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Jul
12
comment The number of partitions of $n$ and the $n$th Fibonacci number.
The main difference would seem to be that the first is correct notation and the second is meaningless. When $\operatorname{mod}$ is used as an infix function from $\mathbb{Z}\times\mathbb{N}^+$ to $\mathbb{N}$ you shouldn't separate it from its first argument. $(P_n \operatorname{mod} F_n) = P_n$ looks like a bizarre way of writing $0 \le P_n < F_n$, but $P_n = P_n \pmod{F_n}$ just looks like a tautology with a typo of $=$ for $\equiv$.
Jul
12
comment Describing all points 4000 miles from the north pole
It's impossible to answer the underlying question without first determining the model of the globe and the metric. I think that you're modelling the globe as a sphere (as opposed to an ellipsoid such as the WGS84) and measuring the 4000 miles in Euclidean 3-space (as opposed to geodesics on the surface of the globe). Is this correct?
Jul
9
comment Difficulty parsing combinatorics exercise
I read this as counting pairs of partitions ($\lambda$ is a partition into distinct parts of $k$, $\mu$ is a partition into distinct parts of $l$) such that $k+l=m$, $\lambda$ has $n$ more parts than $\mu$ with the complication of the $0$s.
Jul
3
comment Finding double root of $x^5-x+\alpha$
And then we have $\gcd(x^5 - x + \alpha, 5x^4 - 1)$ $= \gcd(5x^4 - 1, 4x - 5\alpha)$ $= \gcd(4x - 5\alpha, \frac{3125}{256}\alpha^4 - 1)$, so in order for that GCD to be linear we require $\frac{3125}{256}\alpha^4 - 1 = 0$.
Jun
27
comment Looking for different proofs of “Discrete Liouville's Theorem”.
Can you think of a suitable title which includes some non-LaTeX text? Pure LaTeX titles cause some problems.
Jun
27
comment This should be a piece of cake… right?
It wouldn't have cut more than one piece of cake at once, taking the natural meaning of simultaneously. I've edited to remove that adverb and make other clarifications. There's one remaining ambiguity, which I didn't want to resolve because I didn't know in which direction to resolve it: "the cuts may be done however you want" seems to directly contradict "to make it simple, let's say you have a straightedge and a compass".
Jun
27
comment This should be a piece of cake… right?
If I put the cakes side-by-side and cut through them all parallel to the table in one motion, is that one cut or $n$?
Jun
25
comment The expected outcome of a random game of chess?
@mjqxxxx, I think it's because the chess library considers the game to be over when neither player has enough material to mate, and so breaks the loop, but doesn't consider it to be a stalemate, so it wasn't being counted correctly.
Jun
19
comment A game with numbers from MEMO $2013$
Surely the simplest first two moves for B are $0,0$.
Jun
12
comment How do you create a nonlinear game that the player can always win?
The way to maximise the non-linearity is to minimise the interdependence of the puzzles. The point I'm hoping that you'll take away is that you need to rethink your goal.
Jun
10
comment Hex game winning strategy
It's not clear from the question what the board layout is (you can do ASCII art by putting 4 spaces before each line, and there are online designs for ASCII art hexagons), or what the rules of the game are.
May
21
comment Deterministic Push-Down Automata
Is $U$ a terminal? And as a hint: have you tried building a non-deterministic push-down automaton to recognise this language?
May
21
comment What is one way to prove that there exists no ellipse that matches the exact curvature of the sin wave?
That doesn't rule out the sine wave being less than half of an ellipse.
May
16
comment what is the minimum value of the angles inside these triangles?
I think the question is: what is the smallest angle $\alpha$ such that there exists a dissection of the square into triangles satisfying two properties: that none of the triangles has an internal angle greater than $\alpha$; and that no vertex of a triangle touches another triangle except at a vertex. If so, there's an easy lower bound of 67.5 degrees.
May
6
comment Calculate the Probability for Binary Matrix
I assume that the second sentence means that each element of the matrix is $1$ with probability $p$, but is the third sentence talking about independence of random variables or about linear independence (i.e. the matrix is non-singular)?
Apr
23
comment Repeating cycles in the $3n-1$ problem
Your cycles have a very close link with the cycles of $3n+1$ starting with negative $n$.
Apr
19
comment Return of the lost ant 3D
That such paths exist isn't an problem. Whether or not they have a length is another matter.
Apr
17
comment Maximum number of edges in a (n,n) bipartite graph, that doens't have a complete bipartite subgraph $K_{r,r}$
$c=0$ works for every $r$.
Apr
17
comment Maximum number of edges in a (n,n) bipartite graph, that doens't have a complete bipartite subgraph $K_{r,r}$
There's a trivial solution: let $c=0$. If $r=1$ then that's tight.
Apr
17
comment Prove that there are two frogs in one square.
Harry Dunlop's answer already provides a solution: this is essentially just Hilbert's Hotel backwards.