Rebus Puzzles

Today, on the PUZZLE A DAY website,  I was confronted with the following challenge shown in Figure 1.

Figure 1: Can you solve the two rebus
puzzles that appear below? 

It wasn't clear to me what a rebus puzzle was so I had to look at the answers. It soon became became clear. The top puzzle can be interpreted as "Neither here nor there" and the second as "Pay through the nose". Of course, I'm familiar with this type of puzzle but I wasn't aware that such puzzles went by the name of "rebus". Now I'm the wiser.

I asked ChatGPT the following questions:

What is a rebus puzzle?

A rebus puzzle is a type of word puzzle that uses pictures, symbols, or letters to represent words or parts of words. The puzzle typically presents a series of images, which when deciphered, form a word or phrase. Rebus puzzles are often used in games, brain teasers, and educational settings to challenge people's ability to think creatively and critically.

For example, a rebus puzzle might show a picture of an eye followed by a picture of a heart. The answer to the puzzle would be "I love you," with "eye" representing "I" and "heart" representing "love."

What is origin of the word "rebus" as in "rebus puzzle"?

The word "rebus" comes from the Latin phrase "non verbis sed rebus" which means "not by words but by things." The term was first used to describe a form of representation in which pictures were used to represent words or syllables. In medieval times, rebuses were often used as a form of wordplay in heraldry, where images of animals, objects, and symbols were combined to represent the names or titles of individuals or families. The use of rebuses as a puzzle format emerged in the 17th century, and they have remained popular ever since.

From the same website on March 6th 2023, there are 15 rebus puzzles (shown in Figure 2) relating to Mathematics and Science.

Figure 2: source

At first glance, I have no idea whatever as to how each puzzle encodes the name of a movie but I'll keep working on them. I like the idea of creating rebus puzzles purely out of numbers. Figure 3 shows an example of a rebus puzzle that relies on letters and numbers.

Figure 3: source

It didn't take me too long to figure that one out (safety in numbers). Figure 4 shows a puzzle that conforms to my ideal of only using numbers and mathematical symbols. It comes from a site dedicated to such puzzles.

Figure 4: source

I didn't quite get it. I was thinking along the lines of "the root of ... " and wishing that the number was 666 rather than 66. Then it might have been "the root of all evil" as the number 666 have evil connotations. However, it wasn't "root" but "route" and then of course the answer is simply "route 66". Clever. I may pursue these sorts of puzzles further on Mathematics blog. 

In the Groove

Sometimes you get into a groove, habits form and you don't go beyond the prescribed limits that you yourself have set. For some years, I've been using the Online Encyclopaedia of Integer Sequences (OEIS) when examining the mathematical significance of the number associated with my diurnal age. Now that my numbered days are approaching 26,000, I long ago realised that I was missing out on some important sequences because the number wasn't listed in the OEIS display.

25406 is a good example of what I mean. This number is a member of OEIS A025414:

a(n) is the smallest number that is the sum of 3 nonzero squares in exactly n ways

3, 27, 54, 129, 194, 209, 341, 374, 614, 594, 854, 1106, 1314, 1154, 1286, 1746, 1634, 1881, 2141, 2246, 2609, 2889, 3461, 3366, 3449, 3506, 4241, 4289, 5066, 4826, 5381, 5606, 6569, 5561, 6254, 7601, 8186, 8069, 8714, 8126, 9434, 8921, 8774, 11066, 11574

Clearly 25406 is well short of making an appearance in the above list. However, it is listed in a Table of n, a(n) for n = 1..1000 where it is the 72nd entrant and thus the smallest number that can be represented as a sum of two nonzero squares in 72 different ways. But how to discover this when a search for 25406 using the OEIS search bar does not bring up this sequence?

The secret was to click on Hints and then reading through the contents of this page, one discovers the following little pearl of wisdom:

To search for a single large number in the OEIS, try Google, because Google has searched all the .txt files in the OEIS, and so may do a better job than the OEIS search engine.

Sure enough, this approach throws up a whole lot of other OEIS results that don't appear in the normal search, among them being OEIS A025414. This is a significant discovery because it increases the likelihood of finding an interesting sequence for large numbers such as 25406. The pedagogical principle at play here is to poke around and don't stay stuck within the same comfortable but limited perimeter. This is what had happened to me with the OEIS searching.

CodeCogs

I noticed that Google Docs was using CodeCogs to render LaTeX into image gifs. CodeCogs at https://www.codecogs.com/latex/eqneditor.php provides an interface whereby LaTeX code can be generated quite easily together with an image of what the code would produce if rendered properly (and therefore to be used when the code can't be rendered properly). Here is the interface:

Figure 1: interface that meets the user on CodeCogs

The code can be copied into a document that interprets the LaTeX code and the result is as shown here --> \( \int_{0}^{1}x^3 dx=\bigg [\frac{x^4}{4} \bigg]_0^1\).

The image produced on the site can be simply dragged and dropped:

Alternatively, the HTML code can be inserted into the web page and the image called up from CodeCogs. The result is the same except for the pop up text box that appears when you hover over the image:

These sorts of services are useful for those who lack confidence in creating LaTeX code but there are limitations to what can be produced. There's no substitute for being able to code directly. One site that provides this facility is Overleaf, an online LaTeX editor. Projects that are created can be stored on the site and printed out if desired. The free plan only allows for solo use, no collaborators. There are paid plans that allow for collaboration and provided additional features such as synching with Dropbox and GitHub, full document history, track changes etc. Of course, if you're using a Mac like me, you can download the free program TeXShop to your computer and simply work from there, no online access required. Questions can be posed and solutions hopefully provided on StackExchange.

Free Text Books

I stumbled upon a site offering free text books for high school subjects, including A Levels, O Levels and IGCSE. I'd been looking for ebooks designed for IGCSE courses that my granddaughter will be studying when she enters Grade 10 in July. I found text books for five of the courses that she is studying: English, Mathematics, Business Studies, ICT and Modern History. Unfortunately, these textbooks are not from Cambridge University Press as the school requires but are instead from Hodder Education. They seem just as good and are free, as opposed to paying A$40 to A$50 each for the school mandated books. Here is the URL: https://www.gceguide.xyz/e-books

In a country like Indonesia where the cost of these sorts of text books is very burdensome, it seems perverse to require parents to expend hundreds of dollars when perfectly acceptable, free alternatives are available. Admittedly, the books are not paper copies but it would still be cheaper to ask parents to buy a cheap tablet on which to read the ebooks. One advantage is that students would then not have to lug around heavy textbooks in their backpacks.

Nonetheless, it is useful to me because I found a number of A level Mathematics textbooks that I've now downloaded and added to my already extensive collection of e-textbooks. For example, this advanced level Statistics textbook should be helpful as I'm currently embarked on a revision of this topic.

Google Classroom and Equatio

Google Classroom is apparently going to made available to everybody, not just teachers at registered schools. This change will roll out over the next few weeks, so I'll keep checking and hopefully have a chance to play around with it soon. Google stated:

We see value in bringing technology to people who want to learn, no matter the setting. That’s why we’re opening up Google Classroom to users without G Suite for Education accounts. Now, teachers and students in many different environments can teach or attend classes, manage assignments and instantly collaborate—all with their personal Google accounts. Starting today, these new Classroom users will be able to join existing classes and over the coming weeks, they’ll have the ability to create their own classes as well.

A change is also being made to Google Docs that it offers even better support for creating mathematical expressions. Here is what blogger Eric Curts of Control Alt Achieve had to say:

As a previous math teacher, I have a soft spot for the pains teachers and students can feel when trying to use math symbols and expressions in technology. The same thing can be said in science when typing chemical formulas. Technology seems to play best with normal letters and numbers, rather than exponents, subscripts, fractions, and such. 

Although there are many tools to help address these issues, one I have often recommended has been g(Math). This is an add-on for Google Docs, Sheets, and Forms, that provides a variety of ways to enter and insert mathematical expressions into these files. 

Recently I found out that g(Math) is getting an upgrade with a new name, new features, and new format! The new version is called Equatio and it makes it easier than ever to write with math and science symbols in Google Docs and Forms. Rather than being an add-on, this version of the tool is actually a Chrome extension, meaning eventually it will be able to work in many apps beyond just Docs and Forms. 

This new version is launching April 4, 2017, but you can see below for a sneak peek of this new tool complete with detailed directions, screenshots, and animations (click here).

So just to summarise what Equatio is all about, let's add the following excerpt from the same site that was just quoted from:

Equatio is a Chrome web extension that makes it easier to create and insert mathematical symbols and expressions, as well as scientific formulas when using Google Docs and Google Forms. Because it is a Chrome web extension, you need to be using the Chrome browser on a PC, Mac, or Chromebook to use Equatio. 

Equatio will be available for installation from the Chrome Web Store as of April 4, 2017. There will be a free version of Equatio that works in Google Docs, and a premium version that also works in Google Forms. 

I've been playing around with g(Math) and it makes adding LaTeX code easy but the resulting output is a graphic that can be resized and around which text can be wrapped. This is the same approach as adopted by Google Sites but not Blogger that makes use of Javascript to render the LaTeX characters. Equatio will operate by offering various options:

• The Equation Editor allowing you to enter math and science content with natural language and predictions.
• The LaTeX Editor allowing you to enter LaTeX characters
  
• Handwriting Recognition
• Speech Input
• Symbol Galleries

All mathematical expressions and chemical formulae created using Equatio will be inserted as images that can be resized and about which text can be wrapped.

Displaying Mathematical Expressions

If looking at this post on an android device, the equations will not display properly.
\[\sum_{i=0}^n i^2 = \frac{(n^2+n)(2n+1)}{6}\]It was with some difficulty that I've finally figured out how to display mathematical expressions properly in Blogger. The approach that I've adopted is to paste the following script just above the closing head tag in the blog's template:

The MathJax javascript will then render any LaTeX mathematical expressions, such as "\sum_{i=0}^n i^2 = \frac{(n^2+n)(2n+1)}{6}" (this is the one behind the expression at the top of this post). This site explains how to use the $ sign:

The double $ signs work fine but the single $ signs, that are supposed to give the inline expression, didn't work at first with the script as above. However, after pasting a modified script (see HMTL template), all was well, with both single dollar signs  and backward slash ( ...  backward slash ) being accepted.

Back on January 27th 2016, I made a post about the ASCIIMathML and the associated javascript program ASCIIMathML.js associated with it. Here is the text of that post:

The ASCIIMathML is a markup language that claims to be close to standard mathematical notation, easy to read and easy to type. In combination with a javascript program ASCIIMathML.js it will faithfully render mathematical expressions within HTML pages. I came across it when using a new online content management service called versal. Using the mathematical gadget provided by versal, I found the markup language very easy to use and was quickly able to create relatively complex formulae. This is my public URL on versal. The free version allows for the creation of unlimited public courses whereas the PRO version (at US$50 per year) allows for the creation of private courses, the tracking of student progress and other benefits.

Apparently ASCIIMathML.js has now been incorporated into the latest version of MathJax.js and the latter is:

a cross-browser JavaScript library that displays mathematical notation in web browsers, using MathML, LaTeX and ASCIIMathML markup. MathJax is released as open-source software under the Apache License and, importantly MathJax can display mathematical notation written in LaTeX or MathML markup. Because MathJax is meant only for math display, whereas LaTeX is a document layout language, MathJax only supports the subset of LaTeX used to describe mathematical notation."   Source.

ASCIIMathML is simpler and less verbose that LaTeX but in Blogger at least, the former seems temperamental and doesn't work consistently. The following equation when entered in ASCIIMathML as "x=(-b +- sqrt(b^2 – 4ac))/(2a)" fails to display the formatted equation. However, when entered in LaTeX as "x=\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}", it displays properly:

$$x=\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ A comprehensive list of LaTeX mathematical commands can be found here.

STEP

I just noticed a tweet by James Grimes concerning STEP. Here is what it looked like:

I went to the site to investigate and it looks an interesting and useful resource. I've signed up to the site and this is what my profile currently looks like:

Assignments can be downloaded in PDF format and looking at the first assignment, the site should provided a good source of questions that are compatible with the sort of 4 unit HSC questions that I'm currently working through.

Here's the central STEP question in Assignment 1:

Creating Online Courses and Classes

moodlecloud offers a free service along with paid plans

Despite my retirement from active teaching, I remain interested in education and am engaged in intermittent mathematics tutoring. I'm always on the lookout for ways to extend the tutoring session beyond the hour or so that I spend face-to-face with my student. To this end, I've made use of moodlecloud that provides a free service as well as paid options. Having so far only one student at any given time, I've found that the free service is quite adequate. Even though only 250 MB of storage is available, I've only used a small fraction.

The free service is adequate for those with a small number of students

Of course, it means that you can't upload large video files but you can link to them and that's enough. The free option is an excellent choice for people who work with only a small number of students.

When I was teaching full-time, I used the school-based Moodle installation most of the time but I did dabble with Edmodo and I still have an account. I recently archived my old courses and even posted to the Mathematics Forum. I may begin to experiment with it again. The site has an app that enables access via smartphone or tablet, so that's useful.

My Edmodo profile

I've also been experimenting with a site called Versal, that amongst other things offers an excellent tool for rendering mathematical expressions.

An example of the rendering of mathematical expressions

Of course, there's a hundreds of similar sites out there but these are three that I am using or have used, and they are all free. While writing this I was reminded of an excellent educational blog that I used to reference but haven't done so since leaving teaching. Here is the link. 

This site offers excellent reviews of educational resources

Day 24179

Today I'm 24179 days old and it's a prime day because 24179 is prime. It forms part of a twin prime pair with 24181. It enjoys 16 entries in the OEIS. One of the entries states that it is a member of a sequence of numbers with the property that it is the first member of a twin prime pair whose sum equals the sums of two consecutive smaller pairs of twin primes. The first member of the sequence A225943 is 17 because 17 + 19 = (5 + 7) + (11 + 13). In the case of 24179, we have 24179 + 24181 = 48360 = (12071 + 12073) + (12107 + 12109). That's the most interesting of the entries. It doesn't take too long to discover the smaller primes. Using WolframAlpha, I just searched for primes between 12000 and 12200 and looked at the sums of the various consecutive twin prime pairs.