Mark:
SQUARE MEN
1975
The officer did have 1975 men. When he formed a square measuring 44 by 44, he had 39 men over. When he tried to form a square 45 x 45, he was 50 men short.
CHESSBOARD SQUARES
This turned out to be much easier than I first thought.
?'Oh boy, as soon as I read that I knew there was going to be trouble.'
Let NxN be the dimensions of the chessboard (8 for this problem).
The number of ways to select two squares is C(N*N,2).
There are N*(N-1) ways to select pairs that share a vertical side and N*(N-1) ways to select pairs that share a horizontal side.
Therefore the probability that the selected squares have a side in common is:
2*N*(N-1) / C(N*N,2)
For N=8, the answer is 112/2016 = .0555...
?'It would appear that we differ by a margin of 50%'
The number of ways of choosing the first square is 64. The number of ways of choosing the second square is 63. There are a total of 64 * 63 = 4032 ways of choosing two squares.
If the first square happens to be any of the four corner ones, the second square can be chosen in 2 ways. If the first square happens to be any of the 24 squares on the side of the chess board, the second square can be chosen in 3 ways. If the first square happens to be any of the 36 remaining squares, the second square can be chosen in 4 ways. Hence the desired number of combinations = (4 * 2) + (24 * 3) + (36 * 4) = 224.
Therefore, the required probability = 224/4032 = 1/18. Check mate I think. No, I hope.
A square, whose side is 2 meters, has its corners cut away so as to form an octagon with all sides equal.
What then is the length of each side of the octagon, in meters
STAND?
Hard to
YOU. CLEAR UP