yeah that's pretty clever. i was briefly considering trying something like that but figured the graph theory approach would be more elegant. but alas it doesn't quite workThe idea is stunning in its simplicity. There are four types of moves allowed - R, L U and D. (R-L) must be 3 and (U-D) must also be 3. They are both odd numbers. The total length of the path in this case is 15, which is also odd. The rest of the details can be filled in.
Like all good Math, this idea generalizes pretty well. Given any arbitrary entry and exit points on the edges of the grid, we can determine whether a path covering all the squares exists. We can do the same for a cuboidal grid.
It's also clear why the "gotcha" solution that allows you to come back to the starting point in the 4x4 case circumvents the above argument.
36π
color=lightGrey does most of it (at least for those who don't reskin the forum)We need spoiler tags here BTW. Would make this thread so much more fun.
If all the coefficients are real, how can there be exactly 2 real roots, and hence an odd number of complex roots?
If all the coefficients are real, how can there be exactly 2 real roots, and hence an odd number of complex roots?
If the polynomial could have complex coefficients, how do you go about solving it? There would be no way near sufficient information, right? Because the roots need not occur in conjugate pairs.Yeah something isn't right there, that seems to violate the fundamental theorem of algebra. Filling in the conjugate pairs only gets you to 8 roots.
EDIT: oh one of the real roots is repeated, duh.
Incidentally, allowing the polynomial to have complex coefficients makes no difference, as the FTA relies on the algebraic closure of R being C anyway.
Ah I forgot about that.If the polynomial could have complex coefficients, how do you go about solving it? There would be no way near sufficient information, right? Because the roots need not occur in conjugate pairs.
Ah yeah. Misread it. Sneaky.Repeated roots
It's going to be based on the Rising Sea notes so you can just check those to get a gauge of the difficultyDepends on the difficulty level of the online courses. I was recently doing some Andrew Ng online courses on Coursera (Deep learning and Neural Networks). I hate Python and am a complete noob at it, but it was pretty easy to sit a few hours on a weekend and complete 3-4 weeks worth of material in one go. High level Math is of course a totally different ballgame.