Peter Taylor
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 Mar 1 comment Combinatorial proof of $\binom{n + 1}{2} - \binom{n}{2} = n$ Simplifying the left hand side seems like the long way to do it. The straightforward combinatorial argument is much easier. Feb 18 comment de Bruijn sequence in which order of subsquences doesn't matter If you remove the cyclic component, then 0 0 1 1 would serve, and generalises to $k = 2$ for any $n$. Maybe that would be a more interesting question. Jan 24 comment The hardest game of mahjongg I think you need to address the issue of how to lay out the tiles for arbitrary $n$ before it even makes sense to start thinking about an answer. Jan 15 comment Maximal tiling without any 3-in-a-rows @randomra, if the maximum Hamming weight which can be achieved in an $m\times n$ rectangle is $w$ then that gives an upper bound for the density of any infinite tiling of $\frac{w}{mn}$ by the simple mechanism of superimposing an $m \times n$ grid over the infinite tiling. Jan 9 comment Maximal tiling without any 3-in-a-rows The existence of (non-tiling) moderately large squares with densities > 1/2 (e.g. 15x15 with 114 bits set) indicates that the case analysis would have to work on larger units than those squares. I very much doubt that the number of cases for this type of approach would be computationally tractable. Jan 9 comment Maximal tiling without any 3-in-a-rows I don't understand the argument that cases can be eliminated by the symmetry of swapping 0s and 1s: that's not a symmetry, because 000 is allowed and 111 isn't. I also don't understand what contradiction you see in your first example. Dec 12 comment If $x^n$ is ivertible in a ring show that $x$ is invertible. So to be precise, your question is how to show that, given a left inverse of $x$ and a right inverse of $x$, the two must be the same even in a non-commutative ring? Nov 30 comment Erdős and Szemerédi sums and producs Note: this looks suspiciously like a certain active contest. Nov 9 comment Squares with squares Not only does this not answer the question, but it's not even clear what question it's attempting to answer. You appear to have your own idea of what numbers "of that kind" are, but you haven't defined it, and it's certainly not clear to me why you disqualify e.g. $38^2 = 1444 = 12^2 \times 10 + 2^2$ Nov 9 comment Squares with squares Fair enough. Updated. Nov 7 comment Sign table in $2^k$ factorial experiment Ok, which specific part do you have trouble generalising to the four-factor case? Oct 31 comment Extended Euclidean Algorithm to find multiplicative inverse of two polynomials If a polynomial has no remainder when divided by 1, it has no remainder when divided by 2, and vice versa. Oct 30 comment How many subsets xor to given value? @user128409235, that's awfully specific. Is this for a programming contest or something similar? Oct 13 comment A nice form of a given function @Halbort, I think $\max \{a_i - (a_i \oplus k)\}$ is as simple as it's likely to get. Oct 1 comment A nice form of a given function Is that clearer? Sep 22 comment How to write a double edge in an incidence matrix. ? An edge isn't represented by a column in an adjacency matrix, but by a cell. Sep 19 comment Definite shape of polyominos I can see more than one interpretation of the question, so in order to narrow it down: are you asking how to justify the claim that two polyominos are really the same polyomino if there's a way to rotate and reflect one so that it perfectly covers another? Or are you asking how to index them so that when you're counting by hand you can be sure that you don't have duplicates? Or are both of those interpretations wrong and it's something else? Sep 19 comment What is the probability of having the same (binary) datasets? On the basis of the information given, you do not have enough information to calculate the probability that the datasets are the same. You can calculate $p(X_1 = 0 \wedge X_2 = 0 \wedge \ldots \wedge X_n = 0)$ (where $X_i$ is 0 if the ith vulnerable is unchanged, and 1 if it changes) given $p(X_1 = 0), p(X_2 = 0), \ldots, p(X_n = 0)$, but the only information you've given us is that the "outcomes are randomly changed" - i.e. that $p(X_1 = 0)$ etc. exist. Sep 18 comment Is there any winning strategy? 2015 and Game with marbles!!! A pedantic answer to the question as worded would be "Yes, there is a winning strategy for one of the players". Proof: the game must be finite (taking at most 2015 turns) and cannot end in a draw. Sep 18 comment Partition of ${1, 2, … , n}$ into subsets with equal sums. @donbright, the question asks for a proof that any $m, n, k$ satisfying $m \ge n$ and $\frac{n(n+1)}{2} = mk$ has a solution.