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 12h comment Why is unit circle not sufficient to bound the partial sums? Do we not also require that $n \ge 3$? The cases $n=1,2$ are without solutions. 23h reviewed Close Why is unit circle not sufficient to bound the partial sums? 23h comment Why is unit circle not sufficient to bound the partial sums? It's hard to follow the dense text with which you open the Question. Perhaps everything needed is there, but I could not find in my first few efforts the meaning of "guard these partial sums". Possibly it relates to the figure of a prison guard in the quoted (imaged) "miniature", but it would help your Readers to set the stage in your own words more fully. 1d revised What is number of p-point subgraphs in n-point graph with average t connections? added missing degree 3 figure to picture; adjusted text, esp. to remove disconnected case 1d comment What is number of p-point subgraphs in n-point graph with average t connections? We can simply drop the disconnected cases from the counts. When every vertex in a graph has the same degree, we say the graph is regular. Note that the chess board has many vertices of degree $4$, but vertices on the boundary of the board have degree $2$ or $3$, so these "grid" graphs are not regular. 1d comment No Borel well-order of the reals? 1d revised What is number of p-point subgraphs in n-point graph with average t connections? noted missing degree 3 case and gave its counts prior to adding the picture 1d reviewed Reopen Direct method in the calculus of variations 1d revised What is number of p-point subgraphs in n-point graph with average t connections? added diagram of maximum degree 3 sequences and counts of each of four cases 1d comment there is a unitary $U \in {M_m}$ such that $X = YU$. Why $X$ and $Y$ have the same range? A direct way to approach this, using the invertibility of $U$, is to show each column of $X$ belongs to the column space of $Y$ (evident) and each column of $Y$ belongs to the column space of $X$ (a bit of algebra is needed). 1d revised What is number of p-point subgraphs in n-point graph with average t connections? math and other formatting improvements/punctuation 1d comment What is number of p-point subgraphs in n-point graph with average t connections? A similar kind of Question, restricted to counting non-crossing "rope" paths between two points, was proposed here: How many possibilities to arrange a rope of length $n$ between two points?. The poster of that Question was also disappointed to learn there would be no exact formula, though as I pointed out in my response, if self-intersections were allowed, it would make the counting easier. 1d comment What is number of p-point subgraphs in n-point graph with average t connections? How about the property of being connected? Did you want to include the disconnected $p$-point, $t$-edge subgraphs in your count? 1d revised What is number of p-point subgraphs in n-point graph with average t connections? typo 1d answered What is number of p-point subgraphs in n-point graph with average t connections? 2d comment What is number of p-point subgraphs in n-point graph with average t connections? I will solve the specific problem, but without a "master graph" (like the chess board), I don't know how to define an average case. 2d comment What is number of p-point subgraphs in n-point graph with average t connections? In a simple graph one has the Handshake Lemma, which tells us the number of edges is half the sum of degrees over all the vertices. The title suggests a rather broad topic, while the body of the Question focuses on subgraphs of a particular "chess board" graph. The particular Question (4 edges, 5 vertices) is tractable, because having one more vertex than edges requires any connected subgraph be a tree. Can you clarify what you actually would like to have answered here? 2d comment Conic ( Parabola by looking at the equation ? ) Your notation seems a bit inconsistent, as at first $a=10$ is used for the coefficient of $y^2$, and subsequently for the $x$-coordinate of the focus. This is not an Answer but a plea for a better statement of the Question. 2d comment How many subsets of $S$ are there that contain $x$ but do not contain $y$? You are correct, the number of such subsets is $2^{36}-2^{35}$ or simply $2^{35}$. The answer key is just wrong, assuming the problem is quoted correctly above. 2d comment Must unital ring homomorphism be identity mapping? @user136266: It will probably be clearer to your readers if you refer to the injective ring homomorphism $\mathbb{Z}\to \mathbb{Q}$ as the inclusion map rather than an identity map.