بازیها دَر کَتِ ما نِمیرودْ،نحنُ شُعبْ بسيط ِمن الِشميرانْ،شَرطی دُمینو بازی کرده و روْ کم میکنیم...سایه شما از سَرِ ما کمْ نشود.


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Ari Siletz

Excellent reasoning!

by Ari Siletz on Tue Jul 27, 2010 03:36 PM PDT

hamsade ghadimi, you've hit on the general proof of impossibility. Also kudos to admin for trying the empirical method which provides insights and clues towards a general proof.

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hamsade ghadimi

i don't believe there's a

by hamsade ghadimi on Tue Jul 27, 2010 03:28 PM PDT

i don't believe there's a solution.  a domino can only be placed horizontally or vertically (not diagonal); therefore, each domino occupies one white square and one black.  this is a required condition.  by removing the 2 white tiles, you now have 32 black tiles and 30 white tiles; therefore, the maximum number of dominoes that you can place on the modified chess board is 30.

on the other hand, if the corner tiles from the same side are removed (one white, one black), then you should be able to fit 31 dominos.


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SamSamIIII

The choices

by SamSamIIII on Tue Jul 27, 2010 03:00 PM PDT

 

presented are purposefully flawed & provokative. The choices must be Kiaani Iran(true Iran) vs Pan-Ommatism(post-true Iran). I say provocative since the blogger forces the folks to make a unproductive choice between a purely Iranian entity(crown) and a foreign yet sacred entity(Islam). No, Islam as a private channell of spirituality shall stay and folks dont have to make such a hard choice . Yet they can & shall make a choice on eradicating the native corrupt additions that has turned our culture & language into a mere offshoot of 7th century Medina.

But then I,m not surprised by these choices since they come from the same gent who in another article corrolated & defined the house of Ommah pawn boy,Mr Seyed Mir Hussein Mousavi to no other than true Kiaani icon Kaveh Ahangar . Comedy of history indeed.

Cheers!!!

Path of Kiaan Resurrection of True Iran Hoisting Drafshe Kaviaan //iranianidentity.blogspot.com //www.youtube.com/user/samsamsia


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comrade

The fate of a pawn

by comrade on Tue Jul 27, 2010 02:34 PM PDT

If it is worth it  to remove the two squares, it's certainly worth it to break a tile for future fitting. No pain, no gain!

visit....//www.ipinst.org/


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Immortal Guard

Nice commentary!?

by Immortal Guard on Tue Jul 27, 2010 02:17 PM PDT

It's not a mosque. I see it as a Persian "Khud" i.e. helmet of a Persian warrior!


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admin

I don't think it's possible

by admin on Tue Jul 27, 2010 02:06 PM PDT

if you try it with a 4x4 you'll see it won't work because 2 tiles left over are never going to be next to each other.
Good political commentary with Mosque/Crown.


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