Imagine walking into a room full of strangers. At first, everyone seems completely unrelated, with different names, different lives, and different birthdays. As the room gets more crowded, we of course know that sharing similarities with the strangers in the room becomes more likely.

But just how likely is it that, for example, two people in the room share a birthday?

The Birthday Paradox asks, more specifically: How many people must be in the room so that it is more likely than not that two people share a birthday?

Intuitively, we might imagine it must be well over 100 since there are 365 possible birthdays that each stranger can have. Probability, however, has a habit of ignoring our intuition.

Suppose there are only two people in the room: you, and one stranger. You have your own birthday, so there is a 1/365 ≈ 0.27% chance that the stranger next to you shares your birthday. If we add a third person to the room, we must consider the probability that either of the two strangers share your birthday, or the two strangers share a birthday with each other. This overlap makes our calculations more difficult, and it will only get worse as we add more people.

Instead, there's a cleaner way to find the solution. Let's find the probability P(n) that no two people share a birthday, then subtract it from 1 to get the probability that two people do share a birthday. Once P(n) < 0.5, then 1 - P(n) > 0.5, meaning at this value of n, there is a greater than 50% chance that two people in the room share a birthday.

For the case of only two people, there is a 364/365 chance that the second person has a different birthday than the first person. If we introduce a third person, there are 363 unclaimed birthdays left in the year, so a 363/365 chance that the third person's birthday falls on one of these days. Repeating this process for n people in the room, we get:

If we keep increasing n, we see that the probability P(n) that two people don't share a birthday decreases, meaning the probability 1 - P(n) that they do share a birthday increases.

Below are some values of 1 - P(n) I found using Maple Learn:

Based on these calculations, we see that when 23 people are in the room, it is more likely than not that at least two people in the room share a birthday.

Our question was very specific: we wanted to know what n needed to be in order for the probability of two people sharing a birthday to be > 50%. We found this number to be 23. But, we could extend this question to, say, the probability being 99.9%. This happens when there are 70 people in the room. Consider the animation below to see this visually, where the x-axis represents the number of people n, and the y-axis represents the probability that two people share a birthday, 1 - P(n):

How many people must be in the room to guarantee that two of them share a birthday? By the shape of the graph above, it may be tempting to think that it is impossible to guarantee this, since the probability seems to asymptotically approach 1. However, we must think through this logically.

If 365 people were in the room, it is still possible that their birthdays would all fall on a different day (although it's extremely unlikely). But if we have more people than we have available birthdays, we can say at least two people will share a birthday with 100% certainty. That is, P(n) = 0 when n > 365, so our formula for P(n) above is limited to values of n that are 365 or less. This is a basic principle in mathematics called the "pigeonhole principle", which uses the analogy that if we have more pigeons than we have holes to put them in, then at least one of the holes must have at least two pigeons.

These ideas can be conceptually confusing, but once you go through the mathematical arguments a few times, mentioning the Birthday Paradox in a conversation is a great way to show off your random math knowledge and confuse your peers!

Attached is an interactive Maple worksheet from Maple's Math Apps gallery if you'd like to investigate the Birthday Paradox further: The_Birthday_Paradox.mw. If you're still interested in more info/applications, see this Maple Primes post by John May.

We’ve just launched Maple Flow 2026. And yes, the title of this post doesn’t follow the standard formula for a “new version, new features” announcement.

But this is not only a post about a new version number and new features.

It is about a problem engineers know too well: important calculations trapped inside spreadsheets, scripts and old project files - visible only as cell references, code fragments and hidden logic.

That might be fine when the calculation is being built. It is not fine when the calculation must be reviewed, reused, defended, audited or handed to another engineer.

Maple Flow is built around a simple idea: engineering calculations should be readable, structured, auditable and easy to share.

The new release gives engineers better ways to create those calculations, bring existing work into Maple Flow, and turn legacy material into documentation that actually explains the engineering.

You can now generate Maple Flow worksheets using tools like Codex or Claude Code. This makes it easier to transform ideas into structured technical analyses and turn existing calculations into clear, auditable engineering documents.

Here are a few ways to use this new integration,

Rescue calculations from spreadsheets

Many engineering teams have years - sometimes decades - of calculations sitting in Excel. Those spreadsheets often contain real engineering value, but the logic is fragmented across cells, sheets and references.

Manually translating all that work into a clearer format can feel like too much effort to even start.

With Maple Flow 2026 and an AI coding assistant such as Codex or Claude Code, you can use a simple instruction like:

Convert this Excel spreadsheet to a Maple Flow worksheet

The AI coding assistant then helps transform spreadsheet logic into a structured Maple Flow worksheet with natural 2D math, clearer variable names and calculations that are visible on the page.

This is not just file conversion. It is a way to expose the engineering intent hidden inside legacy spreadsheets and make that work easier to understand, extend and review.

Generate engineering worksheets from a prompt

The same AI-assisted workflow can also help create a new Maple Flow worksheet from a natural language prompt.

Ask for a design calculation, describe the engineering context, specify the assumptions and ask for supporting text, equations, plots or diagrams. The result can be a worksheet that already contains readable math, explanations and visual context.

Stop hiding math in scripts

Maple Flow 2026 also opens up a practical path for moving useful work out of scripts and into worksheets that communicate the engineering more clearly.

For example, a Matlab script built around low-level loops, switch/case logic and numerical iteration can be translated into a Maple Flow worksheet that uses higher-level mathematical constructs.

In one example, a crude while-loop numerical solver became an fsolve call, and switch/case logic became a much more readable piecewise expression.

The point is not simply to change syntax. It is to move from code that performs a calculation to a worksheet that explains the calculation.

I've outlined a few ideas that demonstrate how you can use AI coding assistants to generate Maple Flow worksheets. There's much more you can do.

Of course, this functionality is subject to the same disclaimers as any other AI-generated content:

• Worksheets are generated by and subject to the limitations of your AI model
• your mileage may vary
• and you absolutely need a (human) expert in the loop to check the output.

But as a starting point, this workflow is seriously powerful.

Set up your AI coding assistant to generate Maple Flow worksheets with the instructions here.

Maple Flow 2026 introduces in-worksheet programming, so you can write programs and scripts directly on the canvas and execute them inline with the rest of your worksheet.

You can enter programs with the keyboard or use the new Programming palette, which inserts common templates such as loops and if/else blocks with the right structure and indentation.

Productivity features help keep the program readable:

• automatic closing of procedures, loops and conditionals after the opening line is entered
• intelligent insertion of statement separators when appropriate
• automatic indentation to preserve structure and readability

The existing Code Editor remains the right place for larger scripts. In-worksheet programming is for the logic you want to sit directly beside the math, text and results.

The Mathcad Migration Assistant has also been updated. It now translates much of the content of Mathcad Prime files into Maple Flow format, in addition to the existing support for Mathcad 13, 14 and 15.

Because Maple Flow 2026 now supports in-worksheet programming, the updated Migration Assistant also translates Mathcad programs.

For Mathcad thinking about moving to Maple Flow, the update makes migration faster and more practical. Existing work moves over with more of the structure and intent preserved.

Worksheets now compute faster! The results depend on the specific calculations, but the improvements can be very noticable - these are selected results from our internal benchmarking suite

For engineers working with larger or more complex worksheets, that kind of improvement changes the feel of the product.

Maple Flow 2026 adds more ready-to-use examples to the Application Gallery, including a new Semiconductor section.

The gallery now includes worksheets semiconductor oxide growth, heat sink sizing, cylindrical fin heat transfer, NMOS threshold voltage, MOSFET power loss, photolithography scanner limits and boron implantation stability.

Maple Flow 2026 also improves everyday interoperability with Excel. You can now paste more types of data from Excel into Maple Flow, including data formatted as percentages.

It is a small feature with a practical benefit: less cleanup, fewer interruptions and a smoother path from spreadsheet data to engineering calculation.

The most important thing about Maple Flow 2026 is not any single feature - it is the direction of travel. Engineering work is becoming more connected. Calculations are moving between spreadsheets, scripts, AI coding assistants, design tools and formal documentation.

Maple Flow gives those calculations a place where the math is visible, the structure is clear and the result is ready to share.

So yes, Maple Flow 2026 includes new features. But the bigger story is this: engineers can now move more of their existing work out of hidden logic and into auditable worksheets that communicate the engineering.

Explore the new features, try the examples and experiment with generating worksheets using AI coding assistants. Grab a trial here.

Today is Pi approximation day (22/7) and I will use that as an excuse to share my new favourite expression for Pi:

And this isn't even an approximation! Recently, continuous mathematics has found its own equivalent to the digital hardware NAND gate. In his paper “All elementary functions from a single binary operator”, Andrzej Odrzywołek demonstrated that a single functional primitive can generate the entire standard continuous spectrum of operations. In other words, every single button on a scientific calculator, from addition and subtraction to sines, cosines, and logarithms, can be built using just this one function.

This Maple Worksheet explores how the 'Exp-Minus-Log' ("EML") operator, when paired solely with the constant 1, can be systematically nested to construct basic arithmetic, constants, and complex transcendental functions within Maple.

In essence, he discovered that the binary operator EML, along with the constant 1, forms a basis for the set of standard scientific-calculator operations.

This means that functions like ,  and operations like  or  can be creating by composing EML with itself in clever ways. Some constants and functions are trivial to represent, such as EML(1,1) = e or EML(x, 1) = exp(x), others however, are not...

With a quick one-command tweak, you can get Maple to use the property of the extended reals that

And then with a simple argument about standard branches, you can construct the natural logarithm for real numbers, which immediately leads the constant zero:

You can then expand the tools in your toolbox by creating subtraction with EML, ln(x), and exp(x)

Which then expands the toolbox further by allowing for the construction unary minus from the constant 0 (since -x = 0 - x), and then addition (since a+b=a-(-b))

Since we've constructed addition, subtraction, zero and one, we can technically construct every integer! It would not be very pleasant, and by no means optimal... but you could! Here's 7 for example:

The next step to building all the standard functions is multiplication and inversion. And these use the classic trick by using the fact that x=exp(ln(x)) can help simplify:

These are compositions of exp, addition, ln, and unary minus (all functions constructed previously), which means they can be made with only EML:

It's at this point that I will leave the derivation of division (a/b) and exponentiation (a^b) as exercises for the reader, so I can skip to something a little more complex...

To go beyond the basic operators, you'll need to step into the complex domain by constructing the imaginary constant i. To do this, take ln(-1) = -i*Pi (by using the standard branch) and combine it with Euler's formula

And once again the expression on the left-hand side is made up of operations that were all previously defined, so you can compose EML to get a new constant:

And finally, it's possible to break down the expression for Pi from the start, since it's the product i*ln(-1)

By successfully extracting the mathematical constants i and Pi, I think this demonstrates the complete constructive capability of the EML operator in the complex domain. While the resulting syntax trees become exponentially deep and unoptimized for human readability, they prove that continuous operations do not require a massive, distinct toolbox. Future applications of this uniform binary structure could dramatically simplify symbolic regression and machine learning optimization models. 
Ultimately, the EML operator reveals the remarkable truth that the vast complexity of scientific mathematics can be distilled down to a single, beautiful building block.

Isn't math awesome?

Last week, we launched the Maplesoft Math Success Platform. 
 

This launch reflects a lot of conversations I’ve had over the past year with educators and institutions about what it means to teach and learn math in the age of AI. 

At first, many of those conversations were about visibility. If students were completing homework, quizzes, and other assessments with help from AI, those results became harder to interpret. Did students understand the work, or had they copied down a solution that made sense in the moment without building the understanding needed to do something similar on their own?

That visibility still matters. 

Over time, though, those conversations led to a more nuanced conclusion. The question is not simply how we prevent students from taking shortcuts. It is how we help them develop the mathematical judgment, intuition, and critical thinking they will need in a world where AI is part of how they learn and work. 

In some ways, that has become even more important. When answers are easy to generate, students need to be able to test ideas, recognize when something does not make sense, explain their reasoning, and trust their own thinking. 

That is why I am proud to share the launch of the Maplesoft Math Success Platform. 

Built on Maple, the platform brings together our math technology and extends it with analytics, AI-driven insights, targeted resources, and content expertise to help institutions support math learning in a more complete way. 

It gives instructors and learning support teams better insight into where students are struggling, supports the creation of better questions and learning experiences, helps students move beyond the answer, and helps institutions respond to a world where AI is now part of how students practice, study, and get help. 

You can learn more about the Maplesoft Math Success Platform on our website.

We also wrote more about the thinking behind this launch in our new whitepaper, Math Education in the Age of AI: From Grading Answers to Understanding Student Progress. It looks at why math education needs a new approach in the age of AI: one that helps instructors ask better questions, create learning experiences that build understanding, and use learning signals to see where students need support.

I’d love to hear what you think. How are you seeing AI change the way students learn, practice, and get help in math? And what kinds of tools or approaches do you think will be most important as math education continues to evolve?

A little while ago, I created a video, Engaging and Enlightening Students with Maple Visualizations, that showed a sample of Maple visualizations that would be helpful in teaching math. Doing that allowed me to get reacquainted with some of Maple's plotting features that I hadn't used for a while. As a result, I made a second instructional video for my Maple tips series, Animating a Polyhedron in Maple. 

I chose this topic because I thought it would show several features in Maple that might not be known to all users. I list them below and encourage you to try them out.

• The plots:-polyhedraplot command allows you to create a 3-D plot of a polyhedron, including one of 138 polyhedra that Maple knows about.

• The list of named polyhedra available can be obtained by calling the plots:-polyhedra_supported command.

• The viewpoint option, which allows you to create an animation by varying the viewpoint through a 3D plot, can be used to rotate the polyhedron.

• Finally, the Export feature allows you to save the plot animation as an animated GIF.

Over the past few months, I've created a number of short videos. My intention is to help people use Maple more effectively. I occasionally give workshops introducing Maple and its programming language, and many of the topics come from questions I get from the participants. 

These can be found on our Youtube channel. Here are the ones posted so far.

What is a Workbook?
Creating a Workbook
How to Customize Your Maple Settings with the Options Dialog
What is Maple Transactions?
How to Submit an Article to Maple Transactions
Quotation Marks in Maple
Using Single Quotes to Prevent Evaluation
 

If you find these helpful and have suggestions for future videos, please leave a comment, thanks!

Mathematics often feels precise and deterministic. We solve equations, follow logical steps, and do our best to arrive at exact answers. But sometimes, surprisingly, randomness can also lead us to deep mathematical truths. One of the most famous examples of this idea is a problem from the 18th century known as Buffon’s Needle.

Imagine you have a floor made of long wooden planks placed side by side. The seams between the planks form a set of equally spaced parallel lines across the floor. Now, suppose you take a needle and randomly drop it onto the floor. Sometimes the needle lands entirely on one plank. Other times, it crosses one of the seams between planks, as shown below.

Now here is the curious question posed by the French mathematician Georges-Louis Leclerc, Comte de Buffon in the 1700s:

If we repeatedly drop the needle at random, what is the probability that it crosses one of the lines on the floor?

At first glance, this sounds like a simple probability puzzle. But the answer turns out to involve one of the most famous numbers in mathematics: π.

To keep things simple, assume the distance between the parallel lines on the floor is the same as the length of the needle. We can also imagine that all of our needles are thrown onto the same plank, potentially crossing onto the plank above or below. This configuration is equivalent to throwing the needle onto any plank as long as the planks are equally wide; this modification makes the analysis much simpler.

Every time the needle lands, two things determine whether it crosses a line:

• The distance x from the center of the needle to the nearest line
• The angle θ at which the needle lands with respect to the parallel lines

See a depiction of this below.

To determine the probability of a needle crossing one of these lines, we need to describe what a "random drop" of the needle means mathematically. If the lines are the same length apart as the length of the needle L, then the center of the needle can never be farther than L/2 from the nearest line. Therefore, 0 ≤ x ≤ L/2. Next, we can simplify our domain for θ. The problem is symmetric, so we only need to consider angles between 0 and π/2. Any given half of the needle then has a vertical reach of (L/2)sin(θ).

We will say a needle "crosses" a line precisely when the center lands close enough to a line that one end of the needle can reach across the line. This occurs when x ≤ (L/2)sin(θ).

An important assumption to make is that every pair (x,θ) in the rectangle 0 ≤ x ≤L/2, 0 ≤ θ ≤ π/2 is equally likely. We’re assuming the needle lands with uniform randomness over all vertical positions x and angles θ. This means that the probability of crossing a line is the fraction of this region where the inequalities above hold. That is, 

Probability = (area of favourable region) / (area of total region)

The "rectangle" formed by inequalities has a total area of (L/2) * (π/2) = π*L/4. The needle crosses a line exactly when x ≤ (L/2)sin(θ), so for a fixed angle θ, the allowable x values are 0 ≤ x ≤ (L/2)sin(θ). The favourable area is then:

The probability of a needle crossing a line is therefore:

This result leads to a fascinating idea. If the probability of crossing a line is 2/π, we can rearrange the formula to estimate π itself:

π ≈ 2N / C

where:

• N = the total number of needle drops
• C = the number of times the needle crosses a line

In other words, by performing a simple random experiment and counting how often the needle crosses a line, we can approximate π.

For example, suppose you drop the needle 10,000 times and it crosses a line 6,366 times. Plugging these values into the formula gives

π ≈ (2 × 10,000) / 6,366 ≈ 3.14

With enough trials, the estimate tends to get closer and closer to the true value of π. At the bottom of this post, I attached a Maple worksheet that simulates this phenomenon. Below are results from simulating this result using N = 10, 100 & 1000, respectively. Notice as N increases, our approximation for π tends to become more and more accurate.

Below is a more dynamic simulation from the Maple worksheet to show how the approximation stabilizes as N increases.

What makes Buffon’s Needle so fascinating is the unexpected connection between geometry, probability, and one of mathematics’ most important constants.

π usually appears when dealing with circles (circumference, area, rotation, etc). But in Buffon’s experiment, there are no circles at all. Instead, π emerges from the geometry of all the possible ways a needle can land on a set of parallel lines.

This was one of the earliest examples of what we now call a Monte Carlo method, which is essentially using random experiments to estimate numerical values. Today, similar techniques are used in physics, finance, computer graphics, and machine learning.

One of the best parts of Buffon’s Needle is that you can try it yourself. All you need is:

• A toothpick or needle
• A piece of paper with a sequence of parallel lines, each a distance of the needle's length apart
• A lot of patience

Drop the needle repeatedly (N times), record how many times it crosses a line (C), and compute 2N/C. The more times you repeat the experiment, the closer your estimate will get to π.

After reading about this experiment, I was convinced that mathematics is not only about abstract symbols and formulas. Sometimes, even something as simple as dropping a needle onto the floor can reveal the hidden structure of elements of the universe that we would've otherwise never known were there.

Buffons_Needle_Simulation.mw

Gabriel’s Horn is one of the most famous examples in calculus of how infinity can behave in ways that completely defy our intuition.

The horn-shaped object is created from a very simple curve: y = 1/x for x ≥ 1 (pictured below).

Now imagine rotating this curve around the x-axis. The resulting surface stretches infinitely far to the right while becoming thinner and thinner. Visually, it resembles a long trumpet or horn that continuously narrows to a thickness of zero.

At first glance, nothing about this shape seems particularly mysterious. As x grows larger, the radius 1/x becomes smaller and smaller. It seems reasonable that both the volume contained inside the horn and the area of its surface would remain finite (or at least if the volume was finite, then the surface area would also be finite). After all, the horn gets extremely thin very quickly.

Calculus allows us to test that intuition.

To compute the volume of the horn, we use the disk method. Each slice perpendicular to the x-axis forms a circular disk of radius r = 1/x, each with an area of π*r² = π*(1/x²).

The total volume is the sum of an infinite number of these disc areas with thickness dx. As an integral,

V = π ∫₁^∞ (1/x²) dx.

This is a simple integral that converges to a value of 1. We could use the power or rule or our favourite computing software (I used Maple below).

Hence, V = π ∫₁^∞ 1/x² dx = π*1 = π. This means the horn contains only π cubic units of space, even though it extends infinitely far. 

Now let’s compute the surface area of the horn. For a surface of revolution, the surface area is

A = 2π ∫₁^∞ y √(1 + (y′)²) dx.

Since y = 1/x, we have y′ = −1/x². Substituting into the formula gives

A = 2π ∫₁^∞ (1/x) √(1 + 1/x⁴) dx.

Software like Maple can easily handle this integral. It tells us the integral diverges to infinity.

However, this is difficult to solve analytically. To understand what happens to this integral, notice that for large x, the square root term is very close to 1, since 1/x⁴ can be approximated as 0 as x grows large. This means the integrand behaves roughly like 1/x (it's actually slightly larger than 1/x). But

∫₁^∞ 1/x dx diverges, and ∫₁^∞ (1/x) √(1 + 1/x⁴) dx > ∫₁^∞ 1/x dx, so ∫₁^∞ (1/x) √(1 + 1/x⁴) dx must also diverge. As a result, the surface area of Gabriel’s Horn is infinite.

This leads to the famous, surprising conclusion:

• The horn has finite volume.
• The horn has infinite surface area.

In other words, it could be filled with a finite amount of paint, but it would require an infinite amount of paint to coat its inside surface.

Of course, real paint has thickness, so the paradox disappears in the physical world. Eventually, the horn would become thinner than the paint layer itself. But mathematically, the result is perfectly consistent.

The key idea lies in how quickly the function 1/x shrinks. The cross-sectional area of the disks scales like (1/x)² = 1/x², and the integral of 1/x² converges.

But the circumference of each slice scales like 1/x, and the integral of 1/x diverges.

So as the horn extends outward, the added volume decreases quickly enough to sum to a finite value, while the added surface area decreases too slowly and accumulates forever.

Gabriel’s Horn beautifully illustrates one of the central themes of calculus: infinite processes can produce results that feel deeply counterintuitive.

Volume and surface area seem closely related, but can behave in completely different ways when infinite limits are involved. A shape can stretch endlessly yet still contain a finite amount of space.

This strange object reminds me that mathematics isn’t just about calculating numbers, but is also about exploring the strange and fascinating consequences of simple ideas pushed to their limits.

For decades, Maple has been built around one of the world’s most powerful mathematics engines—helping students, educators, engineers, and researchers explore ideas, solve complex problems, and communicate mathematics clearly.

Maple 2026 builds on that foundation with major advances in the math engine, expanding the kinds of problems Maple can solve while improving reliability and performance.

At the same time, Maple 2026 introduces new AI-powered tools that help you work faster—finding commands, generating visualizations, explaining concepts, and helping you explore ideas. The key difference is that these tools sit on top of Maple’s math engine, so the results are grounded in real computation rather than guesswork.

If you’ve been following along with our recent Mathy teaser videos and sneak peek posts, you may already have seen hints of some of these features. Now I’m excited to finally share them in full.

One of the most exciting additions in Maple 2026 is the new AI Assistant.

AI tools are incredibly useful for exploring ideas, writing code, and learning new topics. But when the mathematics becomes more involved, relying on AI alone can be risky. The Maple AI Assistant brings those productivity benefits into Maple while keeping the mathematics grounded in Maple’s trusted computation engine.

You can ask the AI Assistant questions in natural language and have it help you:

• find Maple commands or formulas
• generate Maple code
• create visualizations
• explain mathematical concepts
• draft examples, worksheets, or reports

Because Maple performs the underlying computations where appropriate, the results are grounded in Maple’s powerful math engine. The AI Assistant becomes a productivity partner that helps you accomplish tasks in Maple faster and more easily, combining the flexibility of AI with mathematics you can trust.

Watch the AI Assistant in action.

Another feature I’m particularly excited about is Document Import.

Many of us have years of mathematical content stored in PDFs, lecture notes, journal articles, slides, or even handwritten pages. Traditionally these documents are static—you can read them, but you can’t interact with the mathematics inside them.

With Maple 2026, that changes.

Document Import allows Maple to convert many document formats—including PDFs, DOCX files, and presentations—into Maple worksheets where the mathematics becomes live and executable. 

The image below illustrates the transformation.

On the left (“Before”), scribbled handwritten notes from a Calculus III lecture were saved in a Word document. The notes include hand-drawn sketches, formulas, and written explanations.

After importing the document into Maple (“After”), the mathematical expressions were recognized and converted into live, editable Maple mathematics. The text was preserved, and the hand-drawn sketches were retained as images. The resulting worksheet supports evaluation, editing, and further computation.

Once imported, you can:

• evaluate expressions
• modify formulas
• extend derivations
• add visualizations
• explore variations of the mathematics

Instead of recreating examples from scratch, you can bring existing material directly into Maple and start exploring.

While the new AI features are exciting, the heart of Maple has always been its mathematics engine—and Maple 2026 delivers significant advances here.

One particularly notable improvement is Maple’s expanded ability to solve linear recurrence equations. Through improvements to the rsolve command and major extensions to the LREtools package, Maple can now solve dramatically more recurrence relations than before, including many third- and fourth-order cases that were previously beyond reach.

In fact, Maple can now fully solve over 94% of the 55,979 entries in the Online Encyclopedia of Integer Sequences (OEIS) that that can be shown to satisfy a linear recurrence relation. These advances reflect ongoing research into linear difference equations and their algorithmic implementation in Maple, continuing Maple’s long tradition of advancing the state of computer algebra.

Beyond recurrence solving, Maple 2026 includes many improvements across its core symbolic and numeric algorithms. Maple’s assumption system has been strengthened to improve reasoning under mathematical assumptions, and enhancements to the simplify, combine, and evalc commands allow Maple to produce more compact and mathematically natural forms for a wider range of expressions.

There are also improvements to Maple’s differential equation solvers, polynomial system solving, and numerical solving routines such as fsolve, along with updates to other foundational parts of the math library used throughout the system.

Taken together, these improvements expand the range of problems Maple can solve and improve the robustness, correctness, and efficiency of the results.

Maple has always offered extensive control over plotting options, but achieving consistent visual styling across multiple plots could require specifying many settings each time.

Maple 2026 introduces Plotting Themes, which allow you to define a plotting style once and apply it across many plots with a single option.

Themes make it easy to maintain consistent visual styles in worksheets, teaching materials, reports, and publications, while still allowing individual plots to override specific options when needed.

The image below shows an example of creating and applying a custom plotting theme. 

Maple continues to be widely used in classrooms around the world, and Maple 2026 includes several improvements designed to support teaching and learning.

The Check My Work system has been enhanced so Maple can recognize a wider variety of valid student solution steps and provide more accurate feedback.

Maple 2026 also improves the generation of similar practice problems, making it easier to create variations of a problem while preserving its mathematical structure.

In addition, Maple’s step-by-step solutions have been expanded to support more types of expressions, helping students better understand the reasoning behind the mathematics they’re learning.

Maple 2026 also introduces improvements for developers building advanced applications, along with performance enhancements across the system.

One particularly interesting addition is the new VectorSearch package, which implements a vector database directly inside Maple.

If you’re not familiar with vector databases, one way to think about them is through recommendation systems like Netflix or Spotify. Each movie or song can be represented by a vector containing thousands of numbers describing its characteristics—things like genre, pacing, or mood. When you watch something, the system finds other items whose vectors are closest to it, which is how recommendations are generated.

With the new VectorSearch package, Maple can store thousands (or more) of vectors and efficiently find the ones most similar to a given vector. This makes it easier to build applications involving machine learning, data analysis, and modern AI workflows directly in Maple.

Maple 2026 also delivers significant performance improvements. For example, operations involving quantities with units have been greatly optimized—some computations now run over 90 times faster, making Maple even more efficient for engineering and scientific workflows.

Maple 2026 also expands the benefits available through the Maplesoft Elite Maintenance Program (EMP). The new benefits include access to additional Maplesoft products and services:

• Maple Learn, the online environment for teaching and learning mathematics
• Maple Calculator Premium, bringing the power of Maple to your phone with full access to features like Solution Steps and Check My Work
• Maple MCP, which allows you to connect Maple’s math engine to external AI tools so they can produce mathematical results you can trust

These additions extend Maple beyond the desktop, giving users powerful tools for learning, teaching, and exploring mathematics across web and mobile platforms, as well as through integrations with external AI tools.

This post only scratches the surface of what’s new in Maple 2026. There are many more improvements across the math library, programming tools, and performance.

To learn more about all the new features and enhancements in Maple 2026, visit the What’s New in Maple page on our website.

We've reached quite a rhythm with Maple Flow - we update frequently, we add lots of improvements and we move fast.

What does this mean for you? It means that the feedback loop between development, the user experience and course correction has a fast time constant.

Without you being loud and vociferous, the feedback loop breaks. So don't be shy - tell us what you want!.

The new 2025.2 update builds on the theme of connectivity with two popular tools - Excel and Python. On top of that, we also have many other features and fixes that you've asked for.

Earlier versions of Maple Flow let you 

• import and and export Excel data from Excel spreadsheets
• run Maple Flow worksheets in Excel with a mouse-driven UI to generate the communication glue

With the 2025.2 update, you can now copy and paste data from Excel into a Flow worksheet.

To be blunt, this is type of cross-application copy-paste behaviour is a no-brainer. It's such a natural workflow.

We've increasignly found that Python is now being used to script the interaction and data flow between different engineering tools. With Maple Flow 2025.2, you can now execute Maple Flow worksheets from a Python script. 

From Python, you can change and export any parameters and results defined in the worksheet

This gives me the dopamine hit of watching CPU utilization spike in the Task Manager (hey..I get my kicks where I can)

You can now do your parameter sweeps more quickly by executing the same worksheet in parallel, changing parameters for every run.

This is easy to set up - no special programming is needed.

• Print Extents can now be set globally for all sessions, or just for the current session.
• Any user-installed fonts used in the worksheet are now respected in the PDF export
• Worksheets execute faster
• The update includes fixes to many user-reported issues

You can install the Flow 2025.2 update via Help > Check for Updates (or if you're not already in the race, then grab a trial here and take Flow for a spin).

We're not pulling back on this aggresive development velocity, but we need you to point us in the right direction. Let's keep the feedback time constant small!

If one of our posts showed up in your social media feed recently, you may have found yourself staring at a giant maple leaf with feet and thinking, “Wait… who (or what) is that?” you’re not alone. 

Yes, that big, cheerful leaf you’ve been seeing is very real. 
And yes, they have a name. 

Meet Mathy. 

We officially introduced Mathy to the world a couple of weeks ago at JMM 2026 in Washington, DC, but their story actually started much earlier. 

Mathy was originally created by one of our developers, Marek Krzeminski, a few years ago as a fun internal character. Over time, they quietly became our in-office, local mathscot, popping up as mini 3D-printed Mathys around the office and even as a custom emoji someone created. 

Then, sometime last year, someone had what can only be described as a bold idea: 

What if we brought Mathy to life? 

And just like that, the giant maple leaf went from concept to costume. 

Mathy is fun, curious, and a little playful. That’s very intentional. That’s what math should feel like. 

We believe math matters. We also believe math should be approachable, joyful, and a place where curiosity is rewarded. Mathy reminds us, and hopefully others, that math doesn’t have to be intimidating. It can be fun, and it can inspire awe. 

I’ll be honest. When we decided to bring Mathy to JMM, I was a little nervous. Conferences are busy, serious places. Would people really want to interact with a seven-foot-tall maple leaf? 

As it turns out, yes. Very much yes. 

Researchers (from postdocs to seasoned academics), educators, and undergraduate and graduate students all stopped, smiled, laughed, and asked for photos. At one point, people were actually lining up to take pictures with Mathy.

Let’s just say: Mathy was a hit. 

How tall is Mathy? 
About 7 feet. They are hard to miss. 

What does Mathy love (besides math)? 
Dancing. Very much dancing. 
You can see for yourself here: Mathy's got moves!

Does Mathy talk? 
You bet they do. 

Now that Mathy has officially been introduced to the world, you’ll be seeing them more often on social media, at events, and in a few other fun places we’re cooking up. 

So if you spot a giant maple leaf dancing, waving, or talking math, now you know who they are. 

If you spot Mathy, don’t be shy, say hi. 

Many problems in mathematics are easy to define and conceptualize, but take a bit of deeper thinking to actually solve. Check out the Olympiad-style question (from this link) below:

Former Maplesoft co-op student Callum Laverance decided to make a document in Maple Learn to de-bunk this innocent-looking problem and used the powerful tools within Maple Learn to show step-by-step how to think of this problem. The first step, I recommend, would be to play around with possible values of a and b for inspiration. See how I did this below:

Based on the snippet above, we might guess that a = 0.5 and b = 1.9. The next step is to think of some equations that may be useful to help us actually solve for these values. Since the square has a side length of 4, we know its area must be 4² = 16. Therefore, the Yellow, Green and Red areas must add exactly to 16. That is,

With a bit of calculus and Maple Learn's context panel, we can integrate the function f(x) = ax² from x = -2 to x = 2 and set it equal to this value of 8/3. This allows us to solve for the value of a.

We see that a = 1/2. Since the area of the Red section must be three times that of the Yellow (which we determined above to be 8/3), we get Red = (8/3)*3 = 8.

The last step is to find the value of b. In the figure below, we know that the line y = 4 and the curve y = bx² intersect when bx² = 4 (i.e. when x = ± 2/sqrt(b)).

Since we know the area of the red section is 8 square units, that must be the difference between the entire area underneath the horiztonal line at y = 4 and the curve y = bx² on the interval [-2/sqrt(b), 2/sqrt(b)]. We can then write the area of the Red section as an integral in terms of b, then solve for the value of b, since we know the Red area is equal to 8.

Voila! Setting a = 1/2 and b = 16/9 ≈ 1.8 guarantees that the ratio of Yellow to Green to Red area within the square is 1:2:3, respectively. Note this is quite close to our original guess of a = 0.5 and b = 1.9. With a bit of algebra and solving a couple of integrals, we were able to solve a mathematics Olympiad problem!

Over the past year, I have spent a lot of time talking to educators, researchers, and engineers about AI. The feeling is almost universal: it is impressive, it is helpful, but you should absolutely not trust it with your math even if it sounds confident.

That tension between how capable AI feels and how accurate it actually is has been on my mind for months. AI is not going away. The challenge now is figuring out how to make it reliable.

That is where Maple MCP comes in.

Maple MCP (Model Context Protocol) connects large language models like ChatGPT, Claude, Cohere, and Perplexity to Maple’s world-class math engine.

When your AI encounters math, your AI can turn to Maple to handle the computation so the results are ones you can actually trust.

It is a simple idea, but an important one: Maple does the math and the AI does the talking. Instead of guessing, the AI can be directed to call on Maple whenever accuracy matters.

Model Context Protocol (MCP) is an emerging open standard that allows AI systems to connect to external tools and data sources. It gives language models a structured way to request computations, pass inputs, and receive reliable outputs, rather than trying to predict everything in text form.

Here is a high-level view of how MCP fits into the broader ecosystem:

MCP lets an AI system connect securely to specialized services, like Maple, that provide capabilities the model does not have on its own.

If you want to learn more about the MCP standard, the documentation is a great starting point: Model Context Protocol documentation

Here is a glimpse of what happens when Maple joins the conversation:

Depending on the prompt, Maple MCP can evaluate expressions symbolically or numerically, execute Maple code, expand or factor expressions, integrate or solve equations, and even generate interactive visualizations. If you ask for an exploration or an activity, it can create a Maple Learn document with the parameters and sliders already in place.

As an example of how this plays out in practice, I asked Maple MCP:

“I'd like to create an interactive math activity in Maple that allows my students to explore the tangent of a line for the function f(x) = sin(x) + 0.5x for various values of x.”

It generated a complete Maple Learn activity that was ready to use and share. You can open the interactive version here: interactive tangent line activity .

In full disclosure, I did have to go back and forth a bit to get the exact results I wanted, mostly because my prompt wasn’t very specific, but the process was smooth, and I know it will only get better over time.

What is exciting is that this does not replace the LLM; it complements it. The model still explains, reasons, and interacts naturally. Maple simply steps in to do the math—the part AI cannot reliably do on its own.

We have opened the Maple MCP public beta, and I would love for you to try it.

Sign up today and we will send you everything you need to get started!

There is still time to register for Maple Conference 2025, which takes place November 5-7, 2025.

The free registration includes access to three full days of presentations from Maplesoft product directors and developers, two distinguished keynote speakers, contributed talks by Maple users, and opportunities to network with fellow users, researchers, and Maplesoft staff.

The final day of the conference will feature three in-depth workshops presented by the R&D team. You'll get hands-on experience with creating professional documents in Maple, learn how to solve various differential equations more effectively using Maple's numerical solvers, and explore the power of the Maple programming language while solving interesting puzzles.

Access to the workshops is included with the free conference registration.

We hope to see you there!

Kaska Kowalska
Contributed Program Co-chair

Imagine standing 365 metres above Toronto on the CN Tower’s EdgeWalk and throwing a baseball. Could you actually land it on third base at Rogers Centre, about 263 metres away?

Sportsnet raised this question, and we decided to put it to the test in Maple Learn, check out this document to see the answer.

Also take a look at the Sportsnet video on the problem, to see why the answer may not be obvious.

In the Maple Learn document, you can adjust the initial speed and angle at which to throw the ball and then visualize its trajectory (without having to throw as hard as Addison Barger).

I was surprised that even in the simplified projectile motion model, that neglects air resistance, AND assuming I could throw at 60mph (a questionable assumption to say the least) I wouldn’t be able to hit the base myself.

I then used Maple to build a more realistic model that would account for air resistance. The equations below model the position of the ball, where y(0) = h₀ is the initial height of 365m and v₀ is the initial speed.

local h0, m, d, rho, g:
	h0 := 365:
	m := 0.145:
	d := 0.072:
	rho := 1.225:
	g := 9.81:

	local eqns, ics:
	eqns := diff(x(t),t) = u(t), 
		    diff(y(t), t) = v(t), 
		    diff(u(t), t)= -Pi/16 * d^2 * rho/m * sqrt(u(t)^2 + v(t)^2) * u(t), 
		    diff(v(t), t)= - g - Pi/16 * d^2 * rho/m * sqrt(u(t)^2 + v(t)^2) * v(t):
	ics := x(0) = 0, y(0)=h0, u(0) = v_initial*cos(theta_initial), v(0) = v_initial * sin(theta_initial):

	local ans, xpos, ypos:
	ans := dsolve([eqns, ics], numeric, output=listprocedure):
	xpos := subs(ans, x(t));
	ypos := subs(ans, y(t));

In the Maple Learn document, you can visualize the difference between the models by comparing the trajectories. The trajectory from the simple model is shown in blue, and the trajectory after accounting for air resistance is modelled in red.

Accounting for air resistance, I’m no longer convinced even Addison Barger could accomplish this challenge.

Check out the Maple Learn document to try for yourself!