Barcodes Here is an example of a UPC-A barcode, taken from Wikipedia: A UPC-A barcode has 12 digits.  The first digit is something that tells how the numbers are generally used – for example, a particular industry might use a certain number for certain kinds of items.  The last twelfth digit is a check digit that can be used to try to tell whether or not the numbers have an error.  This check digit is constructed in a certain way.   UPC-A Barcode Check Digit Calculation The check digit is constructed as follows: We have the first 11 digits: $$ABCDEFGHIJK$$ So let \(L\) be the last twelfth digit.  We sum the…

beforemathjax textbefore https://math.meta.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference Default font size. Different font size. Bold text. bold text. italic text. Italic text. Underlined text. At first, we sample $$f(x)$$ in the \(N\) ($N$ is odd) equidistant points around \(x^*\): \[ f_k = f(x_k),\: x_k = x^*+kh,\: k=-\frac{N-1}{2},\dots,\frac{N-1}{2} \] where \(h\) is some step. Then we interpolate points \((x_k,f_k)\) by polynomial \begin{equation} \label{eq:poly} \tag{1} P_{N-1}(x)=\sum_{j=0}^{N-1}{a_jx^j} \end{equation} Its coefficients \(a_j\) are found as a solution of system of linear equations: \begin{equation} \label{eq:sys} \tag{asdf} \left\{ P_{N-1}(x_k) = f_k\right\},\quad k=-\frac{N-1}{2},\dots,\frac{N-1}{2} \end{equation} \begin{equation} \label{eq:sys2} \tag{asdf2} \{ P_{N-1}(x_k) = f_k\},\quad k=-\frac{N-1}{2},\dots,\frac{N-1}{2} \end{equation} Trying WordPress default latex, no mathjax format inline: inline or not rm \(asdf \rm{rm} jkl \), textrm \(asdf \textrm{rm}…

You are on a game show and presented with 3 doors.  Behind one is a car and behind the other 2 are goats.  You want to choose the door with a car behind it, as if you do so, you win the car.  You choose one door.  Then, the host opens one of the other doors, which reveals a goat behind it.  The host gives you a choice to either switch your door to the other one that’s still closed or keep your original choice.  Should you switch doors?     Answer: If your strategy is to stick to your original choice, your probability of choosing the door with the…

There are 100 prisoners.  An executioner tells them that tomorrow morning, he will line them up so that each of them is facing the back of the head of another prisoner, except for one prisoner at the end of the line.  In other words, prisoner 1 sees the back of the head of prisoner 2 as well as the backs of the heads of prisoners 3-100, prisoner 2 sees the back of the heads of prisoners 3-100, …, prisoner 99 only sees the back of the head of prisoner 100, and prisoner 100 doesn’t see any prisoners in front of him.  The executioner tells them that he will put either…

100 people are in a room.   All 100 of them are perfect logicians. They are told that at least one person in the room has blue paint on their forehead. They are told that once you deduce that you have blue paint on your forehead, the next time that the lights are turned off, leave the room.   All 100 people have actually had their foreheads painted blue (but of course, each of them don’t know this at this point – they can only see the other people’s foreheads).  The light is turned off, then on, then off, on, etc.  What happens?     Answer: So each person sees…

There are 3 people placed in a room.  They all have perfect logic.  The 3 people are told by a host that a number has been written on each of their foreheads.  Each of the 3 numbers are unique, they are all positive, and they relate to each other such that A + B = C (i.e. one is the sum of the other two).  In the room, each person can only see the other two people’s numbers, as they cannot see their own foreheads.   Suppose you are one of the 3 and you see one person with “20” on their forehead and the other person with “30.”  The…