A Riddle of Chess...

#21

ok i still mantain my answer (the multipied amount of numbers) because of what i tried ot explain better in the last part of my previous post.

i wonder why you sum instead of multiplying, remember this is combinatory theory.

an example

you got a coin and you throw it three times how many are the possible combinations i can have?

for the first trow i got 2
for the second i got 2
and for the third i got 2

if i just sum i'd have 6 possibles results wich is wrong

but the possibles results are (side A, side /cool.gif" style="vertical-align:middle" emoid="B)" border="0" alt="cool.gif" /> are 8

AAA
AAB
ABB
BBB
BAA
BAB
ABA
BBA

8 = 2*2*2, that's why i multiply

#22

Fap, you forget that each time you place one, your set reduces by one square and one space.

Each time you actually get a piece in the right place, you can eliminate it from the ones you have to try in the next space! no mulitplying

the first time you lay a black square down, you have 32 to try.
ONE OF THEM WILL FIT, and YOU DON'T HAVE TO TRY IT ANYWHERE ELSE
so the second time, you only have to find the one of the 31 left that will fit.

your set will decrease by one each time.

worst case, by the perameters given (and leaving out higher philosophy) is as such:

You first attempt will be either to lay down the white or black set, let's say black.

You have a 1 in 32 chance of getting it. worst case it takes 32 times.
*you find the one that fits, on the 32nd attempt*

so then for the white,

You have a 1 in 32 chance of getting it. worst case it takes 32 times.
*you find the one that fits, on the 32nd attempt*

then for the second black square:

You have a 1 in 31 chance of getting it. worst case it takes 31 times, NOT 32.
*you find the one that fits, on the 31st attempt*

then for the second black square:

You have a 1 in 31 chance of getting it. worst case it takes 31 times, NOT 32.
*you find the one that fits, on the 31st attempt*

if you just multiply, your set never reduces by the ones you correctly place!!! you get a total number of positions available, but no accounting for reducing the set for each one you correctly place.

so, you have to assume each set of 32 squares will reduce by one, for each square you place.

-----------------

The Long "OBVIOUS" reasoning for the multiplication of two summations of 1 through 32:

for each color, you originally have 32 squares and 32 spaces so you have-

32 squares to try before you get it right
then, after that one is in the right place

31 squares to try before you get it right
then, after that one is in the right place

30 squares to try before you get it right
then, after that one is in the right place

29 squares to try before you get it right
then, after that one is in the right place

28 squares to try before you get it right
then, after that one is in the right place

27 squares to try before you get it right
then, after that one is in the right place

26 squares to try before you get it right
then, after that one is in the right place

25 squares to try before you get it right
then, after that one is in the right place

24 squares to try before you get it right
then, after that one is in the right place

23 squares to try before you get it right
then, after that one is in the right place

22 squares to try before you get it right
then, after that one is in the right place

21 squares to try before you get it right
then, after that one is in the right place

20 squares to try before you get it right
then, after that one is in the right place

19 squares to try before you get it right
then, after that one is in the right place

18 squares to try before you get it right
then, after that one is in the right place

17 squares to try before you get it right
then, after that one is in the right place

16 squares to try before you get it right
then, after that one is in the right place

15 squares to try before you get it right
then, after that one is in the right place

14 squares to try before you get it right
then, after that one is in the right place

13 squares to try before you get it right
then, after that one is in the right place

12 squares to try before you get it right
then, after that one is in the right place

11 squares to try before you get it right
then, after that one is in the right place

10 squares to try before you get it right
then, after that one is in the right place

09 squares to try before you get it right
then, after that one is in the right place

08 squares to try before you get it right
then, after that one is in the right place

07 squares to try before you get it right
then, after that one is in the right place

06 squares to try before you get it right
then, after that one is in the right place

05 squares to try before you get it right
then, after that one is in the right place

04 squares to try before you get it right
then, after that one is in the right place

03 squares to try before you get it right
then, after that one is in the right place

02 squares to try before you get it right
then, after that one is in the right place

There is ONLY ONE SQUARE LEFT AND ONE SPACE LEFT.

#23

the worst case posible is that you test all the combinations and only your last try is the correct, so the right answer is the number of all combinations of pieces ordered porperly
we agree with this?

#24

QUOTE(Fapper @ Oct 27 2006, 08:56 AM) ...so the right answer is the number of all combinations of pieces ordered porperly
we agree with this?

no, because the total number of combinations doesn't account for the fact that when you get a piece right, you will eliminate some of the possible combinations, and that set will get smaller with each piece you correctly place, and so there will also be less probabilities.

#25

i did not forget i reduce the number of pieces left
my answer = 64*32*31*31*30*30 ..... is based on that in fact

i pick one piece out of 64, so i could have picked 64 different pieces so i have 64 possibilities so far, next step i chose one out of the 32 pieces of the other colour, so i got 32 different chances so for every 64 first step posiblities i have 32 on the second so i go 64*32, and on the third i got 31 ence 64*32*31 (pieces of the first colour picked), fourth step 31 so 64*32*31*31, on the fifth 30, and so on

i could have started saying i choose the first so it would have been 32 chances in the first step (but the whole process would have to be multiplied by 2 since i would have the same cases when starting with the other colour)
2*(32*32*31*31*30*30*29*29*...) = 64*32*31*31*30*30*29*29

please read this carefully and tell me where i'm exaclty wrong.

#26

you got 64*32*31*31*30*30*29*29*...... possiblities to make it as it was original, nobody is gonna be telling you that piece is right until you finish the 64 pieces and that's one try, thats the difference beetween our calculations.

#27

What we have here... is a failure to communicate!

Really, this is a matter of timing; i.e. Is the feedback immediate or delayed? If feedback is immediate after each piece placement, then Raum is right because each succesive placement is limited by the prior correct placements (the choices reduce because the possibilities are DEPENDENT).

If feed back is only offered after all the pieces are placed, than Fapper would be right because each possible solution is independent and you have to look at every possible combination (every square in position 1, every square in position 2, etc.) I believe this is what fapper was trying to illustrate with his coin toss analogy.

So why can't we all just get along and wait for Ayhja to learn how to ask a complete question. Until then - you are both right, IMHO

#28

Frankly, it reminds me of the time I picked up a few things at a 7-11. I had selected 4 different items to buy, and was told that the cost was $7.11. I was curious that the cost was the same as the store name, so I inquired as to how the figure was derived. The clerk said that he had simply multiplied the prices of the four individual items. Naturally, I protested that the four prices should have been added, not multiplied. The clerk said that that was OK with him, but, the result was still the same, exactly $7.11. So, can you tell me what were the prices of the four items? .... and this way Raum, you and fapper can both be right because it won't matter if you are multiplying or adding..... LOL

#29

What we have here... is a failure to communicate!

Really, this is a matter of timing; i.e. Is the feedback immediate or delayed? If feedback is immediate after each piece placement, then Raum is right because each succesive placement is limited by the prior correct placements (the choices reduce because the possibilities are DEPENDENT).

see, here is the thing, the board is only oriented by the positioning of the King, in chess. If you know the king starts on the opposite color (white king on black), and you saw the board before it was taken apart, you would know to look for the king's square first. this would mean, you would never have to deal with the one in 64 chance. that's what I was trying to say.

I also in my second reply, said

I presumed in worst case there would be an indicator that it was in the right square ( like a bing bing noise, or a approving nod of some kind.)

but if there is no indicator a piece is right, the person could work on this until their death, and still not solve it.

If feed back is only offered after all the pieces are placed, than Fapper would be right because each possible solution is independent and you have to look at every possible combination (every square in position 1, every square in position 2, etc.) I believe this is what fapper was trying to illustrate with his coin toss analogy.

see, here is the thing, the board is only oriented by the positioning of the King, in chess. If you know the king starts on the opposite color (white king on black), and you saw the board before it was taken apart, you would know to look for the king's square first. this would mean, you would never have to deal with the one in 64 chance, it would be a 2 in 64 if you can later reorient yourself, or if you saw the board before, 1 in 32, with distinct subsets, as long as you knew what color pieces would have been on your side. that's what I was trying to say.

So why can't we all just get along and wait for Ayhja to learn how to ask a complete question. Until then - you are both right, IMHO

i feel confident my answers hold water, and you would still never have a 1 in 64 chance, at most if you had evidence that was in fact designed to the specs of a regulation chessboard, which could only be ascertained by witnessing it beforehand.

#30

So, if I understand you correctly.... Your answer is based on assumptions made because Ayhja is a lousy communicator. However, if he had stated that you were brought into a room with 32 white squares, 32 black squares, and a frame that would accomodate 64 squares.... and then were told that it was once a whole chessboard and needed to be placed back identical to it's original state.... and if you were then told that you would get a pleasurable tingly sensation in your nether regions with the correct solution but each solution would be judged only after all pieces had been placed on the board....

I suspect that the additional information would lead you to a different answer.

So in conclusion, you cannot prove that you are right without first proving that your assumptions were valid.
Good luck with that.

...and you still haven't told me the price of my 4 items...