To celebrate this day of mathematics, I want to share my favourite equation involving Pi, the Bailey–Borwein–Plouffe (BBP) formula:

This is my favourite for a number of reasons. Firstly, Simon Plouffe and the late Peter Borwein (two of the authors that this formula is named after) are Canadian! While I personally have nothing to do with this formula, the fact that fellow Canadians contributed to such an elegant equation is something that I like to brag about.

Secondly, I find it fascinating how Plouffe first discovered this formula using a computer program. It can often be debated whether mathematics is discovered or invented, but there’s no doubt here since Plouffe found this formula by doing an extensive search with the PSLQ integer relation algorithm (interfaced with Maple). This is an example of how, with ingenuity and creativity, one can effectively use algorithms and programs as powerful tools to obtain mathematical results.

And finally (most importantly), with some clever rearranging, it can be used to compute arbitrary digits of Pi!

Digit 2024 is 8
Digit 31415 is 5
Digit 123456 is 4
Digit 314159 is also 4
Digit 355556 is… F?

That last digit might look strange… and that’s because they’re all in hexadecimal (base-16, where A-F represent 10-15). As it turns out, this type of formula only exists for Pi in bases that are powers of 2. Nevertheless, with the help of a Maple script and an implementation of the BBP formula by Carl Love, you can check out this Learn document to calculate some arbitrary digits of Pi in base-16 and learn a little bit about how it works.

After further developments, this formula led to project PiHex, a combined effort to calculate as many digits of Pi in binary as possible; it turns out that the quadrillionth bit of Pi is zero! This also led to a class of BBP-type formulas that can calculate the digits of other constants like (log2)*(π^2) and (log2)^5.

Part of what makes this formula so interesting is human curiosity: it’s fun to know these random digits. Another part is what makes mathematics so beautiful: you never know what discoveries this might lead to in the future. Now if you’ll excuse me, I have a slice of lemon meringue pie with my name on it 😋

References
BBP Formula (Wikipedia)
A Compendium of BBP-Type Formulas
The BBP Algorithm for Pi

     On International Women’s Day we celebrate the achievements of women around the world. One inspiring story of women in STEM is that of Sophie Germain (1776-1831), a French mathematician and physicist who made groundbreaking strides in elasticity theory and number theory. She learned mathematics from reading books in her father’s library, and while she was not permitted to study at the École Polytechnique, due to prejudice against her gender, she was able to obtain lecture notes and decided to submit work under the name Monsieur LeBlanc. Some prominent mathematicians at the time, including Joseph-Louis Lagrange and Carl Friedrich Gauss, with whom Germain wrote, recognized her intellect and were supportive of her work in mathematics. 

     Sophie Germain is remembered as a brilliant and determined trailblazer in mathematics. She was the first woman to win a prize from the Paris Academy of Sciences for her contributions in elasticity theory and was among the first to make significant contributions toward proving Fermat’s Last Theorem. Among her many accomplishments, one special case of Fermat’s Last Theorem she was able to prove is when the exponent takes the form of what is now known as a Sophie Germain prime: a prime, p, such that 2p+1 is also a prime. The associated prime, 2p+1, is called a safe prime. 

     To mark International Women’s Day, I’ve created a document exploring the Ulam spiral highlighting Sophie Germain primes and safe primes, as an adaptation of Lazar Paroski’s Ulam spiral document. The image below displays part of the Ulam spiral with Sophie Germain primes highlighted in red, safe primes highlighted in blue, primes that are both a Sophie Germain prime and safe prime highlighted in purple, and primes that are neither in grey. 

     The document also contains small explorations of these types of prime numbers. For instance, one interesting property of safe primes is that they must either be 5, 7 or take the form 12k-1 for some k≥1. This can be shown from the fact a safe prime q must equal 2p+1 for some prime, p (a Sophie Germain prime), by definition. Then, since q and p are prime, for q > 7 we can determine through contradiction that q ≡ 3 (mod 4) and q ≡ 5 (mod 6), to conclude q ≡ 11 (mod 12) ≡ -1 (mod 12). And so, q = 12k-1 for some k≥1. The Maple Learn document can be found here along with its Maple script. The document also includes a group where you can test whether some positive integer of your choice, n, is a Sophie Germain prime or a safe prime. Alternatively, given n, a button press will display the next Sophie Germain prime greater than n, using Maple’s NextSafePrime command in the number theory package.  

     In mathematics, there is no shortage of interesting rabbit holes to dive into; many of which are the result of past and present women in mathematics, like Sophie Germain, who have persevered despite their hardships. 

Attention Maple enthusiasts! It gives me great pleasure to announce Maple 2024! Maple 2024 brings together a collection of new features and enhancements carefully designed to enrich your mathematical explorations. Maple 2024 is the result of a lot of hard work by a lot of people, and there is far more in it than I can cover here. But I’d like to share with you some of my favorite features in this release.

AI Formula Assistant

The AI Formula Assistant in Maple 2024 is undoubtedly the feature that excites me the most, especially considering how often I’m asked the question: 'When will Maple include AI features?' This assistant serves as your new mathematical companion and will change the way you look up and enter formulas and equations. Driven by advanced AI technology, it presents a range of relevant options based on your search query. Alongside suggestions, you'll also receive detailed explanations for each formula and its parameters so you can select the one you need, and then you can insert the formula into your document at a click of a button, as a proper Maple expression.

NaturalLanguage Package

The Formula Assistant is built on top of the new NaturalLanguage package, which integrates powerful language models like GPT-4 and ChatGPT from OpenAI into Maple. With this feature, you can leverage large language models to process natural language within Maple. Ask the AI to explain concepts, provide additional details, find specific Maple commands, and more. Of course, since this is a Maple package, you can also use it as a basis to build your own AI-powered applications inside Maple. We’re really looking forward to seeing what you will do with it!

Argument Completion
We’ve had a lot of requests from people who wanted Maple’s command completion features to do even more, and I’m happy to say that Maple 2024 delivers. The new argument completion feature in Maple 2024 is poised to significantly enhance your experience with commands. For many users, including myself, not being aware of all the options a command takes is a challenge, often leading me to refer to help pages for clarification. With argument completion, that's no longer a concern. Just enter the command with the help of the existing command completion feature, then automatic argument completion takes over to guide you through the rest. Give it a try with the 'plot' command!

Check My Work

A personal favorite feature that's gotten even better in Maple 2024 is Check My Work. As someone who has tutored students, I vividly recall their stress before exams, often receiving emails and text messages from them seeking last-minute help. At the time, I found myself wishing the students had a way to check their work themselves, so we would all be less stressed! So I was super excited when we added the first Check My Work feature a couple of years ago, and am very happy that it gets better ever year. In Maple 2024, we’ve expanded its capabilities to support problems involving factoring, simplification, and limits.

Scrollable Matrices:
This feature will definitely resonate with many of the engineers in the Maple Primes community. If you're someone who works with worksheets containing large matrices, you've likely wished that you could scroll the matrices inside your document instead of having to launch a separate matrix browser. With Maple 2024, your wish has come true.

Color Bars

And finally, for those of you who appreciated the addition of color bars in Maple 2023 but wanted to see them extended to more 2D and 3D plots, you'll be delighted to know that this is exactly what we’ve done. We’ve also added new customization options, providing you with greater control over appearance.

This is just a partial glimpse of what's new in Maple 2024. For a comprehensive overview, visit What’s New in Maple 2024.

The Lunar New Year is approaching and 2024 is the Year of the Dragon! This inspired me to create a visualization approximating the dragon curve in Maple Learn, using Maple. 

The dragon curve, first described by physicist John Heighway, is a fractal that can be constructed by starting with a single edge and then continually performing the following iteration process:  

Starting at one endpoint of the curve, traverse the curve and build right triangles on alternating sides of each edge on the curve. Then, remove all the original edges to obtain the next iteration. 

This process continues indefinitely, so while we can’t draw the fractal perfectly, we can approximate it using a Lindenmayer system. In fact, Maple can do all the heavy lifting with the tools found in the Fractals package, which includes the LSystem subpackage to build your own Lindenmayer systems. The subpackage also contains different examples of fractals, including the dragon curve. Check out the Maple help pages here: 

Overview of the Fractals Package  

Overview of the Fractals:-LSystem Subpackage 

Using this subpackage, I created a Maple script (link) to generate a Maple Learn document (link) to visualize the earlier iterations of the approximated dragon curve. Here’s what iteration 11 looks like: 

You can also add copies of the dragon curve, displayed at different initial angles, to visualize how they can fit together. Here are four copies of the 13th iteration: 

Mathematics is full of beauty and fractals are no exception. Check out the LSystemExamples subpackage to see many more examples. 

Happy Lunar New Year! 

Curve sketching is an important skill for all calculus students to learn. In an era where technology is increasingly relied upon to perform mathematical computations and representations, maintaining fundamental skills such as curve sketching is more important than ever.

The new “Curve Sketching” collection is now available on Maple Learn. This collection provides background information on the process of curve sketching and opportunities to put this knowledge into practice. By starting with the “Curve Sketching Guide” and “Relationships Between Derivatives” documents, students are exposed to observational and computational strategies for drawing a function and its 1st and 2nd derivatives.

After looking through these documents, students can begin to practice sketching by observing and interpreting graphical properties with the “Sketch Derivative From Function Graph”, “Sketch Second Derivative From Function Graph”, and “Sketch Function From Derivative Graphs” activities:

Once a student has mastered extracting sketching information by graphical observation, they are ready for the next step: extracting information from a function’s definition. At this point, the student is ready to try sketching from a blank canvas with the “Sketch Curve From Function Definition” activity:

This collection also has activities for students below the calculus level. For example, the “Curve Sketching Quadratics Activity”, can be completed using only factoring strategies:

Whether you are a quadratics rookie or a calculus pro, this collection has an interactive activity to challenge your knowledge. Have fun sketching!

Two solstices occur on Earth every year, around June 21^st and December 21^st, often called the “June Solstice” and the “December Solstice” respectively. These solstices occur when the sun reaches its northernmost or southernmost point relative to the equator. During a solstice, the Northern Hemisphere will either experience the most sunlight of the year or the least sunlight of the year, while the Southern Hemisphere will experience the opposite phenomenon. The hemisphere with the most sunlight experiences a summer solstice, while the other hemisphere experiences a winter solstice.

Canada is located in the Northern Hemisphere and this Thursday, December 21^st, we will be experiencing a winter solstice. As the day with the least sunlight, this will be the shortest day of the year and consequently the longest night of the year.

Here in Canada, the sun will reach its minimum elevation during the winter solstice, and it will reach its maximum elevation during the Southern Hemisphere’s summer solstice on the same day. 

How high in the sky does the sun really get during these solstices? Check out our new Maple Learn document, Winter and Summer Solstice Sun Angles to find out. The answer depends on your latitude; for instance, with a latitude of approximately 43.51°, the document helps us find that the maximum midday elevation of the sun, which occurs during a summer solstice, will be 69.99°.

But how is the latitude of a location determined in the first place? See Maple Learn’s Calculating Latitude document to find out how the star Polaris, the center of the Earth, and the equator are all connected to latitude.

Latitude is one of two geographical coordinates that are paired together to specify a position on Earth, the other being longitude. See our Calculating Longitude document to explore how you can use your local time to approximate your longitude.

Armed with these coordinates, you can describe your position on the planet and find any number of interesting facts, such as your solstice sun angles from earlier, the time for sunrise and sunset, and the position of the Moon.

Happy Winter Solstice!

A new “Sudoku Puzzle” document is now on Maple Learn! Sudoku is one of the world’s most popular puzzle games and it is now ready to play on our online platform. 

This document is a great example of how Maple scripts can be used to create complex and interactive content. Using Maple’s built-in DocumentTools:-Canvas package, anyone can build and share content in Maple Learn. If you are new to scripting, a great place to start is with one of the scripting templates, which are accessible through the Build Interactive Content help page. In fact, I built the Sudoku document script by starting from the “Clickable Plots” template.

A Sudoku puzzle is a special type of Latin Square. The concept of a Latin Square was introduced to the mathematical community by Leonard Euler in his 1782 paper, Recherches sur une nouvelle espèce de Quarrés, which translates to “Research on a new type of square”. A Latin Square is an n by n square array composed of n symbols, each repeated exactly once in every row and column. The Sudoku board is a Latin Square where n=9, the symbols are the digits from 1 to 9,  and there is an additional requirement that each 3 by 3 subgrid contains each digit exactly once. 

Mathematical research into Sudoku puzzles is still ongoing. While the theory about Latin Squares is extensive, the additional parameters and relative novelty of Sudoku means that there are still many open questions about the puzzle. For example, a 2023 paper from Peter Dukes and Kate Nimegeers examines Sudoku boards through the lenses of graph theory and linear algebra.

The modern game of Sudoku was created by a 74-year-old Indiana retiree named Howard Garnes in 1979 and published under the name “Number Place”. The game had gained popularity in Japan by the mid-1980s, where it was named “Sudoku,” an abbreviation of the phrase “Sūji wa dokushin ni kagiru,” which means “the numbers must be single”.

Today, Sudoku is a worldwide phenomenon. This number puzzle helps players practice using their logical reasoning, short-term memory, time management, and decision-making skills, all while having fun. Furthermore, research from the International Journal of Geriatric Psychiatry concluded that doing regular brain exercises, like solving a Sudoku, is correlated with better brain health for adults over 50 years old. Additionally, research published in the BMJ medical journal suggests that playing Sudoku can help your brain build and maintain cognition, meaning that mental decline due to degenerative conditions like Alzheimer’s would begin from a better initial state, and potentially delay severe symptoms. However, playing Sudoku will by no means cure or prevent such conditions.

If you are unfamiliar with the game of Sudoku, need a refresher on the rules, or want to improve your approach, the “Sudoku Rules and Strategies” document is the perfect place to start. This document will teach you essential strategies like Cross Hatching:

And Hidden Pairs:

After reading through this document, you will have all the tools you need to start solving puzzles with the “Sudoku Puzzle” document on Maple Learn. 

Have fun solving!

We have just released an update to Maple 2023 to address a couple of issues.

• We’ve had a few reports of people encountering “Kernel connection has been lost” errors, and this update should fix that problem.
• We fixed a problem involving entering math (specifically, the right curly bracket, } ) using an international keyboard.

If you are experiencing kernel connection problems or use Maple with an international keyboard, you should install this update.

This update is available through Tools>Check for Updates in Maple, and is also available from the Maple 2023.2.1 download page. MapleSim users can get this update from the MapleSim Check for Updates or from the MapleSim 2023.2.1 download page, where you will also find an update to the MapleSim Ropes and Pulleys Library.

How much did you weigh when you were born? How tall are you? What is your current blood pressure? It is well documented that in the general population, these variables – birth weight, height, and blood pressure – are normally or approximately normally distributed. This is the case for many variables in the natural and social sciences, which makes the normal distribution a key distribution used in research and experiments. 

The Maple Learn Examples Gallery now includes a series of documents about normal distributions and related topics in the Continuous Probability Distributions subcollection.

The Normal Distribution: Overview will introduce you to the probability density function, cumulative distribution function, and the parameters of the distribution. This document also provides an opportunity for you to alter the parameters of a normal distribution and observe the resulting graphs. Try out a few real life examples to see the graphs of their distributions! For example, according to Statology, diastolic blood pressure for men is normally distributed with a mean of 80 mmHg and a standard deviation of 20 mmHg.

Next, the Normal Distribution: Empirical Rule document introduces the empirical rule, also referred to as the 68-95-99.7 rule, which describes approximately what percentage of normally distributed data lies within one, two, and three standard deviations of the distribution’s mean.

The empirical rule is frequently used to assess whether a set of data might fit a normal distribution, so Maple Learn also provides a Model Checking Exploration to help you familiarize yourself with applications of this rule. 

In this exploration, you will work through a series of questions about various statistics from the data – the mean, standard deviation, and specific intervals – before you are asked to decide if the data could have come from a normal distribution. Throughout this investigation, you will use the intuition built from exploring the Normal Distribution: Overview and Normal Distribution: Empirical Rule documents as you analyze different data sets.

Once you are confident in using the empirical rule and working with normal distributions, you can conduct your own model checking investigations in real life. Perhaps a set of quiz grades or the weights of apples available at a grocery store might follow a normal distribution – it’s up to you to find out!

A new feature has been released on Maple Learn called “collapsible sections”! This feature allows for users to hide content within sections on the canvas. You can create a section by highlighting the desired text and clicking this icon in the top toolbar: . 

“Well, when can I actually use sections?” you may ask. Let me walk you through two quick scenarios so you can get an idea.

For our first scenario, let’s say you’re an instructor. You just finished a lesson on the derivatives of trigonometric functions and you’re now going through practice problems. The question itself is not long enough to hide the answers, so you’re wondering how you can cover the two solutions below so that the students can try out the problem themselves first.

Before, you might have considered hyperlinking a solution document or placing the solution lower down on the page. But now, collapsible sections have come to the rescue! Here’s how the document looks like now:  

You can see that the solutions are now hidden, although the section title still indicates which solution it belongs to. Now, you can 1) keep both solutions hidden, 2) show one solution at a time, or 3) show solutions side-by side and compare them!

Now for the second scenario, imagine you’re making a document which includes a detailed visualization such as in Johnson and Jackson’s proof of the Pythagorean theorem. You want the focus to be on the proof, not the visualizations commands that come along with the proof. What do you do?

It’s an easy solution now that collapsible sections are available!

Now, you can focus on the proof without being distracted by other information—although the visualization commands can still be accessed by expanding the section again.

So, take inspiration and use sections to your advantage! We will be doing so as well. you may gradually notice some changes in existing documents in the Maple Learn Gallery as we update them to use collapsible sections. 

Happy document-making!

A new collection has been released on Maple Learn! The new Pascal’s Triangle Collection allows students of all levels to explore this simple, yet widely applicable array.

Though the binomial coefficient triangle is often referred to as Pascal’s Triangle after the 17th-century mathematician Blaise Pascal, the first drawings of the triangle are much older. This makes assigning credit for the creation of the triangle to a single mathematician all but impossible.

Persian mathematicians like Al-Karaji were familiar with the triangular array as early as the 10th century. In the 11th century, Omar Khayyam studied the triangle and popularised its use throughout the Arab world, which is why it is known as “Khayyam’s Triangle” in the region. Meanwhile in China, mathematician Jia Xian drew the triangle to 9 rows, using rod numerals. Two centuries later, in the 13th century, Yang Hui introduced the triangle to greater Chinese society as “Yang Hui’s Triangle”. In Europe, various mathematicians published representations of the triangle between the 13th and 16th centuries, one of which being Niccolo Fontana Tartaglia, who propagated the triangle in Italy, where it is known as “Tartaglia’s Triangle”. 

Blaise Pascal had no association with the triangle until years after his 1662 death, when his book, Treatise on Arithmetical Triangle, which compiled various results about the triangle, was published. In fact, the triangle was not named after Pascal until several decades later, when it was dubbed so by Pierre Remond de Montmort in 1703.

The Maple Learn collection provides opportunities for students to discover the construction, properties, and applications of Pascal’s Triangle. Furthermore, students can use the triangle to detect patterns and deduce identities like Pascal’s Rule and The Binomial Symmetry Rule. For example, did you know that colour-coding the even and odd numbers in Pascal’s Triangle reveals an approximation of Sierpinski’s Fractal Triangle?

See Pascal’s Triangle and Fractals

Or that taking the sum of the diagonals in Pascal's Triangle produces the Fibonacci Sequence?

See Pascal’s Triangle and the Fibonacci Sequence

Learn more about these properties and discover others with the Pascal’s Triangle Collection on Maple Learn. Once you are confident in your knowledge of Pascal’s Triangle, test your skills with the interactive Pascal’s Triangle Activity. 

On November 11^th, Canada and other Commonwealth member states will celebrate Remembrance Day, also known as Armistice Day. This holiday commemorates the armistice signed by Germany and the Entente Powers in Compiègne, France on November 11, 1918, to end the hostilities on the Western Front of World War I. The armistice came into effect at 11:00 am that morning – the “eleventh hour of the eleventh day of the eleventh month”. 

Similar to how November 11^th – which can be written as 11/11 – is a palindromic date that reads the same forward and backward, last year there was “Twosday” – February 22, 2022, also written 22/2/22. 

Palindromic dates like November 11^th that consist only of a day and a month happen every year, but how long will we have to wait until the next “Twosday”? We can use Maple Learn’s new Calendar Calculator to find out!

To use this document, simply input two dates and press ‘Calculate’ to find the amount of time between them, presented in a variety of units. For example, here are the results for the number of days left until Christmas from November 11^th of this year:

If we return to our original question, which concerns how long we’ll have to wait until the next “Twosday”, we can use this document to find our answer:

You can use this document as a countdown to find out how much time is left until your favorite holiday, your next birthday, or the time between now and any past or future date; try out the countdown document here!

We have just released updates to Maple and MapleSim.

Maple 2023.2 includes a strikethrough character style, a new unit system, improved behavior when editing or deleting subscripts, improved find-and-replace, better mouse selection of piecewise functions and the contents of matrices, and more.  We recommend that all Maple 2023 users install this update.

This update also include a fix to the problem with setoptions3d, as first reported on MaplePrimes. Thanks, as always, for helping us make Maple better.

This update is available through Tools>Check for Updates in Maple, and is also available from the Maple 2023.2 download page, where you can find more details.

At the same time, we have also released an update to MapleSim, which contains a variety of improvements to MapleSim and its add-ons. You can find more information on the MapleSim 2023.2 download page.

Many everyday decisions are made using the results of coin flips and die rolls, or of similar probabilistic events. Though we would like to assume that a fair coin is being used to decide who takes the trash out or if our favorite soccer team takes possession of the ball first, it is impossible to know if the coin is weighted from a single trial.

Instead, we can perform an experiment like the one outlined in Hypothesis Testing: Doctored Coin. This is a walkthrough document for testing if a coin is fair, or if it has been doctored to favor a certain outcome. 

This hypothesis testing document comes from Maple Learn’s new Estimating collection, which contains several documents, authored by Michael Barnett, that help build an understanding of how to estimate the probability of an event occurring, even when the true probability is unknown.

One of the activities in this collection is the Likelihood Functions - Experiment document, which builds an intuitive understanding of likelihood functions. This document provides sets of observed data from binomial distributions and asks that you guess the probability of success associated with the random variable, giving feedback based on your answer. 

Once you’ve developed an understanding of likelihood functions, the next step in determining if a coin is biased is the Maximum Likelihood Estimate Example – Coin Flip activity. In this document, you can run as many randomized trials of coin flips as you like and see how the maximum likelihood estimate, or MLE, changes, bearing in mind that if a coin is fair, the probability of either heads or tails should be 0.5. 

Finally, in order to determine in earnest if a coin has been doctored to favor one side over the other, a hypothesis test must be performed. This is a process in which you test any data that you have against the null hypothesis that the coin is fair and determine the p-value of your data, which will help you form your conclusion.

This Hypothesis Testing: Doctored Coin document is a walkthrough of a hypothesis test for a potentially biased coin. You can run a number of trials on this coin, determine the null and alternative hypotheses of your test, and find the test statistic for your data, all using your understanding of the concepts of likelihood functions and MLEs. The document will then guide you through the process of determining your p-value and what this means for your conclusion.

So if you’re having suspicions that a coin is biased or that a die is weighted, check out Maple Learn’s Estimating collection and its activities to help with your investigation!

With Halloween right around the corner, we at Maplesoft wanted to celebrate the occasion with an activity where you can carve your own pumpkin… using math! 

Halloween is said to have originated a few hundred years back in ancient Celtic festivals, specifically one called Samhain. This was celebrated from October 31st to November 1st to mark the end of harvesting season and the beginning of winter, or the "darker quarter" of the year. Since then, Halloween has evolved into a fun celebration of candy and costumes in many countries!

With that said, here’s my take on the pumpkin carving activity: 

The great thing is, if you mess up, you can always go back; unlike carving pumpkins in real life. My design is pretty simple (although cute), so let’s see what you all can impress us with!

You can also make your own original art and publish it to your channel so that anyone can see your own artistic creations. You can also attend the Maple Conference next week on October 26 and 27, an event filled with two days of presentations from members of the Maplesoft Community. Participants will also be able to see all the artwork submitted for the Art Gallery and Creative Showcase, where you can draw inspiration for your own submissions to next year’s showcase! The conference is virtual and free of charge, and you can register here.

Looking forward to seeing you there!