I've been thinking of 2 math questions recently. Let me just post it here.
The first one is an interesting question. We have been bombarded with Sudoku recently, you know, the addictive puzzle which we need to fill up 9 latin sqaures, with the additional restriction that every row and column cannot have a repeated number. I've been trying to compute the total number of puzzles that can be generated. It turns out that it is quite hard. Moreover, I found on a site that, according to this fellow, the total number of puzzles that can be generated is:
Note that this number is the number of puzzles which can produce a unique solution, not the number of solutions to Sudoku, which is obviously less. (perhaps someone can come up with a number?)
One more thing, it appears that Solving Sudoku puzzles is an NP problem. That is, no efficient algorithm can be used to compute an arbitrary Sudoku Puzzle.
All this information can be found here.
The second math issue is a really fundamental one, and it was pointed out to me by Jasper. Consider this trigonemetric identity:
Sounds familiar right? But did you notice something wrong with it? It appears that this isn't really the definition for cotangent fuction. It lies in the fact that when we subtistitute x=pi/2, or in fact take x to be any odd multiple of pi/2, we get something funny:
cot(pi/2)=0 yet, 1/tan(pi/2) is not defined. Note that tan(pi/2) is undefined. The left and right limits do not match. Although 1/tan(x) as x tends to pi/2 yields zero, this is just the limit and is not an equality. Hence this definition is false!!
The true definition for cot (x) should be