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The Maths Thread

vcs

vcs

Well-known member
Hey Spark, I got it. It's actually far simpler for this particular case and requires no special knowledge at all beyond common sense. :)

Will post the proof shortly.
 
vcs

vcs

Well-known member
The 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.
 
Spark

Spark

Global Moderator
vcs said:
The 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.
Click to expand...
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 work
 
vcs

vcs

Well-known member
Yeah given my CS background and love of graph theory, I spent a long time thinking about spanning trees, Hamiltonian paths, Euler paths etc
 
Spark

Spark

Global Moderator
Howe_zat said:
If all the coefficients are real, how can there be exactly 2 real roots, and hence an odd number of complex roots?
Click to expand...
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.
 
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vcs

vcs

Well-known member
Spark said:
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.
Click to expand...
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.
 
Spark

Spark

Global Moderator
vcs said:
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.
Click to expand...
Ah I forgot about that.
 
Spark

Spark

Global Moderator
For the maybe two or three other people on this forum who may be vaguely interested in this, Ravil Vakil (Stanford), who wrote the oustanding "The Rising Sea" notes on modern algebraic geometry that I've been meaning to get around to working through for ages, is pitching an online course on these notes that I'm going to be joining in on:

https://math216.wordpress.com/2020/05/10/algebraic-geometry-in-the-time-of-covid-launch/

(as an aside, the note about online lectures being more draining is interesting)
 
vcs

vcs

Well-known member
Depends 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.
 
Spark

Spark

Global Moderator
vcs said:
Depends 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.
Click to expand...
It's going to be based on the Rising Sea notes so you can just check those to get a gauge of the difficulty
 
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