0

If 5 squares are chosen at random from a chess board, what is the probability that they lie on a diagonal line?

this is the same question indeed. Answer is given by Mr.Brian M. Scott. But I got a doubt in that.

Why the answer is limited to 64 squares. Don't we need to consider the possibilities of bigger squares also (total 204 squares)?

for example, five 2*2 squares can also be on same diagonal right?

Community
  • 1
Kiran
  • 4,198
  • I guess that depends on the exact definition of the question. If the question considers only the squares of the chessboard and not bigger squares formed by combining 4, 9, 16, ... of them, then the original answer is indeed correct. Also note that you can not fit five $2\times2$ squares diagonally on a chess board. therefore the answer remains the same. – Loreno Heer Aug 25 '15 at 11:22
  • Sure, if they overlap. But I think the default interpretation of "square" in that problem is one of the $64$ chessboard squares. – André Nicolas Aug 25 '15 at 11:29
  • @loreno heer, if it overlap, i believe 22 square can be placed across a diagonal such that diagonal of the 22 square and diagonal of the chess board can come in same line. – Kiran Aug 25 '15 at 11:39
  • @AndréNicolas, if overlapping and bigger squares can also be considered, how could one start solving it? for chess, it might be easy to count. But in general, say, If 5 squares are chosen at random from an n*n grid, what is the probability that they lie on a diagonal line? – Kiran Aug 25 '15 at 11:41
  • 2
    One would first need a precise statement of the problem, and in particular what choices are allowed. For the same bunch of $5$, can one use different sizes? For single chess-type squares, and $n$ by $n$, the strategy used by Brian Scott works smoothly. – André Nicolas Aug 25 '15 at 13:45
  • @Kiran: Please don't ask such wide-ranging new questions in comments. This is a question about the answer to the other question and its interpretation. If you're interested in the problem of counting squares in the geometric sense, please post that as a new question. – joriki Aug 25 '15 at 15:47
  • 6
    @joriki, thx for the comments. will do so hereafter. for this also, i posed a new question http://math.stackexchange.com/questions/1409047/probability-that-5-square-lie-along-a-diagonal-line-modified – Kiran Aug 25 '15 at 17:24

1 Answers1

2

I believe this is a misunderstanding. The English word "square", when applied to things like chess boards, is usually not meant in the literal geometric sense of an equilateral rectangle. A "square" in this context is one of the (in this case $64$) places on which pieces may be placed.

joriki
  • 238,052
  • This is particularly obvious in the context of "square chosen from a chessboard", where clearly the board is viewed as a collection of squares to choose from. – Marc van Leeuwen Aug 25 '15 at 11:29
  • Sure, the interpretation matters. I have seen a question here to count number of squares in chess board. while learning that, my answer was 64 and then realized i was wrong and answer is 204 – Kiran Aug 25 '15 at 11:37
  • @Kiran: If you provide a link to the question perhaps it could be pointed out what clues in the two questions might help you detect which of the two meanings of "square" is intended. – joriki Aug 25 '15 at 15:45
  • 1
    @joriki here is the link http://math.stackexchange.com/questions/116970/number-of-squares-in-n-times-n-chessboard – Kiran Aug 25 '15 at 17:32