Recent content by Michael Redei

I read the MS page about the downgrade "rights", and they left me rather curious. Suppose I buy a Windows 8 machine and (for reasons that are nobody's business but my own) want to downgrade it to Windows 2000. Would I then be violating some law? Could I be sued by Microsoft? Or do I have the...

Your notation is somewhat unconventional, but I think you mean the following: Given an ordered set of the first N counting numbers, i.e. ##S = (s_1\,s_2\,s_3\,\ldots\,s_N)##, you want to find ##X## "swappings" of the form ##(f_1\,t_1),\ldots,(f_X\,t_X)## such that their product will take the...

It might look better as 12! / (7! 4!), or even better as $$\frac{12!}{7! \cdot 4! \cdot 1!} = 3960.$$ 12! / 7! * 4! would be (12! / 7!) * 4! = 2 280 960

Q2 looks fine to me now. Q3 contains a red herring, since first we're told that there are 3 identical red mugs, and then they should be regarded as one unit by being kept together. Imagine dumping the 3 red mugs in one large red box, then you have only 12 objects to arrange: the red box, 4...

Your third equation says ##ac+bd=1##, i.e. ##a^2+b^2=1##, so if you assume ##b\geq0## (since you want ##2a+2b## to be maximal), you get ##b=\sqrt{1-a^2}##. This means you need to find a value ##a## for which ##2a+2b = 2a+2\sqrt{1-a^2}## is maximal. Do you know how to find such an ##a##?

Add ##a##c to both sides of the second equation, and you get $$ ad + ac = ac + bc \iff a(c+d) = (a+b)c = (c+d)c. $$ One possibility now would be ##c+d=0##, but then you'd also have ##a+b=0##, and so ##a+b+c+d=0##. So suppose ##c+d\neq0##, then ##a=c##, and from the first equation you also...

Q1 looks fine. I haven't checked Q2, but Q3 is way off. How do you get that result anyway?

They look correct to me.

Just offering people two choices to compare and then trying to extrapolate the results into a kind of "global" value for each may lead you straight to Condorcet's paradox: http://en.wikipedia.org/wiki/Voting_paradox

And according to their website, they are

That doesn't really say a lot. What does such a multiplication f(z) = ze2πi/3 look like geometrically? If you sketch 1 and f(1) in the complex plane, how could you describe the geometrical operation that takes you from 1 to f(1)? How about 1/2 and f(1/2)? How about 1+i and f(1+i)? If you can...

I think you're missing an "e" in your second "relevant equation" and that you mean a − z2 = (z2 − z1) ei2π/3 instead. If that's right, you might try to interpret geometrically what multiplying a complex number by ei2π/3 means.

I do a lot of mental arithmetic, just for fun, not because I'm really interested in the answer. One "trick" I use is the following one for multiplying numbers that are close to each other: Suppose you want to multiply 13 by 17. (That's "close".) Their average is 15, so 15² = 225 is close to...

You can simplify that expression. It will become easier if, instead of multiplying terms (i-n), you multiply (n-i) instead. You'll need some correcting factor though.

I've removed the bulk of my last post.