Techy odds and ends

We've all seen those status updates from friends stating that they got 9 out of 10 answers right to a so-called impossible test claiming that " Nobody will get more than 60% of these right!" or something similar, haven't we? Let's look at how these things work, what they do and why they do it. Before going any further, help yourself to a decent dollop of cynicism or you're going to be very disappointed in social media. Remember the golden rule: if it looks too good to be true, then the chances are that it is not true. If there is the remotest possibility that something happening on social media could be something other than what it looks like, then it is not what it looks like and someone is trying to pull a fast one for profit in some way. Are the people setting these quizzes doing it out of the goodness of their hearts for the enrichment of the human race? Of course not. They're doing it to make money . So how do they make their money? What is their...

This is a little foray into number theory and presents one method of calculating the value of \(\pi\). It involves using a program to perform iterative calculations, so in order to minimise the impact of compounded rounding errors, it makes sense to use a computing device that operates with sufficient precision. I chose the SwissMicros DM42 for this because it uses Thomas Okken's Free42 Decimal under the hood, which offers around 34 digits of precision. The DM42 is also extremely fast, particularly so if connected to a USB port while operating because it more than triples the CPU clock compared to battery-powered operation. Back in 1735, the Swiss mathematician, astronomer (and a few other things besides) Leonhard Euler stated, and proved in 1741, that as the number \(n\) tends towards \(\infty\), the probability that two natural numbers \(a\) and \(b\) less than or equal to \(n\) are coprime, i.e. that \(a\) and \(b\) share no common divisor other than \(1\), tends towards ...

Am I a good photographer? Am I heck! But I do want to learn, and think the only way to learn techniques that are equally valid both in digital photography and in traditional 35mm film photography is to use a reflex camera with interchangeable lenses. A compact job won't cut it. A camera phone? Forget it! So, bit of a mid-life crisis :) I splurged out some of my savings on a Canon EOS 800D back that came with an 18-55mm zoom lens. I also bought a couple of different lenses to use with it. The 800D is not a professional camera. It's amateur but high-end amateur. I didn't want to spend huge amounts of money on something that will only be a passing fad, but then again I wanted to put enough into it to give me the best chance of seeing good results and being able to monitor progress. The sensor is not a full-frame sensor, it's an APS-C sensor but the results are still pretty good. The lens that came with the camera in a kit is an "EF-S 18-55m f/4-5.6 IS STM...

A while back, another member of the MoHPC forum mentioned a program he'd written for the HP-41C that allows the user to manage multiple alpha registers. I responded saying that I had written something similar many moons ago that behaved like a LIFO (last-in-first-out) stack rather than an indexed array of datasets and said that I'd look it up. I have no idea whatsoever what I did with my little utility so I decided to rewrite it, purely and simply. So here it is. NB: This program creates and manages a data file in Extended Memory called "ASTACK". If you already have a file of that name, it will be deleted! Note also that running these programs uses Flag 01 and trashes registers R07-R10. Finally, since this uses Extended Memory, you will need to run this on a 41CX or SwissMicros DM41, or on a 41C or 41CV with the "X-Function" module. The size of the data file created in Extended Memory depends on the depth of the alpha stack that you want to crea...

The "N Queens" problem is a puzzle where you take an NxN chequers or chess board (it's usually 8x8 but this works for the general case) on which you have to place N queens in such a way that none of them threaten or are threatened by any of the others. Reminder: in the game of chess, a queen can move as many squares as she likes horizontally, vertically or diagonally but cannot jump over any of the other pieces on the board like a knight. This means that any piece in her direct line of sight on the same line, in the same column or on a diagonal to her is fair game. With this in mind, it is clear that there is only one solution to the problem on a 1x1 board, i.e. a single square (you just plonk a queen on it and you're done), and that there is no way to organise two queens on a 2x2 board or three queens on a 3x3 board without them threatening each other. For a 4x4 or greater board, there is more than one solution. I thought I'd have a go at solving th...

The program downloadable from this page is a numerical solver. It attempts to find the zeroes of a function $f(x)$, i.e. it attempts to find $x$ such that $f(x)=0$ There are several methods for this. One that requires little computing power is Newton's Method, or the Newton-Raphson method, a description of which can be found on Wikipedia . We have no way of knowing the exact value of $f'(x)$ on the HP-41CX used here because that machine has no symbolic math capabilities, so we calculate an approximation of $f'(x)$ by taking a small value $\epsilon$ and calculating: \[ f'(x) \approx \frac {f(x+\epsilon)-f(x)} \epsilon \] This then allows us to calculate the next value of $x$ to use and display so that we see the calculator homing in on the zero that we're looking for: \[ x_{n+1} = x_n - \frac {f(x_n)} {f'(x_n)} \] Once we find a value of $x$ such that $|f(x)| \lt \epsilon$, we stop and that value of $x$ is deemed to be a zero of $f(x)$. The s...

Numerical integration is an operation that many older machines are not able to perform without add-on libraries (e.g. Math Pac for HP-41C) or without writing a program from scratch to do it. There are, of course, some exceptions to this; Hewlett Packard's HP-15C and HP-42S come straight to mind, as does the Casio fx-180P, for example, but these machines are actually the exception, not the rule. There are many methods used to calculate definite integrals, that is integrals between known boundaries yielding a numerical result. One method that is a good trade-off between precision and complexity of application is Simpson's Rule , which attempts to find a quadratic of the form $a\cdot x^2+b\cdot x+c$ that fits or is close to our function. Finding the integral of this polynomial is very easy: \[ \int_{x_1}^{x_2} \! (a\cdot x^2 + b\cdot x + c) \, \partial x = \frac a 3 (x_2^3-x_1^3) + \frac b 2 (x_2^2-x_1^2) + c\cdot (x_2 - x_1) \] Basically, Simpson's Rule states: \[ ...