Monday, April 02, 2007

Dominoes, chess and time.

I have a challenge. Before you get excited let me warn you that you may have come across what I'm going to write. (most likely if you're egm and Jake!). But this is where the the bit about time comes in! If you have time, then why not eh?

So the challenge/activity which I'm proposing is as follows.

1) Get a chess board with two diagonally opposite corners missing. This may not be possible if you have a wooden board, so just stick some blue tack on the corners diagonally opposite each other. If you haven't managed to locate a chess-board you could always Google a picture, I suppose and then edit the corners out on paint, then print it. (You might not have to print it, but that depends on what kind of a person you are- personal preference).

Normally the area of the chess board would be 64 squares, however with the two squares removed the area is now 62 squares.

2) This is where the dominoes join the party. I can understand that you probably don't have dominoes (I didn't) and they may not be the same size as two squares of the chess board. So here's your chance to be creative. Get an A4 paper, a ruler, pencil and scissor and get making some dominoes (31 dominoes each the right size to cover two adjacent squares)! This is the bit where doing it on MS Paint mind we handy- but then make sure that you save the chessboard picture (with 62 squares) separately and don't save any edited versions.

So here's what you should have at this stage (if I've not been clear).

I've yet to cut the beautifully drawn out dominoes, but you get the idea. It's either that I had far too much time on my hands, or I was up for the challenge!

Now the challenge: I previously said that the area of the chess board is 62 squares, and since we now have 31 dominoes who's area is also 62, we should be able to cover the board with all 32 dominoes. So can you cover the chess-board with the 31 dominoes? Please try it - all you have to do is fit them together (if you haven't already!). If anything is unclear let me know. No cheating!

I've tried and here's one of my attempts. (In case you didn't believe me :p ). So how's it going, managed to do it yet? I did spend a while doing it and after some time I realised that I wasn't patient enough to persevere! I'm such a failure.

Now you're faced with two options. One being that you can carry on trying and trying and trying and trying and trying and trying and trying and trying and .... (hmm, not sure whether this should be a finite amount or not. Perhaps it should be finite, but everyone is different so I'll leave it open for you to decide. Though it's finite (ahhhh) in my case!). It doesn't matter which attempt you're on, but keep trying!

Option two being, that you think that maybe it's not possible! But hang on a minute- we have 31 dominoes, 31 x 2= 62 which is the number of squares of the board. So it is possible. Then you go back to option one and well try again... . This cycle can repeat for an 'infinite' amount of times (if you're unlucky that is), but say that at you're xth time of going back to option one and you finally decide that you've had enough. Now you know it's NOT possible- your heart is screaming this to you, but your mind is slightly reserved. So what are you going to do about it? You could if you had the time, write down all the possible ways of arranging the dominoes, and then say that it's not possible. But is that enough? Would you be satisfied with that?Maybe you would and then you'll throw the dominoes away, look mockingly at the chess board and sneer at it in victory. Or should I say relief. You've got your consolation goal, but it doesn't matter that the other team have scored an infinite amount of goals. You have one and that's what's important.

Well done BTW, for scoring that goal. It would have taken a lot of time indeed (unless you used a computer which I'll classify as 'cheating?') and you deserve your cup of tea. Heck you deserve a medal. However are you satisfied with that goal. You can see the replays in your mind - in slow motion. But are you content? Maybe you are, in which case you hang the chess board on the wall as your medal. If you're not then why not? Why do you feel that it might still be possible, and that you might have missed doing it a certain way- the correct way! I can't answer this, but maybe you feel that computing all the possible ways is not enough to satisfy your heart. Your heart, which if it isn't satisfied that this task is impossible, will quickly return to option one that is the trying again and again and ..... Now you obviously don't want to do that, so how can you can convince yourself that this whole challenge is actually not one that can be completed.

Now if you've followed the post up to here, you might be feeling very annoyed with your mind. How dare is sit there, up to something and not helping you. What's it know which your heart doesn't? Why won't you heart just stop (not beating of course!) but stop blubbering about the solution being possible, and give your mind a chance to talk? Damn it, you know that if your mind was in control then you won't still be trying to convince yourself about the solution not existing. So when does your mind take control? It all depends on who are you and how many times you've been through this. For the more times you've been through this the more experience you'll have about when your bloody heart will shut up!

So now your mind is in control. No solution exists, it says and you agree. You don't question this, you believe it to be true. You could naively ask your mind 'why' but after your first time of asking you'll learn that your mind is cold and calculating. It's like the black riders from LOTR-but if will always be there reminding you of the fact that no solution exists, and so allowing you to explore why the solution is not possible. Unlike the warmth which was generated from your heart when you scored, the mind radiates eerie power. If you're still with me, have you convinced yourself that no solution exists? If not then I'm afraid it's back to option one for you! If you're convinced that no solution exists, can you convince me? I don't want to carry on doing option one and I don't quite believe that you've worked out all the possible ways of doing it! Please, tell me why no solution exists! Because only by 'convincing' me and everyone else will you really be satisfied. (My heart is being a jerk, is my excuse).

Once again, if you're still with me then please do enlighten me. I could spoil your fun and rather than you telling me why this is not possible, I could tell you. But come on, is that what you want? You see once you are able to convince me that it isn't possible, your heart comes back to life. The warmth returns. Your minds control of you disappears. It retreats once again, waiting for you to call upon it again, to give you stability and control when it's needed. Now the real goal celebrations start! Your mind no longer seems like a know it all, your heart doesn't feel any strain. Drinks all round. The solution doesn't exist.

[What was meant to be a post posing a challenge and possibly the answer, has become .. well I don't know! What do you make of it? I look forward to hearing your answers, but don't worry if you're the one who wants enlightenment, then that'll be tomorrow! Oh and I'm not mad- really, just got slightly carried away!]


egm said...

You're right... read about it, so it'd be cheating! :)

Jean-Noël said...

Getting all the materials ready seemed like too much work so I started by assuming that no solution existed.

Surely the area of a domino is equal to the area of two squares so any domino covers one white and one black square. This implies that we need 31 white and 31 black squares. But since we removed two opposite corners we removed two squares of the same colour. So no solution is possible.

I find this argument convincing enough to return to revising stats.

Jake said...

I liked that puzzle because normally when we are given puzzles with a parity argument; we are so used to the parity being even/odd numbers as opposed to anything else - it makes you think a bit more.

beans said...

Jean-noel: Bravo! Same rules apply- next 24 hours you're known as Sir Jean-noel. Enjoy!

You got the nail on the head. No matter how you arrange the darn dominoes, you'll always have two black ones which are empty, because we have 32 black squares and only 30 white squares.

But I'm upset to read that you'd rather revise stats than cut up dominoes! :D

Jake: Yeah it did make me think, but unfortunately not the right way! Neat answer though, going to try it on the family!

And egm, did you actually use a chessboard to see if it was possible? :D

Jean-Noël said...

I wasn't really implying that stats is more fun than cutting up dominoes ;-) But unfortunately revising for exams does include revising things that I find boring and utterly annoying. But then it's a price most of us have to pay ;-)

KTC said...

Having already seen this possibly a number of times, I didn't cut anything up as I knew the answer already. You might call know both the impossibility and the prove before reading the problem cheating though. :)

beans said...

Well since this is your first comment on my blog, I guess I can overlook your cheating this time round. :D (Don't want to scare you away!)

Do you want me to quote your previous post? :D But I know what you mean about stats being a pain, and if you did start cutting up dominoes then you might never have stopped! So I guess it's a good thing that you chose to suffer. ;) I wish I could choose the same path as yourself!

egm said...

Nope, I didn't try it out cause I had none handy, and like Jean-Noel, I didn't want to spend time cutting up dominoes!

beans said...

Well all I can say is that you don't know what you've missed out on! :D
My dominoes are now supporting colour (thanks to my little sister!).:o

Ah well, I guess you guys have better things to do with your time, but what I don't know! ;)