The Trachtenberg Method: Several Examples to Explore
Updated on January 31, 2024
7 min read

The Trachtenberg Method: Several Examples to Explore

There is an array of mind tricks to calculate and multiply digits. The speediest way of mental calculation is known as the Trachtenberg method. It is a complex set of formulas that comprises readily encoded actions, so once you’ve learned it, a range of maths computations will not be a grueling task at all.

On the contrary, you will be amazed at how promptly your brain will design output. One may surely wonder why you did not discover this tactic earlier. Being in a Nazi camp, a young man named Jakow was just trying to occupy his own mind. And this is how our scientific world has gained these unique and extraordinary mathematical algorithms that allow us to multiply digits impressively quickly and with no fuss.

Understanding Trachtenberg System and Why It Is Useful to Learn Speed Math Tricks

Let’s imagine that one’s brain has long been habituated to perform some type of brain exercises, like sudoku or chess, or that creating mind maps for remembering new things has become a snap. It is evident that with memory aids, it is possible to add mental power, keep some deterioration signs to a minimum, and redeem mind inactivity for its total recharging and renewal. Yes, it definitely takes time to become proficient in solving puzzles or making mental associations with complex concepts to memorize them more efficiently. However, having visible progress and receiving sufficient cognitive stimulation, your brain works in a relaxed mode, even performing previously tricky tasks. So if you feel that hitherto incorporated mind tactics have become easy for you, it means that it is the right time to seek something totally new.

Novelty is essential for our brains. New experiences of any sort are always accompanied by a boost of dopamine. Such a compound as dopamine is recognized by our brain as a reward, so we feel a sense of enhanced satisfaction. And as a result, our learning capacity becomes stronger too. Remember that everything is interconnected and works for one goal — to become more perceptive and savvier regarding our cognition. For example, exploring such psychological phenomena as the Mandela effect or a rare condition, such as a blind mind’s eye, also known as Aphantasia, when one lacks imagination and cannot visualize, can definitely shake up your worldview and perception. And you will awaken your imagination, grow intellectually, and broaden your vision. Such knowledge has never been a mind baggage or hindrance. On the contrary, it allows you to better understand the world around you. New knowledge encourages us to attain more, and it assists in bringing our cognitive systems out of a dormant state.

Learning the Trachtenberg method can be one of those new experiences vital for our brains. Unlocking how to perform multiplications through well-established operations can come in handy for those who strive to become sharp as a tack and astute, cultivate shrewdness, and generally speed up a range of cognitive functions. The benefit of the Trachtenberg method is that you don’t need to memorize tables or multiplication charts to perform multiplication. You just need to know how to pick the right rule for the action. Thus, the system offers an established set of operations when you multiply by 11. When the multiplier is 12, this set of actions will be completely different, and so on. The disadvantage is that there is no one golden rule for operation with any number; rather, the rule changes depending on the factor. However, templates within the Trachtenberg system are easy to decrypt and remember. This is how anyone can perform operations without the help of paper and pencil, just using the power of the mind and obviously memory mechanism. The method comprises additions as well as division.

Let’s check the example of several actions within the pattern of direct multiplication with 2 digits multipliers. The first rule to keep in mind is when doing direct expansion using 2 digits multipliers within the system, it is necessary to apply leading zeros. The rule for the leading zeros is to put as many zeros in front of the multiplicand as there are digits in the multiplier. Let’s take 3456 as an example. And now we need to multiply this number by 24 — it is two digits multiplier. What must be done here? You have to put two leading zeros in front of the multiplicand. If you want to have 3 digits multiplier, then three zeros must be written. And the same situation with 4 digits numbers.

Now, imagine we want to multiply 61 by 32. It is necessary to write down the next formula: 0061*32. The next step is to imagine we draw a red line connecting “1” from our multiplicand and “2” from our multiplier, creating the outer pair. And it means we will multiply  2*1. The inner pair we will create, with the help of an imaginative blue line and, is what we call an inner pair. In this case, “3” from our multiplier will be connected with no number in the multiplicand. It means we have nothing to do here. These lines will be used as we go through our operation of multiplying. The lines always go from the unit’s digit of the multiplier to the digit of the multiplicand that will actually be written within the pattern, and each line keeps moving across. Thus, the next outer pair will be 2*6, and the inner pair is 3*1.

Following the actions of the operation, there is a need to write down obtained digits in the correct positions to do the multiplication. A set of information through the direct multiplication calculation will be written, while some sets will be left in mind. The sequence of actions really matters to get the correct results. Creating outer and inner pairs, you can use your fingers and point to the number in each pair as you calculate to help keep track. Many easy-to-understand tutorials will guide people through each step. Just try it, and you will see that there are no complexities beyond, even if, at first sight, it seems quite complicated.

Methods to Learn the Trachtenberg System

As you can already understand, to learn different patterns of the system, you need concentration, a desire to learn a new skill, time, and a few quite useful tips on where to start exploring the Trachtenberg system and how to make progress:

One: know the important vocabulary within the system

Definitions that we already know are multiplier, multiplicand, as well as in some patterns; you will have to come up with outer and inner pairs. When it comes to special tactics in the Trachtenberg system for using smaller numbers between 2 and 12, it is also pivotal to understand what are the neighboring digits. Thus, in 7363, 3 doesn’t have a neighbor, while 6 has 3 as its neighbor.

Two: each task needs a rule

No matter if you have to multiply by 3, 5, or 9, you have to apply the proper algorithm of actions. There is no one-size-fits-all magic formula for every number. There is always something to discover when dealing with such speed calculation.

Three: pick a number and start small

You already roughly understand what direct multiplication looks like. Now, you have to pick any number, for example, 11. And carefully go through each step the established algorithm offers for this specific multiplier. Don’t beat your brain out to grasp all rules at once. If necessary, go through the same action several times to make it stick to your memory. So let’s multiply 3325*11. The universal rule for 11 states that we have to add each digit to its immediate right (neighbor). Remember that to the first digit 3 and the last digit 5 we add zero because these positions never have a neighbor. What we get there:







As a result, we get 3325*11=36575

First, you can write it down, and then you can do it mentally. A useful reminder: knowing about the specifics of the encoding mechanism and about what part of the brain controls memory, you will also understand that remembering algorithms of speed math requires rehearsal and constant training to register this data and transform it into long-term memories which you can recall whenever it will be needed.

Four: check out the counter-intuitive features of the system

As with any other system, there is a probability of anomalies that can be applied to certain operations. The task is not to miss them to avoid mistaken outcomes from mental maths.

Five: apply mnemonics to encode more rules

Quick multiplication with the help of the direct method, well-designed formulas for each smaller multiplier, e.g., 2-12, two-finger math — there are so much exciting categories within one system. How to memorize them? Train your mind-mapping skills to place formulas on the imaginative route and recreate them for a vast range of operations. Don’t be afraid to embrace this challenge.


How fast is the Trachtenberg method?

As soon as you unlock the entire system, multiplication actions will be quite fast. In addition, the experience of learning the available formulas is fun and valuable for overall cognitive power.

Is the Trachtenberg system good?

Apart from acquiring basic maths skills, the journey of exploring the system will definitely be fascinating and not boring. Moreover, it yields some hidden benefits, as performing such calculations supports the growth of new brain connections that will be stronger than ever.

What is the best way to do mental math?

Such a system as the Trachtenberg method is a powerful way to calculate speedily, as well as to keep your memory and mind sharp!

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