Year: 2025

How Online Math Learning with Wiris Transformed UNED

Post author By Aida Alsina
Post date June 25, 2025

Breaking Barriers in Online Math Learning with Wiris

In the ever-evolving landscape of online education, ensuring that mathematical and scientific content is easy to create, share, and evaluate is a challenge.

Universidad Nacional de Educación a Distancia (UNED), Spain’s leading distance-learning university, recognized this challenge early on. With thousands of students across various disciplines, UNED sought a robust solution to enhance its teaching and evaluation processes, particularly in mathematics and science-related subjects.

To tackle these challenges, UNED turned to online math learning with Wiris, integrating WirisQuizzes and MathType into their educational workflow. Having been a valued client of Wiris for over 15 years of the company’s 25-year journey, UNED has continuously trusted in Wiris’ math solutions to support its evolving digital education needs. We sat down with Mari Carmen García Llamas, a professor in the Economics and Tourism Bachelor’s Degrees, and Raúl Morales Hidalgo, Head of the E-learning Platform at UNED, to understand how these tools transformed their math teaching and assessment methods.

The Challenge: Teaching Math in a Digital Environment

Before implementing Wiris’ solutions, UNED faced significant obstacles in creating and communicating mathematical content effectively in an online setting.

From a technical perspective, their biggest challenge was integrating a math-editing tool into their custom learning platform. Faculty members struggled with writing mathematical formulas in forums, assignments, and quizzes resulting in a poorer learning experience for students.

From a teaching perspective, professors required an intuitive solution to easily insert mathematical expressions into quizzes and ensure smooth interaction with students.

Carmen explains:

“Without a tool to write mathematical symbols, explaining calculations was a big challenge. Whether it was an integral or an algebraic equation, it was always difficult to pinpoint where a student had made an error.”

Additionally, UNED needed an assessment tool that allowed mathematical expressions throughout a quizz, from problem statements to answer options. The absence of such a tool limited quiz and assignment creation.

The Solution: Enhancing Online Math Learning at UNED with Wiris

To overcome these challenges, UNED explored available solutions and found WirisQuizzes and MathType to be the most comprehensive and well-integrated tools for their needs. Their decision to implement Wiris’ solutions was based on several key factors:

Seamless integration with OpenLMS that they use in-house

UNED initially discovered MathType through faculty recommendations and was particularly drawn to its integration with Microsoft Office. However, since they had a custom in-house platform, they needed a solution that could be fully embedded.

Wiris’ technical team collaborated with UNED to integrate MathType and WirisQuizzes into their platform, ensuring a smooth transition and enhanced functionality.

User-friendly for professors and students

With online math learning with Wiris, professors could now write and edit formulas effortlessly across various platforms, including forums, quizzes, and assignments. This eliminated previous barriers in math-based communication between students and teachers.

Carmen highlights the impact:

“We needed a tool that was just as easy to use for the teacher as it was for the student. Wiris makes it possible to create math-rich quizzes where every question is fully customizable.”

Encouraging Active Learning by Breaking Away from Memorization

One of the standout features of WirisQuizzes is its ability to introduce randomized mathematical expressions into quizzes. This ensures that each student receives a unique version of an exercise, discouraging memorization and promoting active learning. This was highlighted by Carmen when she stated:

“WirisQuizzes allows us to create quizzes where every exercise is unique. The level of flexibility it provides is amazing.”

A welcome surprise: the power of WirisQuizzes

While UNED initially focused on MathType, they soon discovered that WirisQuizzes offered additional benefits. As they expanded their digital assessment strategies, WirisQuizzes proved to be an unexpected but valuable tool in enhancing student engagement and practice opportunities.

Raúl shares his excitement:

“When we started making changes to our platform, we were pleasantly surprised by WirisQuizzes. It felt like an unexpected gift, and it has been an incredibly powerful one.”

The Implementation Process: A Collaborative Approach

To successfully integrate online math learning with Wiris into their learning environment, UNED, along with Wiris’ technical team, followed a structured implementation process:

Understanding institutional needs

UNED’s faculty and technical teams worked with Wiris to assess their specific requirements, particularly the need for platform integration and math-based assessments.

Customization and integration

Wiris’ technical experts customized the tools to align with UNED’s platform requirements, ensuring a smooth user experience. Furthermore, the team at Wiris assisted in resolving any doubts as well as providing tips and training on how to use WirisQuizzes efficiently.

Deployment across OpenLMS

The integration was successfully implemented on OpenLMS, allowing professors to seamlessly create and edit formulas within quizzes and discussion forums.

Learning Math Online with Wiris – The Results

Following the implementation of online math learning with Wiris, UNED experienced measurable improvements in teaching efficiency and student engagement:

Greater flexibility for teachers in math-based quiz creation.
Improved student engagement through randomized exercises.
Seamless math communication in forums and assignments.
Quick technical support ensures smooth day-to-day usage.

Carmen notes:

“Wiris allows us to create diverse quizzes with distinct options, making every assessment dynamic and interactive.”

Students also benefit from interactive, dynamic learning experiences, where every exercise is different, allowing them to practice multiple problem variations in a single topic.

Future Plans: Looking Ahead with WirisQuizzes

After experiencing the benefits of WirisQuizzes, UNED is eager to see how the tool evolves and continues to support its online math learning needs. While it is currently used for continuous self-evaluation, faculty members appreciate its capabilities and will continue integrating it into their teaching workflows.

Carmen shares:

“More than one of my colleagues has reached out to me about the potential of Wiris’ solutions. I believe we are only beginning to explore what’s possible.”

Rounding Off: UNED’s Endorsement of Online Math Learning with Wiris

With over 15 years of using Wiris products, UNED strongly endorses WirisQuizzes and MathType for their ease of use, powerful assessment features, and seamless integration.

Raúl emphasizes:

“We have always recommended Wiris. That’s why we’ve been with you for 15 years already.”

Try Wiris for Your Institution

UNED’s success story showcases how online math learning with Wiris can revolutionize digital math education.

If you’re looking to improve math-based assessments in your institution and want a seamless math-writing experience for students and teachers, don’t delay—contact us today.

Request a Demo Today

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AI for Math: The Smart Way to Create Quizzes Faster

Post author By Aida Alsina
Post date June 12, 2025

Creating engaging and varied math quizzes can be a surprisingly time-consuming task. Teachers often face the dreaded blank page, struggling to come up with fresh problems that match different levels, topics, and student needs. But that’s about to change. With AI for math making its way into classrooms, quiz creation just got a powerful upgrade.

Meet LearningLemur: Your AI Question Generator

LearningLemur is an intuitive math platform integrated with Google Classroom that provides customizable quizzes, automatic grading, and personalized feedback, saving educators time and enhancing student learning. Its extensive library of customizable quizzes and exercises fosters engagement and mastery, while detailed analytics help educators track progress and identify areas for improvement. From simplifying assignment creation to delivering instant feedback, LearningLemur transforms classrooms into dynamic and interactive learning spaces, making math more accessible and engaging for all students.

Now, Wiris has introduced a new feature within LearningLemur: a math question generator that understands your needs and generates ready-made quizzes in seconds. It’s designed to address one of the most common pain points for educators: how to start creating math assessments efficiently without sacrificing quality.

Here’s what LearningLemur’s new AI-powered tool offers:

Describe the type of questions you want.
Receive an instant set of customized math problems.
Edit, refine, and assign them directly to your students.

Forget about starting from scratch. This tool helps you work smarter, not harder, all within a platform that is intuitive and easy to navigate, even for teachers with limited tech experience.

How does it work?

Using this tool is simple and straightforward. Teachers input the number of questions, select the question type (e.g., open answer, multiple choice) and finally provide a prompt (e.g., “Make fractions sum problems for students of grade 12. Ask them to be simplified.”).

The AI handles the rest, offering draft content that aligns with your instructions. And because it’s powered by the trusted Wiris correction engine, you maintain full confidence in the mathematical accuracy.

Why This AI Feature Matters for Educators

The LearningLemur AI question generator isn’t just about saving time. It’s designed to help teachers focus on what really matters: pedagogy, clarity, and curriculum alignment. While the AI handles the generation, teachers can shape the output by ensuring pedagogical alignment with curricular goals, adjusting the complexity and style of each question, and refining the language and clarity for the specific group of learners. This tool does not replace your expertise but accelerates your workflow, allowing you to dedicate more time to effective teaching.

Tips for Writing Effective Prompts

Crafting the right prompt is crucial to getting the most out of the LearningLemur AI for math tool. Think of it as briefing a colleague or delegating to a student teacher. Be clear, be specific, and include the following:

Specify the exact math topic you want to assess in your students. For example, indicate whether you want exercises on fractions or polynomials.
Define the educational level or age of your students to tailor the difficulty. Questions should be accessible yet challenging, matching the knowledge and skills of your learners.
Add any constraints. If you have specific requirements, such as simplifying answers or using decimals only, be sure to include them. This ensures the generated questions meet your pedagogical or exam criteria.
Optionally, use word problems to add context and generate relatable scenarios. Adding context or real-life situations makes problems more engaging and easier to understand for students. For instance, framing a problem in the context of shopping, travel or science helps connect math to everyday life and increases motivation.

Example Prompts

“Create 6 word problems on solving linear equations for Grade 9. Include at least one with negative coefficients and require students to simplify x.”

“Create 10 questions on simplifying algebraic expressions for 11-year-old students. Include at least one question involving the distributive property and another with factoring.”

“Create 5 questions for first-year university math students on calculating the rank of a matrix using elementary row operations.”

Use Cases in the Real Classroom

The LearningLemur math question generator adapts across levels and contexts:

For daily practice, generate variations to reinforce a concept.
For formative assessment, tailor difficulty and format.
For homework, instantly provide differentiated sets by level.
For exam prep, create question banks by topic or skill.

And since you can edit anything, it integrates seamlessly into any teaching style.

Diverse Learning Needs: Easily Tailor Content to Students’ Needs

From the teachers’ perspective, LearningLemur offers great flexibility, as educators with limited time or resources in creating math problems can now generate high-quality content more quickly. These advantages are particularly evident for teachers managing large classes, as they can personalize content while reducing preparation time.

Additionally, LearningLemur enables teachers to address the diverse needs of their students by creating customized quizzes and exercises tailored to different skill levels and learning requirements. Teachers can also assign targeted homework tasks, allowing students to strengthen specific competencies where they may need additional support.

Empowering Teachers Through Intelligent Automation

AI for math isn’t about replacing educators: it’s about empowering them. LearningLemur is a clear example of how AI question generators can reduce friction in the creative process while leaving full pedagogical control in teachers’ hands.

Whether you’re battling writer’s block or trying to generate multiple versions of a quiz quickly, this tool gives you a solid head start.

Try the AI of LearningLemur now

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How to Calculate the Area Between Two Curves?

Post author By Aida Alsina
Post date June 5, 2025

Step-by-step Solution to an Oxford Access Exam Problem.

The Oxford University access exams are famous for their difficulty and for challenging students with mathematical problems of high complexity. In this article, we will focus on a 2020 exam problem that involves calculating the area between two curves. Through a clear and detailed analysis, we will break down each step necessary to approach and solve this mathematical challenge.

If you are interested, feel free to also visit our other article, “How to solve an equation of degree 16”, where we explore another challenging Oxford exam problem.

The problem statement is as follows:

A line is tangent to the parabola y=x² at the point (a,a²), where a>0.

The area of the region bounded by the parabola, the tangent line, and the x-axis equals

(a) a²⁄₃, (b) 2a²⁄₃, (c) a³⁄₁₂, (d) 5a³⁄₆, (a) a⁴⁄₁₀

This is a classic calculus problem that, as we will see, requires the use of both derivatives and integrals. Don’t worry; we’ll break it down into three clear and simple steps. Let’s get to it!

How to Find the Equation of the Tangent Line?

Before starting with the calculations, let’s visualize the problem. Below, you’ll see a graph illustrating the parabola y=x² and the tangent line at the point (a, a²), where a>0. In this graph, we can see how the tangent line touches the curve of the parabola at a single point without crossing it.

Now, to proceed, we first need to find the equation of the tangent line at that specific point. Since the tangent touches the parabola at point (a, a²), we need to determine both its slope and its equation.

This is where the derivatives come into play. The slope of the tangent at any point on the parabola is given by the derivative of the function y=x². We start by calculating the derivative: dy/dx=2x

This means that the slope of the tangent at any point on the parabola is 2x. To find the slope at the tangent point (a, a²), we substitute in the derivative:

Therefore, the slope of the tangent line at point (a, a²) is 2a.

Now that we have the slope, we can write the equation of the tangent line using the formula for the line in its form:

y – y₀ = m (x – x₀)

Where m is the slope of the line, and (x₀ , y₀) is the point through which the line passes. In our case, the point of tangency is (a, a²), and the slope is 2a, so we substitute these values in the formula for the line:

y – a² = 2a (x-a)

y – a² = 2ax – 2a²

y = 2ax – a²

So, the equation of the tangent line at the point (a, a²) is:

y = 2ax – a²

Find the Intersection Points

Now that we have the equation of the tangent, we need to find the points of intersection between this tangent and the x-axis, that is, when y=0. This will allow us to determine the limits of the region whose area we want to calculate.

The area to be calculated is marked in orange in the graph below.

Substituting y=0 in the tangent equation and solving for x we obtain:

0= 2ax – a²

2ax = a²

x = a/2

So the tangent cuts the x-axis at point

P=(a/2, 0 )

Calculate the Area Between the Parabola, the Tangent and the X-axis

Once we have the equation of the tangent and we know the points of intersection, we can proceed to calculate the area of the region bounded by the parabola, the tangent and the x-axis. For this, we use the definite integral, which allows us to find the area between two curves.

The area we want to calculate is obtained by subtracting two regions. In the first graph, you can see the total region we are interested in. However, part of this area is specifically delimited in the second graph, which we must subtract to be left with only the part we are looking for.

Let us calculate both integrals:

a) Integral of graph 1:

definite integral of x squared from 0 to a. It equals the integral from 0 to a of x squared dx, which is equal to x cubed over 3 evaluated from 0 to a, resulting in a cubed over 3.

b) Integral of graph 2:

Now that we have solved all the integrals, we subtract the results to obtain the total area:

In conclusion, the area of the region bounded by the parabola y = x², the tangent y = 2ax – a² and the x-axis, is:

a sup 3 / 12

Therefore, option c is the correct answer

And so we come to the end of this fascinating problem! If you enjoyed this analysis or found it useful to better understand the solving process, feel free to share it with other math enthusiasts!

Now we challenge you: do you dare to calculate the area formed by the curve y = x³ and its tangent at the point (1,1)? Share your solution or your ideas in the comments. We’d love to hear your approach!

Try WirisQuizzes to generate this exercise or others!

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Engaged Students, Better Results: La Salle’s Math Learning Platform With WirisQuizzes

Post author By Aida Alsina
Post date May 30, 2025

From Textbooks to a Personalized Digital Math Learning Platform

When La Salle San Ildefonso set out to replace printed math textbooks with digital tools, they needed more than just online worksheets—they needed a complete math learning platform. With WirisQuizzes integrated into Moodle, their teachers built a collaborative system that supports independent learning, personalized feedback, and scalable assessment.

What started as one teacher’s initiative has become a model adopted across multiple La Salle campuses, reshaping how students engage with math.

The Challenge

When La Salle San Ildefonso first decided to phase out physical math textbooks, they faced a critical question: How do we offer high-quality math learning in a wholly digital format without sacrificing feedback, rigor, or accessibility?

As teacher Javier Melchior explained, early digital tools offered little beyond binary right-or-wrong grading

“The only feedback we could give was whether the answer was right or wrong. That wasn’t enough”, Javier explained. ”What students needed was guidance – not just the solution, but the how behind the solution”.

This insight became a turning point for La Salle’s approach to digital education. It wasn’t just about digitizing content but about ensuring that students could learn actively and reflectively within a new system.

At the same time, the shift to digital demanded a scalable infrastructure for:

Creating large volumes of exercises quickly.
Providing real-time, personalized feedback.
Supporting teachers across varying levels of technical proficiency.
Giving students the tools to practice independently and meaningfully.

Other platforms like Google Classroom, while helpful for document sharing, lacked the sophistication for in-depth math evaluation and formative feedback.

The Solution

That’s when WirisQuizzes came into focus—not just as a plugin but as the core engine behind La Salle’s custom-built math learning platform.

Led by Javier and math teacher Guillermo Pérez González, the department began building a shared bank of exercises directly within Moodle using WirisQuizzes. These weren’t just questions—they were intelligent, dynamic exercises that could offer step-by-step feedback, track partial understanding, and regenerate with different data for unlimited practice.

“We discovered the true power of WirisQuizzes when we realized we could break down a problem into parts,” said Javier. “It’s not just about getting the final answer. If a student does the first 20% right, the platform can recognize and score that.”

WirisQuizzes became more than an assignment tool for students—it was their personal study assistant. “Now they ask us, ‘Is there a quiz for this topic?” Guillermo shared. “It’s become a natural part of how they study and prepare.”

The Implementation Process

La Salle’s implementation of WirisQuizzes was a grassroots initiative that grew into an institutional innovation. Here’s how it evolved.

Assessment

The shift began nearly a decade ago when La Salle removed printed textbooks for some math levels. With a digital-first model on the horizon, Javier started exploring alternatives. “I found WirisQuizzes online by chance,” he said. “And after testing its capabilities, I pitched it to my colleagues. From there, it just took off.”

Customization

Using Moodle’s system-level question bank, the teachers created hundreds of questions organized by topic: polynomials, functions, geometry, probability, and more. Every question included custom feedback. “Almost all our questions show the solution process,” said Javier. “This makes it easier for students to learn independently—even their parents can follow along.”

Importantly, this work was collaborative. Teachers built and reused each other’s content. “We never assigned categories formally,” said Guillermo. “If someone had free time, they’d jump in and add to the bank. It’s a shared resource for everyone.”

Deployment

WirisQuizzes was deployed through Moodle across the school. Teachers could:

Select questions from a shared bank.
Build quizzes quickly.
Offer time-limited assessments or practice exercises.
Use quizzes for classwork, homework, or even during substitute hours.

The school also exported question banks to other La Salle campuses in Tenerife, Madrid, and Gran Canaria. Adoption varied depending on each site’s IT support and teacher training, but the system was designed for easy export/import.

Training

While many core users were math or STEM teachers with programming experience, others found the “programming” aspect intimidating. Guillermo acknowledged this gap. “Some teachers hear ‘programming’ and think of C++,” he joked. “But it’s not like that. They’d see how simple it is if they just took one step.”

To address this, La Salle and Wiris discussed tailored training sessions that start simple, focusing on quiz-building basics before moving into more advanced logic and scripting.

The Results

La Salle’s math learning platform, powered by WirisQuizzes, delivered transformative results across several dimensions.

Improved Student Autonomy & Motivation

Students began voluntarily requesting quizzes to practice new topics. “It’s like they have a personal tutor,” said Javier. In one memorable example, he logged in on a weekend to find nine students had already completed a practice quiz and scored a perfect 10. “They were competing healthily. It became fun.”

Collaborative Efficiency Among Teachers

The math department created a bank of over 600–700 questions, reusable across grade levels and subjects. This significantly reduced prep time while increasing question quality and consistency.

Parental Engagement and Transparency

Because quizzes included feedback, parents could follow along, even if they weren’t strong in math. “Some told us they were thrilled to see their child working independently,” Guillermo shared.

Formative Assessment at Scale

Teachers used quizzes to diagnose student progress week by week. “In Bachillerato, I might run three quizzes weekly,” said Guillermo. “And they all count toward the grade. It keeps everyone working consistently.”

“WirisQuizzes isn’t just a quiz engine,” Javier added. “It’s a full feedback system. It’s the backbone of our math learning platform.”

The Future of WirisQuizzes at La Salle

La Salle continues to expand the use of WirisQuizzes beyond math:

They’ve developed questions for economics, technology, and even geography.
Some teachers are exploring how AI-based feedback could provide personalized remediation paths.
A growing number of educators across La Salle’s network are joining upcoming training sessions to learn how to build questions and customize feedback.

They’ve also expressed interest in exploring remote proctoring, integrating with other LMS platforms like Google Classroom, and co-developing content with other schools.

Rounding Off

La Salle’s transformation is a shining example of how a school can build its own collaborative, intelligent, and scalable math learning platform. By integrating WirisQuizzes into every level of instruction—from the classroom to home study—teachers empowered students with autonomy, parents with transparency, and the school with data-driven teaching strategies.

What started as one teacher’s experiment has become a system-wide initiative that is changing how La Salle teaches and how students learn.

👉 Want to build your own feedback-driven math platform?

Get in touch today

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Towers of Hanoi: A Math and Programming Challenge

Post author By Aida Alsina
Post date May 21, 2025

Towers of Hanoi: A Math and Programming Challenge

The Towers of Hanoi is one of the most iconic puzzles in the field of mathematics and computer science. With its simple structure and transparent rules, this problem has captivated mathematicians and amateurs alike, making it an intriguing challenge since its creation.

According to the legend, the Towers of Hanoi were conceived by a group of monks in an ancient temple in India. In the center of this temple, three golden pillars stood, upon which rested 64 disks of various sizes, arranged in the first pillar in an orderly fashion: the largest disk at the base, with progressively smaller disks stacked on top, until reaching the smallest disk, placed at the top.

The monks’ objective was to move all the disks from the first pillar to the third, using the central pillar as an auxiliary. To accomplish this task, they had to follow a strict set of rules:

Only one disk can be moved at a time.
A disc can never be placed on top of a smaller disc.

Legend says that when all the disks were correctly transferred to the last pillar, the world would come to an end. However, how close is the end of the world, really? How long would it take to complete this challenge?

This challenge was officially introduced by the French mathematician Édouard Lucas in 1883 as part of his research in number theory and quickly became a popular problem.

Next, we will analyze the solution of this problem from a mathematical perspective, breaking down its key principles and explore how it transforms into an interesting programming challenge.

Recursion as a Solution to the Puzzle

The challenge of the Towers of Hanoi is not only in moving the disks from one pillar to another following the strict rules but also in calculating how many moves are needed to solve the problem of moving n disks from the origin pillar to the destination pillar.

The most efficient way to approach this problem is through recursion, a fundamental technique in programming. Recursion allows the problem to be divided into smaller subproblems, where the solution of each subproblem leads to the solution of the original problem.

In the case of the Towers of Hanoi, recursion applies naturally, and we can divide the problem of moving n disks into three steps:

Move the first n – 1 disks from the origin pillar to the auxiliary pillar, using the destination pillar as the auxiliary.
Move the largest disc (the disc n) from the origin pillar to the destination pillar.
Move the n – 1 disks from the auxiliary pillar to the destination pillar, using the origin pillar as an auxiliary.

We can repeat this pattern recursively, reducing the number of discs with every step until we reach a base case where only one disk remains, which we just directly move.

In mathematical terms, the minimum number of moves to solve the problem with n disks equals the number of moves needed to solve it for n – 1 disks, plus one to move the largest disk, plus the number of moves needed to solve it for n – 1 disks again. This formula yields the result:

TH( n ) = TH( n – 1 ) + 1 + TH( n – 1 ) = 2TH( n – 1 ) + 1

Where TH( n ) represents the number of moves needed to solve the problem with n disks.

From the above formula, we will substitute each TH( i ) by its corresponding expression until we reach TH( 1 ) . Then, we will replace TH( 1 ) by 1, since, with only one disc, it is possible to move it directly to the destination pillar, which justifies that the minimum number of moves in this case is 1.

Let’s see the expression developed:

We first substitute TH( n – 1 ), and we obtain:

TH( n ) = 2TH( n – 1 ) + 1

TH( n ) = 2( 2TH( n – 2 ) + 1 ) + 1 = 2²TH( n – 2 ) + 3

Now we substitute TH( n – 2 ):

TH( n ) = 2²TH( n – 2 ) + 3

TH( n ) = 2²( 2TH( n – 3 ) + 1 ) + 3 = 2³TH( n – 3 ) + 7

If we continue substituting in this way, we will arrive at a general formula:

TH( n ) = 2^k TH( n – k ) + (2^k – 1)

Where k is the number of steps backward in the recursion.

For k = n – 1 we arrive at TH( 1 ), the base case:

TH( n ) = 2²TH( n – 2 ) + 3

We know that TH( 1 ) = 1, then we obtain:

TH( n ) = 2^{n – 1} · 1 + ( 2^{n – 1} – 1 ) = 2ⁿ – 1

We have shown that the number of moves needed to solve the Towers of Hanoi with n disks is:

TH( n ) = 2ⁿ – 1

Solving the Puzzle Using Mathematical Induction

Another way to approach the solution to the Towers of Hanoi puzzle is to use mathematical induction, a technique used to prove that a statement is true for all natural numbers, based on two fundamental steps: the base case and the inductive step.

In this case, the goal is to show that to move disks from the first pillar to the third pillar, 2ⁿ – 1 moves are required. Let’s break this process down with a demonstration by induction.

Base case:

Suppose we have only one disk. In this scenario, it is clear that only one move is needed to move the disk from the source abutment to the target abutment. That is, for n = 1, the number of moves required is 2¹ – 1 = 1, which is true.

Inductive step:

Now, suppose that the statement is true for n = k, i.e., that to move k disks requires 2^k – 1 moves. What we must prove is that the statement is also true for n = k + 1.

To move k + 1 disks, we first need to move the upper disks from the origin pillar to the auxiliary pillar, using the destination pillar as an auxiliary. According to our inductive hypothesis, this will take 2^k – 1 moves.

Next, we move the larger disk (the disk k + 1) from the origin pillar to the destination pillar, which requires just 1 move.

Finally, we move the k disks from the auxiliary pillar to the destination pillar, using the origin pillar as an auxiliary, which will also take 2^k – 1 moves, according to our inductive hypothesis.

Therefore, the total number of moves required to move k + 1 disks is:

(2^k – 1) + 1 + (2^k – 1) = 2^k+1 – 1

Thus, we have shown that if the statement is true for n = k, it is also true for n = k + 1.

As we have observed, it is true for 1. By mathematical induction, we can conclude that for n disks, the minimum number of necessary moves is 2ⁿ – 1.

Going back to the legend, if the monks were extremely fast and could move a disk in just one second, we could calculate how long it would take to complete the challenge with 64 disks. According to the formula, the total number of moves would be 2⁶⁴ – 1, approximately 18.4 quintillion moves. With each of these taking just 1 second, this gives us 18.4 quintillion seconds. Converting these into years, we get that the total time would be approximately 581.4 billion years. So, if the legend turned out to be true, the end of the world would still be a long way away!

CalcMe formula We note that the function describing the number of moves needed to solve the Towers of Hanoi is exponential, meaning that it grows rapidly as we increase the number of disks. In the following graph, we can visually see how this function spikes, clearly illustrating the exponential growth.

graphic formula

This visual support allows us to confirm, once again, that completing the challenge with 64 discs would take an extraordinary amount of time.

If you liked this mathematical challenge and you were surprised by its solution, don’t hesitate to share this article! And if you dare, test your skills by solving the Towers of Hanoi puzzle: will you manage to move the disks in the exact number of steps? Let us know your experience!

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AI in Mathematics: The Revolution in Scientific Writing

Post author By Aida Alsina
Post date May 6, 2025

Artificial Intelligence (AI) is a field of computer science that seeks to develop systems able to perform tasks that traditionally require human intelligence, such as logical reasoning, problem solving, and learning. Based on advanced algorithms and machine learning models, AI can analyze large volumes of data, identify complex patterns and adapt autonomously to new contexts.

In mathematics, these capabilities are transforming the way knowledge is generated, verified, and communicated. From automating demonstrations to generating detailed explanations, AI is facilitating the work of researchers, teachers, and students. In addition, its ability to process natural language is improving the writing of mathematical papers, making texts more accessible, accurate and structured.

In this article, we will explore how AI in math is revolutionizing mathematical writing and how Wiris’ tools are part of this advancement.

The Role of AI in math and the Creation and Editing of Mathematical Content

Writing scientific papers in mathematics has always been a challenge due to the complexity of their notations and symbols. Traditionally, mathematicians have relied on handwriting or the use of complex typesetting languages. However, with the advent of AI, the process of creating and editing scientific documents has evolved exponentially.

Artificial intelligence has brought multiple benefits to the writing of mathematical papers, facilitating both the writing and the editing and publication of academic papers. Below are some of the main ways in which AI is impacting this area.

Handwriting recognition

One of the most significant advances has been the development of systems capable of interpreting and digitizing handwritten notes. Traditionally, mathematicians wrote their equations and proofs on paper, which was time-consuming if they needed to transcribe them into a digital format. With AI-based handwriting recognition tools, such as those built into MathType, it is now possible to directly handwrite equations into our touch devices and automatically convert them into editable digital text.

This not only saves time but also reduces transcription errors and allows you to work more efficiently. In addition, this type of technology is especially useful for students and teachers, as it facilitates the conversion of notes into organized documents without the need for manual typing.

Formula automation

In the creation of mathematical articles, writing formulas accurately is essential. Traditionally, this required specialized tools that, while powerful, could be complex and demanded a deep understanding of their syntax and structure, such as LaTex.

Advancements in technology have enabled the development of tools like MathType, which simplify equation writing without the need for coding. Instead of memorizing commands and code structures, users can simply enter their formulas through an intuitive graphical interface. This democratizes access to mathematical writing, allowing more people, regardless of their technical expertise, to create high-quality mathematical documents.

Another key aspect is the automatic correction of errors in equations. AI in math can identify inconsistencies in formula writing and suggest real-time corrections, preventing mistakes that could compromise the validity of an academic paper.

Optimization of editing

Another benefit that AI has brought is the optimization of the editing of mathematical documents. In the past, mathematicians and scientists had to manually check their documents for errors in equations, notation and text structure. Today, there are AI algorithms embedded in editing programs that can analyze the consistency of mathematical expressions, suggest improvements in writing and detect inconsistencies.

AI-based writing assistants, such as those integrated into advanced text processors, can help improve the clarity and readability of mathematical articles. In addition, they provide stylistic recommendations to give the paper a logical and easy-to-follow structure, which is crucial in academic paper writing.

The integration of AI into mathematical writing is not just an evolution but a revolution that is transforming the way we create, edit, and share scientific knowledge. Tools like Wiris, with their powerful capabilities, are making mathematical writing more accessible, efficient, and error-free. As these technologies continue to advance, they will further bridge the gap between human intuition and machine accuracy, making the work of researchers, teachers, and students alike easier. If you found this exploration interesting, share it with your peers and join the conversation about the future of AI in math.

If you’d like to find out more about our products, please feel free to get in touch with our Sales team at sales@wiris.com for more information.

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Wiris in the Classroom: Learn and Practice with Unlimited Exercises

Post author By Aida Alsina
Post date April 16, 2025

Learning mathematics requires more than just understanding theories: the key to mastering complex concepts and honing skills lies in continuous practice. However, traditional methods often limit the learning process: once the student has solved a problem, the challenge disappears, and exams often repeat the same questions, thus losing their value as an assessment tool. Imagine having the possibility to generate unlimited exercises with random variables, adapted to the needs of each student. With Wiris, this is now possible.

In addition, it has an automatic correction functionality which means that while the exercise is being generated, the answer is calculated simultaneously. This allows instant feedback, making it easier to learn and correct mistakes immediately.
For teachers, this tool offers a key advantage: the possibility of generating an infinite number of exercises, with each one being unique. In this way, teachers can create varied and personalized learning experiences, ensuring that students face new challenges every time they practice.

For their part, students enjoy constant practice which allows them to reinforce their skills without the worry of repeating the same exercises. In addition, this ability to generate unlimited exercises gives them excellent preparation for exams.

Practical example: Solving a system of linear equations

To illustrate how this tool works and how it can be used in the classroom, we will focus on solving systems of linear equations. We chose these types of problems because they represent a common mathematical challenge faced by all students and offer the possibility to explore different problem-solving methods. In addition, systems of equations can have various characteristics, such as being compatible, incompatible, or indeterminate, allowing students to work with a wide range of situations.

Let us imagine that we have the following system of linear equations with three unknowns:

We are going to solve it using Gauss’s Method. The first step is to write the system in the form of an augmented matrix:

Now, we will apply elementary operations to reduce the matrix to its row echelon form.

Therefore, we obtain:

From equation 3 of the system, we find the variable z:

Unlimited generation of systems of linear equations with random variables

Now that we have solved this system, for further practice, we would like to be able to generate new systems of equations of the same form but with random values.

These systems would take the following form:

Below, we will show you a code example to implement this functionality and generate given random systems of equations. With Wiris, it is totally possible!

r() := random(-4,4)

sol = [r() with i in 1..3]

[a,b,c] = sol

repeat

A = [[r() with i in 1..3] with j in 1..3]

until determinant (A) != 0

b = A * sol

ec = {}

for i in 1..3 do

ecaux = A.i * [x,y,z]

ec = append(ec, ecaux = b.i)

end

{ec1, ec2, ec3} = ec

r() := random(-4,4): This function generates a random number between -4 and 4.

sol = [r() with i in 1..3]: Here, three random values are generated that will represent the solutions of the system of equations.

[a,b,c] = sol: The generated values are assigned to the unknowns of the system.

repeat…until determinant(A) != 0: This block ensures that the coefficient matrix A is invertible (i.e., its determinant is not zero), which guarantees the system has a unique solution.

b = A * sol: The result vector b is calculated by multiplying the coefficient matrix A by the solutions sol

for i in 1..3 do: In this cycle, the three equations are generated, using each of the rows of the matrix A and the solutions sol.

{ec1, ec2, ec3} = ec: Finally, the three generated equations are stored in the variables ec1, ec2, and ec3, ready to be used in new exercises.

Using WirisQuizzes and implementing this code as a question, we can obtain the following results:

Reloading another system with the symbol “=” generates a new set of equations, as shown below:

For more details, you can consult the complete product documentation here.

If you have found this article useful, we would love you to share it with other colleagues, teachers or students. We would also be delighted to hear your opinion and any suggestions you may have.

Check Out WirisQuizzes

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What is the Birthday Paradox?

Post author By Aida Alsina
Post date April 8, 2025

The Fascinating Probability of Sharing a Date

The birthday paradox is a mathematical concept that invites us to challenge our intuition. A paradox, in essence, is a statement or result that, although seemingly contradictory or contrary to common logic, proves to be true when analyzed closely. In the case of the birthday paradox, the scenario is simple but puzzling: how many people must be in a room for the probability of at least two sharing the same birthday to be greater than 50%?

The result will be surprisingly low, but do you dare to answer before you continue reading to reveal the answer?

The most intuitive way to approach the paradox would be to think that it takes at least 183 people, half the days of the year, for the probability of two people sharing the same birthday to be greater than 50%. However, the reality is very different.

To understand how this probability is calculated, let us recall that the basic formula of any probability consists of dividing the favorable cases by the total cases.

In this case, rather than calculating the probability of a coincidence directly, it is easier to first calculate the probability that no two people share a birthday. After determining this probability, subtracting it from 1 (that is, calculating the complement) gives the probability that at least two people share a birthday.

To calculate the probability that no two people share a birthday, we will proceed constructively, evaluating scenario by scenario.

If there is only one person in a room, there is no chance that they will share their birthday with someone else since no other people are present. Therefore, the probability that their birthday is unique is 100%, i.e., 365/365.
Now, if a second person comes in, the probability that their birthday will be different from the first person’s is 364/365, since there are only 364 days remaining in which they can be born to avoid a coincidence.
If a third person is added, the probability that their birthday does not match anyone else’s is 363/365 since there are now two busy days.

Therefore, for n people, all the above conditions must be met simultaneously: that the second person does not share a birthday with the first, that the third person does not share a birthday with the previous two, and so on. This implies that the probability that none of the n people share the same birthday is the product of the individual probabilities:

And finally, we obtain:

Analyzing probabilities for different values of n

Let us examine the results for different values of n:

For n=10 we obtain that P (at least one coincidence) = 11,61%

CalcMe representation of the equation

This calculation has been created using CalcMe

For n=15 we obtain that P (at least one coincidence) = 25,03%
For n=23 we obtain that P (at least one coincidence) = 50,05%
For n=50 we obtain that P (at least one coincidence) = 96,53%
For n=60 we obtain that P (at least one coincidence) = 99,22%

As we can see from the examples above, to achieve a probability of more than 50%, only 23 people in the same room are needed. This result may seem surprising, but it makes sense if we consider that with 23 people, 253 different pairs can be formed, and each of these pairs represents a chance for two people to share the same birthday. Even more surprising is that to have a probability greater than 99%, only 60 people are needed, demonstrating how the number of possible combinations grows rapidly with every new person added.

This is the graph of the probability distribution:

Graph showing the probability of two people sharing a birthday, with the number of people on the X-axis and the probability on the Y-axis, expressed as a percentage.

In contrast, if instead of calculating the probability of coincidence between any pair of people we consider the probability that someone in a room of n people (excluding you) specifically shares your birthday, the calculation is different. This probability is given by:

For n = 22, this probability is about 0.059 (5.9%), which is quite low. In fact, there would need to be at least 253 people in the room for this probability to exceed 50%.

The birthday paradox is a fascinating reminder of how our intuition can fail when confronted with mathematical logic.

Now that you know the birthday paradox and its key concepts, can you calculate the probability that exactly k people within a group of n people share their birthday? We’d love to hear your results in the blog comments! If this article surprised you, share it with your friends so that they can learn more about this mathematical paradox. What’s more, if you’re interested in trying these calculations for yourself, have a look at CalcMe.

Try CalcMe

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Gamification in Math Learning: Does it Improve Learning?

Post author By Aida Alsina
Post date March 25, 2025

Gamification in Math Learning: Benefits & Challenges

Struggling to keep students engaged? Gamification, the use of game mechanics in learning, helps increase motivation, comprehension, and retention. The abstraction of concepts and the difficulty in linking them to real-life situations can generate disinterest, frustration and even anxiety in students. In this context, gamification has emerged as an innovative strategy that aims to transform the learning experience by integrating game elements into the educational process.

But what exactly is gamification, and what impact does it have on the teaching of mathematics? According to Deterding et al. (2011), it is defined as the use of game design elements in non-game contexts to enhance the participation, motivation, and engagement of individuals. In the educational setting, this involves not only the incorporation of game mechanics such as challenges, immediate feedback, points, and rewards, but also the creation of more dynamic and interactive experiences that encourage exploration and active learning.

Throughout this article, we will explore the existing evidence on the effectiveness of gamification in math education and see how interactive games like LearningLemur can benefit students’ learning.

Why Gamification Works: Key Benefits for Math Learning

Gamification has proven to be an effective tool for improving the learning experience in various disciplines, and mathematics is no exception. Its impact goes beyond making classes more entertaining; it influences key aspects such as motivation, understanding of concepts, and the development of cognitive and social skills. Below are some of the main benefits supported by research.

Increased Motivation and Engagement

One of the main challenges in teaching mathematics is to maintain students’ interest, especially when they perceive the exercises as repetitive or difficult. Gamification transforms these tasks into interactive and stimulating experiences. According to a study by Hamari, Koivisto, and Sarsa (2014), the implementation of gamified strategies has been shown to significantly increase motivation and academic performance in various disciplines, including mathematics.

Improved Comprehension and Retention of Concepts

Games allow students to interact with mathematical concepts in a hands-on, visual way, facilitating deeper understanding. Instead of memorizing formulas without context, students can experiment with problems in dynamic scenarios. A study by Wouters et al. (2013) indicated that gamified environments not only improve knowledge retention but also strengthen the ability to apply what is learned in different situations.

Cognitive and Social Skills Development

Beyond learning mathematical concepts, gamification promotes essential skills for critical thinking and problem-solving. Solving challenges within a gamified environment requires students to analyze situations, explore different strategies, and adapt their approach based on the results obtained. In addition, many gamified experiences include collaborative dynamics that foster communication and teamwork. It also encourages resilience by allowing students to learn from their mistakes without the fear of conven

tional academic failure.

Mathematics Anxiety Reduction

Math anxiety is a common problem that can affect students’ performance and attitude toward the subject. Presenting problems in a playful and less structured format can reduce the pressure associated with formal learning. According to Sailer et al. (2017), gamification helps to generate a more positive relationship with mathematics, allowing students to face challenges without fear of failure. Immediate feedback and reward systems reinforce individual progress, helping to build confidence and autonomy in learning.

Gamification Challenges: What Educators Need to Know

While gamification offers multiple benefits in mathematics learning, its effectiveness depends largely on how it is designed and implemented. Some key factors to consider for successful implementation are:

Designing appropriate content: Creating gamified activities aligned with curricular objectives requires time and planning on the part of educators.
Balancing Play and Learning: It is essential that the fun of the game does not overshadow the educational content, ensuring that students acquire the intended mathematical skills. To avoid this problem, it is key to design activities where progress in the game depends on mastery of the mathematical concepts, ensuring that the mechanics reinforce the content rather than distract from it.
Adaptability: Each student learns at their own pace and in a different way. An effective gamified approach must be flexible and allow for customization according to learners’ skill levels and learning styles.

Strategies for Effective Gamification in Mathematics Learning

To maximize the potential of gamification in math, educators and developers must carefully structure their approach. Here are some key strategies to ensure effective implementation:

Setting Clear Learning Objectives

Gamification must have a clear educational purpose to be effective. Before incorporating game elements, it is essential to define precise learning outcomes. Every challenge, reward, or interactive element should directly contribute to developing students’ understanding of mathematical concepts. When games are designed to align with curriculum goals, students remain engaged while making measurable academic progress.

Incorporating Interactive Games for Engagement

Engaging students through interactive math learning is one of the most effective ways to enhance learning. Digital platforms and apps provide adaptive learning experiences that offer students challenges suited to their individual skill levels. Features such as leaderboards, badges, and achievements create a sense of accomplishment, motivating students to persist in solving mathematical problems while reinforcing key concepts.

Providing Instant Feedback and Progress Tracking

Timely feedback is crucial in the learning process. One of the major advantages of gamification in math learning is the ability to provide instant feedback. When students receive immediate results on their performance, they can quickly identify mistakes, adjust their approach, and strengthen their understanding. Many digital platforms include progress-tracking tools, allowing both students and educators to monitor learning trends, address weaknesses, and celebrate achievements.

Encouraging Collaboration and Healthy Competition

Gamified learning is not just about individual progress—it can also foster teamwork and social interaction. Incorporating cooperative challenges and team-based problem-solving enhances mathematics learning by encouraging peer-to-peer discussions and shared strategies. At the same time, friendly competition through scoreboards and ranking systems can inspire students to push themselves further, increasing motivation and engagement in a healthy, constructive way.

Tools and Resources for Gamifying Mathematics Learning

With advancements in technology, numerous tools are available that support gamification in math. These resources offer diverse ways to create a more dynamic learning experience, helping students grasp mathematical concepts more effectively:

Digital Learning Platforms: Websites and apps that integrate math practice with games to teach mathematical concepts through fun and challenging activities.
Classroom Gamification Kits: Both physical and digital resources are designed to help teachers implement game-based learning strategies in their lessons.
Virtual and Augmented Reality: Immersive experiences that provide students with visual and hands-on interactions to better understand abstract mathematical ideas.
Customizable Gamification Plugins: LMS-integrated tools that allow educators to personalize their teaching by adding game-based elements to their curriculum.

Try Gamification in Mathematics Learning with Learning Lemur

The benefits of gamification in mathematics learning are clear: increased motivation, improved comprehension, and enhanced engagement. If you’re ready to experience the power of interactive games in education, explore Learning Lemur. Our platform allows educators and students to immerse themselves in engaging, game-based mathematics learning experiences.

Create a free account today and discover how gamification can revolutionize the way you learn mathematics!

Start Using Gamification for Free

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What is the Collatz Conjecture?

Post author By Aida Alsina
Post date February 28, 2025

What is the Collatz Conjecture?

Mathematical conjectures are riddles that challenge the human mind: seemingly simple problems that, despite having been verified in millions of cases, still lack a proof. They are seemingly intuitive questions that hide unexpected complexity.

The fascinating thing about conjectures is that, in many cases, they can be empirically verified i.e., verified by an enormous number of numerical examples. However, this verification does not satisfy mathematicians, who seek a theoretical proof, a solid proof that validates the statement in its entirety. Without such a proof, the conjectures remain open challenges.

The Collatz conjecture is a perfect example of this type of mathematical mystery. It was proposed in 1937 by Lothar Collatz and its statement is as follows:

Choosing any positive integer we will apply the following steps:

➔ If the number is even, we will divide it by 2.

➔ If the number is odd, multiply it by 3 and then add 1.

➔ Repeat the process with the new number obtained.

The conjecture says that, no matter what number you start with, you will always get to number 1. Once you get to 1, the process repeats indefinitely: 1 →4 →2 →1

Formally, this can be written as the function f=N →N defined as

Two illustrative examples of the Collatz Conjecture

Let’s look at a couple of simple examples.

Let’s take the number n=6 and apply the steps of the Collatz conjecture:

6 is even so we divide by 2: 6/2=3
3 is odd so we multiply by 3 and add 1: 3·3+1=10
10 is even so we divide by 2: 10/2=5
5 is odd so we multiply by 3 and add 1: 3·5+1=16
16 is even so we divide by 2: 16/2=8
8 is even so we divide by 2: 8/2=4
4 is even so we divide by 2: 4/2=2
2 is even so we divide by 2: 2/2=1

Once we reach 1, the process is repeated: 1 →4 →2 →1.

So we observe that starting with n=6 the conjecture is satisfied.

Let us now take the number n=21 and apply the steps of the Collatz conjecture:

21 is odd so we multiply by 3 and add 1: 21·3+1=64
64 is even so we divide by 2: 64/2=32
32 is even so we divide by 2: 32/2=16
16 is even so we divide by 2: 16/2=8
8 is even so we divide by 2: 8/2=4
4 is even so we divide by 2: 4/2=2
2 is even so we divide by 2: 2/2=1

Again, we arrive at number 1.

Both examples have reached the number 1, but in a different number of steps. In fact, although 21 is a larger number, it reached 1 in fewer steps than 6. This conjecture has been tested for an incredible number of numbers, up to more than 2ˆ60 = 1.152.921.504.606.846.976 cases, without finding any counterexample. However, it still remains a mystery whether there is a number that does not satisfy the conjecture.

Graphical representations of the Collatz conjecture

The directed graph of orbits is a visual representation that facilitates the understanding of the behavior of numbers under the rules of the Collatz conjecture. In this graph, each number is represented as a node, and the connections between them show the steps followed by the conjecture process. When following the sequence of a number, the nodes are connected by arrows that indicate how the number is transformed at each step. Although even numbers are generally omitted from the representation for simplicity, the graph illustrates how the numbers “orbit” around certain cycles, such as 1 →4 →2 →1.

In the image above, we can see an example of the directed graph of orbits, where we have highlighted in black the numbers 3 and 21. The number 3 refers to the sequence of 6, since in this graph the even numbers are omitted to simplify the visualization.

We note that 21 quickly reaches 1, since it only passes through even numbers on its way, as we saw in the previous example. On the other hand, the number 6 first becomes 3, then pass through the number 5, which will finally take it to 1 after several passes through even numbers.

It is also interesting to note that the number 9, although relatively small, follows a longer sequence: it goes through the numbers 7, 11, 17, 17, 13 and 5, and finally reaches 1, after a total of 19 steps.

Below is a graph in which the X-axis shows the different initial integer values, while the Y-axis represents the number of iterations required for each number to reach the value 1.

In summary, the Collatz conjecture is an apparently simple problem that, to this day, still has no formal proof. Although it may seem a statement of little relevance or practical utility, this conjecture has applications in various fields, such as number theory, cryptography, algorithm analysis and artificial intelligence. In these fields, the study of complex sequences and their behavior under specific rules can offer valuable insights for solving larger problems and understanding mathematical patterns.

Have you been fascinated by the simplicity and mystery of the Collatz conjecture? Now it’s your turn: take the number 27 and start the sequence. How many steps will you need to get to 1?

If this challenge got you, don’t keep it to yourself! Share it with others who are curious about mathematical riddles and find out together who can solve the challenge the fastest.