Ok my question is what is the maximum number of tile we can make in the 2048 game assuming we were really lucky and got all 4 number tiles and got the new squares exactly where we needed them?
A game called  is making rounds on social media. I am trying to determine the maximum score attainable for this game.  Let's assume WLOG that only 2s are returned (if 4s are possible the max score is doubled).  First, I'll relax some of the rules to create a game, $\mathcal{G}$, that can achieve a score at least as high as the original, 2048 game.  A run $R$  of $\mathcal{G}_n$ consists of a sequence of moves $m_1,\ldots,m_k$ which manipulate a multiset, $\mathcal{S}$ of maximum size n.  Each $m_i$ is either:   a put $p$, which inserts a 2 into $\mathcal{S}$, or a merge, $\alpha_i$, which replaces two $2^i$ elements from $\mathcal{S}$ with a single $2^{i+1}$ elt.   I allow sequential moves of the same type because 2048 allows multiple merges or no-merge turns.  The score of a position is the sum of $\mathcal{S}$.  What is the maximum score possible on a game of order n?  Note: The final position must have only unique values, else a merge and put would be possible in increase score.  I claim it is $\sum_{i=1}^{n} 2^i$, but couldn't come up with any good induction arguments.  It's pretty easy to prove that an element $2^k$ is producible in a game $\mathcal{G}_k$, but proving an upper bound for production got a bit hairy, and also doesn't cover the full space.
reports:  Two mathematicians have uncovered a simple, previously unnoticed property of prime numbers [...]. Prime numbers, it seems, have decided preferences about the final digits of the primes that immediately follow them. Among the first billion prime numbers, for instance, a prime ending in 9 is almost 65 percent more likely to be followed by a prime ending in 1 than another prime ending in 9.  Do these insights make breaking keys based on primes more mathematically plausible in a shorter amount of time? I'm trying CTest in CMake in order to automatically run some of my tests using make test target. The problem is CMake does not "understand" that the test I'm willing to run has to be built since it is part of the project.  So I'm looking for a way to explicitly specify this dependency. I am a economics major undergraduate student who will apply for grad schools in the coming semester. I think I have quite a strong research experience in mathematical statistics (asymptotic statistics to be specific), but, unfortunately, not much in economics itself. So I am curious if economics departments in graduate schools will care about my research experience in statistics?  Maybe a broader question is if grad schools care about applicants' research experience in a different (maybe even unrelated) field at all? I guess in my case statistics and economics are still quite correlated, but what about other more distant combinations of target departments and previous research experience? I would like to install Ubuntu  for almost 120 users. But it's very difficult in using USB and drives.  Any suggestions? Programmatically determining amount of parameters a function requires - Python How do I turn off some formats in GeoServer's WMS? I have published a couple of layers on my Geoserver.  To be complaint with the T&amp;C of my data provider, I need to block the access of some of those layers as Vector data. Hence I have already blocked the WFS services on these Layers.  I came to know that even on the WMS service, you can get Vector Data, if you use GeoRSS and some of the .   How do I block GeoRSS, SVG &amp; UTFGrid as formats in the WMS Service? How to allow page break inside a float environment? I'm using new float enviroment to show some examples in my document. This  enviroment contains tcolorbox which provides a gray background. The main text in the enviroment is a mixture of plain text and code blocks (in lstlisting env). Each float should have it's own caption and label.  In style file:  \usepackage{newfloat} \DeclareFloatingEnvironment[name={Example}]{suppfigure}   In the main document:  \begin{suppfigure}[!h] \caption{Some code example}\label{suppfigure1} \begin{tcolorbox}[arc=0mm, left=1pt, right = 1pt, boxrule=0mm,colback = {shadow-gray}]   Text  \begin{lstlisting}  int i = 0;  \end{lstlisting}  Some more text  \begin{lstlisting}  String.Format("{0,10:0.0}", -123.4567);   \end{lstlisting}  \end{tcolorbox}  \end{suppfigure}   How can I allow page break inside the suppfigure to extend my example to multiple pages? Tail sum for expectation In , the tail sum formula for expectation is introduced for a nonnegative (0,1,...) discrete random variable $X$:  $$E(X) = \sum_{i=0}^\infty P(X &gt; i).$$   I wonder if there is a similar formula for nonnegative continuous random variable $X$:  $$E(X) = \int_0^\infty P(X &gt; x) dx?$$  If no, are there some conditions for it to hold? And how can it be proved?  Here is my thought:  If the cdf $F$ of $X$ is bijective, then $X=F^{-1}(U)$ for some random variable $U$ uniformly distributed over $[0,1)$. So $$E(X) =     \int_0^1 F^{-1}(u) du.$$  To prove the tail sum formula, it suffices to prove $$\int_0^1     F^{-1}(u) du = \int_0^\infty P(X &gt; x) dx.$$ But I am stuck here.   What's more, is the condition that the cdf $F$ of $X$ is bijective really necessary for tail sum formula to hold? Can tail sum formula be generalized to a random variable that is not necessarily nonnegative?   Thanks! Perpendicular Chord parabola If $r_1,r_2$ be the length of the perpendicular chords drawn through the vertex of a parabola $y^2=4ax$, then show that $$(r_1r_2)^{4/3}=16a^2(r_1^{2/3}+r_2^{2/3})$$ Calculating geometry for feature class with x/y and Lat/Long values in different projections? Having trouble with Windows XP Read Only Settings on a folder I am having trouble with a folder that needs to be read-write for iTunes to open. For a while it was read-write, but it seemingly inexplicably changed to read-only now. When I try to change it back from that, it says it does it, but as soon as I try to open iTunes or the properties for the folder it will be listed as read-only again.  If anyone knows how I can change the folder from read-only so that it will be permanently read-write it would be incredibly helpful.  How can one change the properties of a file/folder from read-only to read-write in Windows XP? Reading Photoshop PSD Files with Layers (There are a few existing questions on Photoshop alternatives, but none seem to focus on reading PSD files specifically)  Say I'm a programmer working with a web designer and would like to implement a design she's produced as a PSD file using Photoshop. It's a fairly complicated design using overlays/animating layers, so the ability to examine the PSD file and turn individual layers on and off is necessary.   Is there an app available in Ubuntu that can read PSD files created by Photoshop?  Alternatively, is there a more linux-friendly file format that Photoshop can export to and still preserve layers? How to get a UK visa refund I applied online for a visa application to the UK. I'm from Sri Lanka. Yesterday I cancelled the appointment, I changed my mind. I will not go to the visa appointment. Now my question is how I get my refund? Java: Sorting user-input array upon return Let $a,b$ be positive integers.  When $$k = \frac{a^2 + b^2}{ab+1}$$ is an integer, it is a square.    Proof 1: (Ngô Bảo Châu): Rearrange to get $a^2-akb+b^2-k=0$, as a quadratic in $a$ this has two values: $a$ and $kb - a = (b^2-k)/a$. (The second root is determined in two different ways from the expansion $(x-r_1)(x-r_2) = x^2 - (r_1 + r_2)x + r_1 r_2$.)  Now suppose we have $a,b$ such that $k$ is an integer but not a square, by the investigation about roots we have that the second root is a nonzero integer since $k,b,a$ are integers and $k \not = b^2$, futhermore it is positive which is easily seen from its defining equation.  WLOG assume $a \ge b$ so that the second root is strictly smaller than $a$. This leads to a decent, replacing $a$ with the second root.    Proof 2 (Don Zagier): Apply reduction theory (specifically, Sätze 1 and 2 of Section 13 of my book on quadratic fields) to the quadratic form $x^2 + kxy + y^2$, which is the unique reduced quadratic form in its equivalence class.    Note that Proof 2 is pretty much the same as Proof 1 when written out in explicit detail, but I could not read Zagier's book because I cannot read German.  I would like to know more approaches to this and other alternative proofs of this result if possible! Thanks in advance.  I would also be interested in related problems (especially easier ones of a similar nature) and texts which cover the reduction theory in English. The Problem  Currently if I flag a post, I cannot check the status of the flag unless I have prior helpful flags. This means, for example, that a new user who wants to check the status of his first flag cannot do so while it is pending (active).  This feature request is based on my experience on MSO, where I wanted to check the status of my flag but could not get to the summary page without cheating (tweaking my SO link to work with MSO). It also stems from this question, where the OP indicates he had the same issue:   My Proposal  If I have flagged any post, then show me a "flags" link (red arrow in image below). If I have any helpful flags, then the number of helpful flags should be displayed (blue arrow). If I have not flagged anything, then neither should be displayed (as with current site functionality). How general is the Lagrangian quantization approach to field theory? It is an usual practice that any quantum field theory starts with a suitable Lagrangian density. It has been proved enormously successful. I understand, it automatically ensures valuable symmetries of physics to be preserved. But nevertheless the question about the generality of this approach keeps coming to my mind. My question is how one can be sure that this approach has to be always right and fruitful. Isn't it possible, at least from the mathematical point of view that a future theory of physics does not subscribe to this approach? How can I expand a partition into non adjacent free space using GParted? My dual-boot machine has partitions as follows:   sda1 50 GB NTFS sda2 15 GB unallocated sda3 20 GB ext3 sda4 20 GB ext4   I would like to allocate the unused space on sda2 to my Ubuntu partition on sda4, hopefully leaving the other two partitions untouched, leaving this configuration:    sda1 50 GB NTFS sda3 20 GB ext3 sda4 35 GB ext4   Is this possible? If so, how can it be done, preferably using GParted? Word that means "outdated name" Is there a word that means "outdated name"? For example:   Record, although very little music is on vinyl Film, although most movies are digital Horsepower, although no one uses horses as a metric anymore Phone, although it's mostly used to access the web   and so on. A teacher wants to randomly form two teams of 5 students from a group of 5 girls and 5 boys for a sports activity. A teacher wants to randomly form two teams of 5 students from a group   of 5 girls and 5 boys for a sports activity. Two of the girls, Ann and   Alice, are selected as team leaders. Find the probability that one   team has exactly 3 girls.   If i do this way $$\dfrac{{3 \choose 2}{5\choose 2}{1\choose 1}{3\choose 3} \times 2!}{{8 \choose 4}{4 \choose 4}} = \dfrac{6}{7}$$  Which turns out to tally with the answer. But i got one big question, is my way correct. Do i really need to times $2$ here? Please see my link below to my previous question cause the previous question make me doubt if i should times $2$.