Mark S.
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 12h comment References on a game with white and black stones That paper helps a lot! One thing I want to check is if you're still interested in the game where the starting position has an even number of symbols, because I think the paper doesn't discuss that at all (with Black going first). Either way, there's still more interesting stuff to say about this game, though. 12h revised Integral of $\vec \nabla f(x)$ added tag 1d comment How do the roots of “$x^2 + bx + c$” change as $b$ is kept constant and $c$ is changed? @Michelle, "feeling unintelligent": Unfortunately, things like "exploring in mathematics by making convenience-driven choices" are not emphasized early on. And keep in mind that I'm not a math peer; I'm a math teacher. Also, since your question had little context (this is probably why the question is closed), I had to make a guess at what level to pitch my answer at. (Not to mention that answers are not just for the question asker.) If you just keep on asking questions, you'll keep learning. 1d comment How do the roots of “$x^2 + bx + c$” change as $b$ is kept constant and $c$ is changed? @Michelle, sorry about any confusion; I'm happy to answer questions about my answers. If I had just defined $S=\sqrt{|b^2-4c|}$ and $R=-b$, then by the quadratic formula, the roots in the real case would be $(R\pm S)/2$ and the roots in the complex case would be $(R\pm iS)/2$. Only to avoid writing "$/2$" in so many places, and for no other reason, I made the choice to define $r=-b/2$ and $s=(\sqrt{|b^2-4c|})/2$ instead. The $1/2$ comes into the definition of $s$ because I chose it to simplify the algebra that followed. 1d comment References on a game with white and black stones I've been working here and there on this since you posted it; there's definitely some interesting patterns, but I don't yet have a complete collection of conjectures. I'll probably give up and post my partial results by the end of another week. Where did you come across this game? Or is it your own invention? 2d comment How do the roots of “$x^2 + bx + c$” change as $b$ is kept constant and $c$ is changed? @Michelle Consider similar situations with numbers. $\sqrt{3}=\sqrt{|3|}$, since $3>0$. And $\sqrt{-2}=i\sqrt{2}=i\sqrt{|-2|}$ since $-2<0$. So any square root of a real number is related to the square root of its absolute value, possibly with a factor of $i$. Using this same idea, I figured that introducing $\sqrt{\left|b^2-4ac\right|}$ would help in understanding. To avoid having to write something like $s/2$ all the time, I defined $s$ to be $\frac12\sqrt{\left|b^2-4ac\right|}$. Incidentally, "exploratory algebra" was not a typo. There are many different choices you can make when exploring. Feb 6 answered What is the name for the point where a non-smooth transition occurs Feb 6 comment Converting programming logic to mathematical notation @BugHunterUK, "how to write algorithms in math": as Ian said, pseudocode is fairly common. You can find examples in many papers/books on algorithms in graph theory, for instance. "express solutions to problems in math": Most solutions to problems in math are not algorithms. And when they're close to code, they're often closer to code in a pure functional language than a procedural language. Feb 6 answered Is there a name for dividing a set into pieces, some of which may be empty? Feb 5 comment How to define “being inside of something” in the context of topology? @holistic, convex hulls come from "computational geomtery", although they can sometimes be closely tied to "linear algebra". For time, you may want to say something like "as time increases from 0 to 1, the percentage of the volume of the apple that is a subset of the convex hull of the bag increases from 0 to 1". Feb 5 comment mathematical symbol for a new member replacing a member in a set This contains the answer I was thinking of writing, but is twice as well-written as mine woild have been. Nice job! Feb 5 comment How to define “being inside of something” in the context of topology? The kinds of examples of being inside you refer to are not really places for topology to shine. Homeomorphism, the concept of "these are the same as far as topology is concerned" does not preserve things like "food being in a shopping bag" (though arguably "food touching a shopping bag"). If you're looking for a mathematical idea that's not necessarily from Topology, perhaps "inside" is close to "is a subset of the convex hull", but that's a geometric concept, not a topological one. Feb 4 comment Converting programming logic to mathematical notation It's a common error, but "discreet" and "discrete" have very different meanings. I suspect your example isn't something that's going to have an interesting mathematical translation. In math there's a lot of pseudocode and proofs of things like "checking up to sqrt(n) is enough". It's not like there's a separate mathematical way to write every algorithm (aside from I guess writing it in a functional language) Jan 28 awarded Explainer Jan 28 comment General clarification for derivative notation @AlanL, did this answer your question? If so, you may want to accept it. If not, let me know if anything I said is unclear or if there's some aspect of the question I did not address. Jan 28 comment Sharing a pepperoni pizza with your worst enemy I am now hesitant to split a pizza with you. Jan 27 comment discrete mathematic Question that i need help with please Try to use Stars and Bars as described at en.wikipedia.org/wiki/Stars_and_bars_%28combinatorics%29 (or some other technique from your textbook/class notes, if applicable) and post your thoughts and attempts. Jan 27 revised How do the roots of “$x^2 + bx + c$” change as $b$ is kept constant and $c$ is changed? Added picture of 3D plot of how the solutions vary as c varies and relevant algebra Jan 26 revised Stacking circles added "packing problem" tag Jan 25 comment Placing stones on vertices of polygon @MJD Nice exposition in that blog post! If you're not familiar with Octal Periodicity, you may want to see page 11 in these notes to see how to confirm that the Nim values for "rows of dots" are eventually periodic.