Beware of Bad Math Tricks!

I view many mathematical posts (e.g., on Instagram and TikTok) to expand my knowledge of different aspects of mathematics, such as learning what Subfactorials, Double Factorials, and Primorial Factorials are, and what it really means when you divide fractions. I am frustrated with many of these posts that present a mathematical trick, hack, or shortcut for the following reasons.

Share a mathematical trick, hack, or shortcut without explaining the concepts behind it.
- Some prioritize speed of getting the answer, which leads to common pitfalls like misapplying “Please Excuse My Dear Aunt Sally” (PEMDAS) by treating multiplication before division.
- Relying on the “butterfly method” for adding or subtracting fractions.
Failing to provide the answer or the way to find it when asking you to solve it.
Failing to provide the necessary assumptions for the mathematical trick, hack, or shortcut to work.
Demonstrating a mathematical method that has little to no value for learning mathematics.

These tricks, hacks, or shortcuts can create “phantom” competence, causing students to struggle with more complex mathematics later. There needs to be a balance between conceptual and procedural repetitions. If students can’t explain why a tip or trick works, they shouldn’t use it.

The main focus of a post should be to inform and enjoy mathematics and, as a result, inspire readers to explore the subject further, potentially leading to more research. By sharing these mathematical tricks, the aim should be to help audiences recognize and appreciate the power and beauty of mathematics.

Here are some math tricks, hacks, and shortcuts to possibly avoid and why.

Solving Simultaneous Linear Equations Tricks

mathswalaamitsir posted “Solving Equations Tricks.” This post presented a “trick” method for solving the following linear equations.

Solving for x.

Numerator

3x + 2y = 7  Step 1
 x + 2y = 3

minus

3x + 2y = 7  Step 2
 x + 2y = 3

Denominator (Determinate)

3x + 2y = 7  Step 3
 x + 2y = 3

minus

3x + 2y = 7  Step 4
1x + 2y = 3

$x\,\,=\,\,\frac{\left( 7\times 2 \right) \,\,-\,\,\left( 3\times 2 \right)}{\left( 3\times 2 \right) \,\,-\,\,\left( 2\times 1 \right)}\,\,=\,\,2$

Solving for y.

Numerator

3x + 2y = 7  Step 5
 x + 2y = 3

minus

3x + 2y = 7  Step 6
1x + 2y = 3

Denominator (Determinate)

Steps 3 & 4 above

$y\,\,=\,\,\frac{\left( 3\times 3 \right) \,\,-\,\,\left( 7\times 1 \right)}{\left( 3\times 2 \right) \,\,-\,\,\left( 2\times 1 \right)}\,\,=\,\,0.5$

Solving The Linear Equations Using Matrices

Solving AX = B, where

A is the 2×2 matrix of x and y coefficients
X is x and y, and
B is 7 and 3

$A\,\,=\,\,\left[ \begin{matrix}3& 2\\1& 2\\\end{matrix} \right]$

$|A| =\,\,\left( 3\times 2 \right) \,\,-\,\,\left( 2\times 1 \right) \,\,=\,\,4$

$A^{-1}\,\,=\,\,\frac{1}{|A|}\left[ \begin{matrix} 2& -2\\ -1& 3\\ \end{matrix} \right]$

$B\,\,=\,\,\left[ \begin{array}{c} 7 \\ 3 \\ \end{array} \right]$

$X\,\,=\,\,A^{-1}B$

$X\,\,=\,\,\frac{1}{4}\left[ \begin{matrix} 2& -2\\ -1& 3\\ \end{matrix} \right] \left[ \begin{array}{c} 7\\ 3\\ \end{array} \right]$

Completing the calculation for X, we see that we get the same result as the steps presented in the post Solving Equations Tricks!

I would advise not using this trick without first understanding the underlying concept of solving systems of linear equations using matrices. What I like about this trick is that by solving the linear equations in this manner you are in reality performing multiplication using the inverse matrix in the numerator, and calculating the determinant of the matrix in the denominator of both x and y.

Square Roots

_mathtricks__ posted “Solving Equations Tricks.” This post presented a “hack” method for solving the square root of a number using the following steps.

Add the sum of the digits under the radical, e.g., the square root of 196 = 1 + 9 + 6 = 16.
Subtract 2 (square root) from the result in step 1, e.g., 16 – 2 = 14, and you get the value of the square root of that number. 14² = 196.

$\sqrt[2]{25}\,\,=\,\,\sqrt[2]{2+5}\,\,=\,\,7-\,2 =\,5$

$\sqrt[2]{64}\,\,=\,\,\sqrt[2]{6+4}\,\,=\,\,10-\,\,2 =\,8$

$\sqrt[2]{196}\,\,=\,\,\sqrt[2]{1+9+6}\,\,=\,\,16-\,\,2 =\,14$

However, if we try this hack with find the square root of 121 we do not get the correct answer 11.

$\sqrt[2]{121}\,\,=\,\,\sqrt[2]{1+2+1}\,\,=\,\,3-\,2 =\,\,1$

An uninformed or even informed user might assume that the hack works for all cases of finding the square root of a number that has a square root. I haven’t put much effort into discovering which values this trick applies to, and I don’t plan to spend a lot of time exploring further.

Multiplication

Multiplication Trick

unstoapablestudy0111 posted “Multiplication Tricks.” This post presented a short cut method for multiplying a two-digit number by a single-digit using the following steps.

Multiply the second digit (e.g., 7) of the two digit number (e.g., 77), by the one digit number (e.g., 5).
Write the result with a space between the digits, e.g., 3_5.
Add the two digits of the product (e.g., 3 + 5 = 8) and place that sum in the blank space, e.g., 385.

Therefore, using the above method we would get the following.

77 x 5 = 3_5 = 385
88 x 3 = 2_4 = 264
66 x 4 = 2_4 = 264
33 x 6 = 1_8 = 198

So, I tried this trick as follows.

37 x 6 = 4_2 ≠ 462, but actual results should be 222

An uninformed or even an informed user might assume that the hack works for all cases of multiplying a two-digit number by a single-digit number. However, closer examination shows that this trick ONLY works when multiplying a two-digit number that is a multiple of 11 by a single-digit number.

Multiplication of Any Two Digit Number

lian_wang2 posted “Multiplication of Any Two Digit Number.” This post presented a method for multiplying a two-digit number by a two-digit using the following steps with the expression 83 (multiplicand) x 45 (multiplier).

Multiply the tens digits of the multiplicand and multiplier and place the product underneath the expression, e.g., 8 x 4 = 32.
Multiply the ones digits of the multiplicand and multiplier and place the product after the value from step 1, e.g., 3 x 5 = 15 to get 3215.
Multiply the ones digit of the multiplicand and tens digit of the multiplier and place the product underneath the value from step 2 as shown below, e.g., 3 x 4 = 12.
Multiply the tens digit of the multiplicand and ones digit of the multiplier and place the product underneath the value from step 3 as shown below, e.g., 8 x 5 = 40.
Add the results of steps 2 through 4 as shown below.

83 x 45
 32     (Step 1)
 3215   (Step 2)
  12    (Step 3)
  40    (Step 4)
-------
 3735

What am I missing? Is this method faster? Is it better than long multiplication, and in what way? Easier to remember? Does it help someone understand two-digit multiplication better? Does it only work when the addends are two digits? Why do I need to know this method? These are the kinds of questions everyone should ask when evaluating any mathematical trick, hack, or shortcut.

Let’s use long multiplication to evaluate the same expression and compare with the “trick” above.

Using long multiplication offers a structured, step-by-step way to multiply large, multi-digit numbers by breaking them into smaller, manageable parts. Its main benefits include improving calculation accuracy, enabling efficient handling of complex problems, enhancing understanding of place value, and providing a reliable method for multiplication.

Addition

farzad_raufi_2026 post presented a short cut method for adding two, two-digit numbers using the following steps with the expression 93 (addend) + 39 (addend).

Using the second addend, add the tens and units digits together (e.g., 3 + 9 = 12) and place the tens digit of the sum in the hundreds place, and the units digit of the sum in the units place.
Added the tens and units digits from step 1 (e.g., 1 + 2 = 3) and place this sum in the tens place of the sum in step 1.

93 + 39 = 1_2 (Step 1)
93 + 39 = 132 (Step 2)

This seems to work only when the addends are two-digit palindromes. I am not sure of the purpose of this approach, why it is used, or where it can be applied.

superdrawings2025 post presented a short cut method for adding two, three-digit numbers using the following steps with the expression 786 (addend) + 786 (addend).

 786
+786
----
14
 16
  12
----
1572

I am not sure of the purpose of this approach, why it is used, or where it can be applied.

The above “tricks” might be related to learning mathematics, although their specific relevance isn’t immediately clear to me. I don’t know whether the addition tricks mentioned above are part of Vedic Maths tricks, Chinese math tricks, or tricks from another country’s math system.

Can You Find The Next Number?

IQ GROW GEN posted “Only 1% People Can Solve This Number Series! Can You Find the Next Number?” This post presented the following number series 4, 6, 12, 14, 28, 30, ? Then asked the questions, What Comes Next?

Answer: 60

I used this opportunity to really explore sequences because many posts that quiz you to find the next number in a sequence don’t give an answer. (See the What Comes Next? webpage.) Sometimes it would be helpful if the person posting would include the answer (e.g., 60) and the sequence they used. I see each post as a way to learn.

Conclusion

I could go on and on about the many “bad” math tricks, hacks, and shortcuts I see every day, and wonder how they can be used when teaching students and in everyday life. I am not even sure why I still look at them, as I get frustrated by the lack of information presented that helps me understand the concept and how to apply it. Additionally, is it better to memorize the mathematical trick, hack, or shortcut, or is it better to understand the underlying concept? Most education focuses too much on memorization (knowledge) because recall is valued more than reasoning (easier to test and get a number to rate a person on how well they can regurgitate the information), and information is treated as the goal rather than the starting point. What is needed is the progression from knowledge to understanding, and then wisdom — from information to insight to sound judgment. Each plays a different role in how a person learns, thinks, and acts.

Here are some words of wisdom from some other sources.

“So instead of telling students “tricks are bad,” I propose we show students the limits of that trick, and help them understand why the trick stops working where it does. This might seem obvious. But there’s a huge difference between telling a student “don’t use FOIL, tricks are bad” and showing them its limits with a thoughtful set of problems.” [dkane47]

Tina Müller, M.Sc. Mathematics, Leiden University, stated, “Doing mental arithmetic quickly has nothing to do with mathematical understanding. There are tricks to do it, and the rest is a lot of practise. Being able to multiply 14-digit numbers fast makes for a nice party trick or will enable you to enter fast-calculation competitions. But just like being good in spelling competitions doesn’t make you a good writer, being good in fast-calculation competitions doesn’t make you a good mathematician.”

“In math, there’s no shortcut to true understanding. While tricks and tips may offer temporary relief, they undermine students’ ability to develop critical thinking and problem-solving skills. By emphasizing conceptual understanding, particularly through methods like multiplicative thinking and Structures of Equality, we can empower students to make sense of math, not just memorize it. Ultimately, the goal is not quick solutions but lasting comprehension that equips students to tackle more complex mathematical challenges with confidence.” [russo]

And don’t get me started when someone presents a math problem using the following wording!

Most People Get This Math Question Wrong!
Only 1% Can Solve This Math Problem!
Only a genius …

STOP the madness!

References

MATH is FUN

“Matrix Calculator.” Math Is Fun. Accessed February 23, 2026. https://www.mathsisfun.com/algebra/matrix-calculator.html.

“Solving Systems of Linear Equations Using Matrices.” MATH is FUN. Accessed February 23, 2026. https://www.mathsisfun.com/algebra/systems-linear-equations-matrices.html.

Other

“5 Easy Math Tricks We Should No Longer Be Teaching Students.” Third Space Learning. January 6, 2023. https://thirdspacelearning.com/us/blog/math-tricks-bad-habits-we-teach-students/.

When you multiply by ten you just add a zero. Does this make you cringe? It should as it simply isn’t true. And we know why, don’t we? If you add a zero you’re adding nothing, therefore the number doesn’t change. Try telling that to a child who has worked it out for themselves without you ever having to utter the forbidden phrase. As elementary teachers, we need to model good conceptual understanding in all our math lessons, rather than relying on easy math tricks that work but don’t help children to have an understanding of the ‘why’ and the ‘how’. Here are the 5 most common bad habits I’ve identified in my role as an elementary math team leader; these are the easy math tricks teachers should no longer be using, and some ways to explain the math instead.

Ansari, Manir (unstoapablestudy0111). “Multiplication Tricks” Instagram. Accessed February 23, 2026. https://www.instagram.com/reels/DEUjGT5OQTh/.

Bhatia, Ankita (_mathtricks_). “Math Hack // Square Root Trick.” Instagram. Accessed February 23, 2026. https://www.instagram.com/p/C3-S-hOxFmC/.

dkane47. “Tricks, and What To Do About Them.” Five Twelve Thirteen, February 18, 2023. https://fivetwelvethirteen.wordpress.com/2023/02/17/tricks-and-what-to-do-about-them/.

“Inverse Matrix Calculator.” matrix RESHISH. Accessed February 23, 2026. https://matrix.reshish.com/inverse-matrix/.

IQ GROW GEN. “Only 1% People Can Solve This Number Series! Can You Find the Next Number?” Quora. Accessed February 26, 2026. https://qr.ae/pCj7aW.

mathswalaamitsir. “Solving Equations Tricks.” Instagram. Accessed February 23, 2026. https://www.instagram.com/reels/DVDGCz3kmik/.

“Matrices – solving two simultaneous equations.” 2009. mathcentre. https://www.mathcentre.ac.uk/resources/uploaded/sigma-matrices8-2009-1.pdf.

One of the most important applications of matrices is to the solution of linear simultaneous equations,
On this leaflet we explain how this can be done.

“Mathematical Investigation of Two Digit Palindromes.” June 7, 2021. https://math1089.in/2021/06/07/mathematical-investigation-of-two-digit-palindromes/.

To most outsiders, modern mathematics is unknown territory. Its borders are protected by dense thickets of technical terms; its landscapes are a mass of indecipherable equations and incomprehensible concepts. Few realize that the world of modern mathematics is rich with vivid images and provocative ideas.

Meyer, Dan. “What Do You Do When a Math Trick Gets to Class Before You Do?” Mathworlds, February 16, 2023. https://danmeyer.substack.com/p/what-do-you-do-when-a-math-trick.

Raufi, Farzad (farzad_raufi_2026). “NO TITLE” Instagram. Accessed February 25, 2026. https://www.instagram.com/reels/DVGfDbak475/.

Russo, Julie (russo). “Why Tips and Tricks Don’t Work in Math.” structureofequality.com, October 30, 2024. https://structureofequality.com/2024/10/30/tips-tricks-in-math/.

Stapel, Elizabeth. “Solving a System by Substitution — Explained!” Purplemath. Accessed February 23, 2026. https://www.purplemath.com/modules/systlin4.htm.

The method of solving “by substitution” works by solving one of the equations (you choose which one) for one of the variables (you choose which one), and then plugging this back into the other equation, “substituting” for the chosen variable and solving for the other. Then you back-solve for the first variable.

superdrawings2025. Instagram. Accessed February 25, 2026. https://www.instagram.com/reels/DRuIfByj-0C/.

Wang, Lian (lian_wang2). “Multiplication of Any Two Digit Number” Instagram. Accessed February 25, 2026. https://www.instagram.com/reels/DSJWX0cCNCe/.

Notes

Chinese Math Tricks

Chinese arithmetic traditionally relies on two complementary systems: mental/visual strategies rooted in place-value thinking (abacus and mental abacus) and written algorithms using place-value notation (counting rods historically, modern column algorithms). Below are the principal methods used to add, subtract, multiply, and divide large numbers, with concise descriptions and practical steps.

Abacus (suanpan) and mental abacus

Physical suanpan:
- Layout: each rod has two upper beads (heaven beads, value 5 each) and five lower beads (earth beads, value 1 each). Each rod = one decimal place.
- Addition/subtraction: operate bead adjustments per column from right (units) to left, using complements and carries; combines single-digit bead moves to perform multi-digit operations efficiently.
- Multiplication/division: use repeated-add/subtract strategies, place shifting, and decomposition of multiplier/divisor into convenient parts. Skilled users use finger patterns to perform long multiplications and divisions that resemble written algorithms but executed on beads.
Mental abacus:
- Practitioners visualize the suanpan and manipulate imagined beads to compute. Enables rapid multi-digit arithmetic and strong place-value awareness.
- Typical learning: beginners practice with physical abacus, then graduate to visualization drills (e.g., representing numbers on imaginary rods, performing addition/subtraction sequences).

Written column methods (school arithmetic)

Addition:
- Write numbers in columns aligned by place value (units under units, tens under tens).
- Add column-wise from right to left, write the digit result and carry the floor(divide by 10) to the next column.
- Example: 3 7 5 + 8 6 9 — units 5+9=14 (write 4 carry 1), tens 7+6+1=14 (write 4 carry 1), hundreds 3+8+1=12 → 1244.
Subtraction:
- Column-wise from right to left. If minuend digit < subtrahend digit, borrow 1 (10 in that place) from next left column; reduce that column by 1.
- Example: 1002 − 378: units 2−8 → borrow 1 from tens/hundreds chain, etc., yielding 624.
Multiplication (long multiplication, standard algorithm):
- Multiply multiplicand by each digit of multiplier starting from rightmost digit; write each partial product shifted left according to the multiplier digit’s place; sum partial products.
- Use single-digit × multi-digit products with carries handled per column.
- Example: 237 × 46 → compute 237×6 = 1422; 237×4 (tens) = 9480; add → 10902.
Division (long division, column-style or chunking):
- Traditional long division: determine how many times the divisor fits into the leading part of the dividend, subtract the product, bring down next digit, repeat. Track quotient digits left to right.
- Chunking (partial quotients): subtract large easy multiples of divisor (e.g., 10×, 100×) from dividend iteratively, accumulate quotient; useful for mental/teaching approaches.
- Example: 10902 ÷ 46: 46 fits into 109 two times (92), remainder 17 bring down 0 → 170 fits 3 times (138), remainder 32 bring down 2 → 322 fits 7 times (322) → quotient 237.

Historical methods: counting rods and suanfa texts

Counting rods:
- Used a positional system on a board; rods arranged in alternating vertical/horizontal patterns to represent digits 1–9 and place value.
- Algorithms for four operations were geometric manipulations of rod configurations; efficient for large integers and fractions.
Mathematical classics (e.g., Sunzi, Nine Chapters on the Mathematical Art):
- Present systematic rules and worked examples for operations, extraction of square/cube roots, fractions, and solving linear systems. Procedures align with modern column algorithms but expressed in concrete procedural rules and problems.

Decimal fractions and large numbers

Decimal (place-value) notation used in modern China; decimal point handled identically to Western notation.
For very large numbers, use scientific notation in higher education and scientific work; everyday large-number calculations use calculators, computers, and spreadsheets.

Pedagogy and cognitive advantages

Curriculum emphasizes place-value mastery, number sense, and column algorithms in primary grades, with abacus training common in some schools and extracurricular programs.
Abacus training accelerates mental arithmetic fluency and fosters strategies for decomposing numbers and handling carries/borrows mentally.
Modern classrooms supplement algorithms with number talks, estimation, and use of calculators for verification.

Practical tips for learning these methods

Learn place-value thoroughly (units, tens, hundreds).
Master single-digit operations and standard multiplication table (1–9).
Practice column addition/subtraction until carries/borrows are automatic.
Learn long multiplication by practicing partial products and aligning shifts.
Learn long division by practicing both standard long division and chunking (partial quotients).
Use abacus practice to build mental imagery for carries and multi-digit operations, then transition to mental abacus exercises.
Work through classical worked examples (e.g., from “Nine Chapters” examples or modern textbooks) to see step-by-step procedures on large numbers.

Summary

Contemporary Chinese arithmetic uses the same place-value column algorithms taught worldwide, supplemented in many places by abacus/mental-abacus training for speed and intuition. Historical methods (counting rods, classical procedural texts) shaped these algorithms but the practical techniques for adding, subtracting, multiplying, and dividing large numbers are the column algorithms and abacus methods described above.

Google Search. “Chinese Math Tricks.” Accessed February 26, 2026. https://share.google/aimode/SJT6YcK9buDoAiWtL.

Quora. “How Do the Chinese Learn to Add, Multiply, Subtract, and Divide (Big) Numbers?” Accessed February 25, 2026. https://www.quora.com/How-do-the-Chinese-learn-to-add-multiply-subtract-and-divide-big-numbers.

Long Multiplication

Long multiplication, also known as vertical multiplication, is a method used to multiply large numbers by arranging them vertically. This technique is particularly useful for multi-digit numbers, as it helps in keeping track of place values and simplifies the multiplication process. The method involves breaking down the numbers into parts, multiplying each part, and then adding the results together to find the final product.

Palindromes

A number is called a palindromic number if it reads the same both forward and backward. In other words, it has reflectional symmetry across a vertical axis. Few palindromic number are 11, 121, 31413. English sentences can also be palindromic, like do geese see god or murder for a jar of red rum or step on no pets.

Rule of Thumb

A “rule of thumb” is a broadly accurate guide, principle, or practical method based on experience rather than precise, scientific calculation. It serves as a mental shortcut for making quick, general decisions. The phrase rule of thumb refers to an approximate method for doing something, based on practical experience rather than theory.

“Phantom” Competence

“Phantom” or “illusion of competence” refers to the false belief that one has mastered a skill or understands a topic simply because they have spent time reviewing it, or, more recently, because AI tools have enabled them to produce work beyond their actual skill level. It is a gap between perceived proficiency and real capability.

Vedic Maths Tricks

Vedic Maths is a collection of ancient mathematical techniques and principles originating from the Vedas, the ancient Indian scriptures. It offers a unique approach to solving complex calculations efficiently and accurately. This is an ancient wisdom-based technique that transforms the complex calculations of numbers into an easy process. These techniques were mentioned in ancient Vedas by the then saints. Hence, the name is given as “Vedic Maths”.

“5 Powerful Vedic Maths Tricks to Improve Your Calculating Speed.” INTERVAL LEARNING, April 17, 2025. https://www.intervaledu.com/blogs/5-powerful-vedic-maths-tricks-to-improve-your-calculating-speed/.

GeeksforGeeks. “Vedic Maths Easy Tricks.” GeeksforGeeks, November 6, 2023. https://www.geeksforgeeks.org/maths/vedic-maths/.

Manjit. “25+ Vedic Maths Tricks In Simplified Version.” Vedicfeed, August 31, 2017. https://vedicfeed.com/vedic-maths-tricks/.

Rupesh, Mamta. “The Ultimate Guide to Vedic Maths for Beginners: Fast & Easy Tricks to Get Started Today.” The Vedic Maths – A Collection of Techniques/ Sutras to Solve Mathematical Problems, August 30, 2025. https://thevedicmaths.com/the-ultimate-guide-to-vedic-maths-for-beginners-fast-and-easy-tricks-to-get-started-today/.

Are you finding math tricky and hard? Don’t worry. Maths is not as tough as you thought when you understand the concepts clearly using maths tricks and shortcuts. Learn such tricks and shortcuts using the Vedic Maths for Beginners guide. Vedic Maths is derived from ancient Indian scriptures called “Vedas”, and helps you do the mathematical calculations quickly without pen and paper using simple tricks. In this blog, we will learn what Vedic Maths is and some easy tricks for performing math calculations quickly at school and in daily life.

Vedantu. “8 Vedic Maths Tricks: Calculate 10x Faster.” Vedantu, April 6, 2019. https://www.vedantu.com/blog/vedic-maths-tricks.

Videos

Bad Internet Math Propaganda

The beauty of numbers! If you multiply something by 1, it doesn’t change the value of the expression! This isn’t beauty – it’s chicanery! We look at this piece of bad internet math propaganda and discuss a cute trick to convert between miles and kilometers using the Fibonacci sequence.

@andrelifehack
Lifehack of multiplication😎#lifehack #lifehackvideo #multiplicationtricks
♬ Paris – Else

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