In 2009, when Usain Bolt smashed the world record in the 100 metres sprint in 9.58 seconds, the world wondered what was the fastest possible speed that anyone could run. Could anyone run 100 metres in 9.5 seconds? 9.0 seconds? 2.0 seconds?
Just as an athlete pushes their body to its limits, a competitive mental calculator pushes their brain’s processing power to its limit. But what is that limit? How fast can the brain do Mental Math?
Earlier this year, Marc Jornet Sanz set yet another massive new record in Multiplication – solving ten eight-digit multiplications of the form \(56025884 \times 83169091\) in 162.3 seconds! But could anyone reach 160 seconds? 120 seconds?
And last year, Yusnier Viera regained the world record for calendar date calculations by correctly calculating the weekday for 140 random dates in just 60 seconds. Could anyone reach 150 dates? 200 dates?
Measuring Processing Speed
The calculation speed of the human brain can be compared with the processor speed of a computer, measured in bits per second.
10 bits per second means essentially that the computer or person can make 10 basic binary decisions every second, for example toggling every 0 and 1 in a 10-digit binary number like 0111010011 to the other number:
01110 10011
10001 01100
Unlike a computer, the processing power of the human brain is very difficult to calculate. However, psychologists have done some clever experiments to determine how fast humans can think. Several decades ago, the Hick-Hyman Law was established:
The time taken to perform a single operation is proportional to the logarithm of the number of choices available.
Using this, Psychologists recently found an estimate of around 60 bits per second for the maximum processing speed of the human brain. Others have established that everyone’s brain has a similar processing speed, although those with a higher IQ are slightly faster (E. Roth (1964)).
The Fastest Mental Calculators
So how does this compare with the best mental calculation performances? Do the best calculation masters achieve such speeds when they dedicate their focus to Mental Math?
Let’s see the processing speed for three of the fastest calculation performance of all time. The full Mathematics for each is available by expanding the spoilers.
Adding a Single Number
Consider a calculation of the form \(ab + c\) (e.g. \(47 + 8\)), which occurs very frequently in Mental Math.
Expand to read mathematical derivation
This requires two steps:
recalling from memory the answer to the sum \(b + c\) (here: \(7 + 8 = 15\))
if the answer is 10 or higher, finding the next number after \(a\) (here: \(4 + 1 = 5\))
There are 55 different possibilities for sums \(b + c\), if e.g. \(4 + 5\) and \(5 + 4\) are considered the same. However we can always ignore “0” in an addition, so if we consider all additions with 0 as being the same, then we have only 46 different possibilities. The precise calculation of entropy is more complex as we need to work with probabilities, but as a compromise it will be convenient to use a value of 50 for the number of possibilities for the addition. Note that the same processing complexity exists whether the mental calculator uses soroban/abacus methods, or a direct numerical method.
Converting this into bits, this represents \(log_{2} 50 = 5.64\) bits
For the second part, There are ten numbers (0-9) that we might need to increment, and this is required about 50% of the time (this ignores a very small contribution for calculations like 9994 + 8, where multiple digits need to be incremented to get 10002). So the average number of bits required is about \(\frac{1}{2} log_{2} (10) = 1.66\) bits.
Adding a single digit to any number requires about 7.30 bits of processing.
Addition of Ten 10-digit Numbers
To calculate this, the mental calculator essentially needs to perform 100 single digit additions (this is true whether they perform the addition with single-digit columns or multiple-digit columns as the top competitors do), for a total of 730 bits of processing.
The world record is currently held by Jeonghee Lee – who can calculate one of these problems in just 10 seconds! This is 73 bits per second – slightly higher than the “60 bits per second” approximate estimated maximum for a typical human brain, as we would expect from the world champion.
Multiplication
When multiplying two 8-digit numbers, such as \(abcdefgh \times stuvwxyz\), using the standard cross-multiplication algorithm:
63 additions of the product (e.g. 24 from \(3 \times 8\)) to the running total for the current digit (e.g. \(24 + 191\))
Similar to the analysis for single-digit addition, there are about 50 possibilities for the single-digit multiplication, so each requires about 5.64 bits of processing.
The additions in the second step typically involves adding a small 2-digit number onto a larger number. This is two single-digit additions and approximately one increment. Adding these together, each 2-digit addition here requires about \(5.64 + 5.64 + 3.32 = 15.00\) bits of processing.
In total for one multiplication of 8-digit numbers, the amount of processing needed is \(64 \times 5.64 + 63 \times 15.00 = 1306\) bits of processing.
So how does Marc Jornet Sanz’s new record compare to this? His average time per question was 16.23 seconds, or 80 bits per second!
Calendar Dates
The standard algorithm for determining the day of the week for a given date is fairly complicated, but top-level mental calendars will typically learn hundreds of shortcuts to reduce the calculation to the following:
Expand to read mathematical derivation
Find the code \(D\) (from 0-6) for the date (e.g. 30th June => 3)
Find the code \(Y\) (from 0-6) for the year (e.g. xx19 => 4)
Find the code \(C\) (from 0-6) for the century (e.g. 20xx => 0)
Recall the remainder from memory when \(D + Y\) is divided by 7 (call this number \(R\))
Recall the remainder from memory when \(R + C\) is divided by 7 (this number represents the correct day of the week)
Performing a correction for leap years
As there are 366 possible values for \(D\), 100 for \(Y\), and 5 for \(C\) (in competitions the dates are between 1600 and 2099), the processing required for obtaining these codes is \(log_{2} 366 + log_{2} 100 + log_{2} 5 = 17.48\) bits. Experienced Mathematicians will see why all code-based algorithms will produce a similar result here, even if splitting the date into 2 or 4 parts.
As there are 15 different possible additions with numbers from 1-6 (noting that e.g. \(3 + 5 = 5 + 3\)), with a further possibility that one number is 0 an is ignored, the processing required for an addition like this is \(log_{2} 16 = 4.00\) bits (most people can work modulo 7 directly, so no need to explicitly calculate a remainder). Top competitors will typically require about 2 of these additions per question, depending on their personal database of codes.
For dates in January or February in a leap year, a further quick correction is needed, but this affects only 4% of dates so doesn’t greatly change the number of bits required.
Putting all of this together, the amount of processing required for a single calendar date is about \(17.48 + 2 \times 4.00\), so about 26 bits, after rounding up to account for the leap years.
The world record is held by Yusnier Yiera – with a huge 140 dates correct in one minute. This is \(140 \times 26 = 3640\) bits in 60 seconds, or 61 bits per second.
Conclusion
These 3 statistics give more evidence that the ballpark figure of 60 bits per second is an reasonable value for the maximum processing speed of the human brain. Although of course as the workings of the brain are complex and not as well-understood as a computer, all of these statistics are simply estimates. Certain tasks and certain individuals may permit slightly higher or lower thinking speeds.
3 thoughts on “Fastest Possible Processing Speed of the Human Brain”
Hello! It would be nice to have a discussion about square roots. This discipline has been always mysterious to me, in terms of the world record timings. I use
the duplex method (see https://forum.artofmemory.com/t/duplex-method-for-square-roots/32913 or from page 335 in The Mental Calculator’s Handbook), which is a small variation of the method explained in this blog (https://worldmentalcalculation.com/mental-square-roots-algorithm/); I use the double of the first digit solution as the divisor and not work with decimal numbers. My average time for each
six-digit radicand square root, with result of eight significant digits, is 20
seconds. I find astonishing the computation of such square root in 6 seconds, as
the current world record requires. Is this possible with the duplex method? Maybe an analysis of the number of bits per second could give an insight into the methods used by the world record holders.
I show my sloppy computations on the bits (I am not confident on these
computations, so maybe there are mistakes):
1) The duplex of a number is the largest cross when squaring such number by
cross multiplication. For example, the duplex of 5134 is 2x(5×4+1×3)=46, the
duplex of 513 is 2x(5×3+1^2)=32, etc. For an n-digit number, there are like n/2
single-digit multiplications and n/2 two-digit sums. Each single-digit multiplication
needs 5.64 bits of processing, and each two-digit sum needs 15.00 bits of
processing (see the mathematical derivation in the section “multiplication” above). Then n/2*(5.64+15.00)=10.32*n bits of processing. As the square root up to eight significant digits implies seven duplexes of one to seven digits, we have 10.32*(
1+2+3+4+5+6+7)=288.96 bits. If we take into account the double in the duplex
definition (double of a two or three-digit number), we have approximately 300 bits of processing for all duplexes.
2) On the other hand, the divisor is twice the first significant digit of the result. There are seven divisions in the root computation (usually a 2 or 3-digit number divided by a small 2-digit number). Suppose the division 157/18. If we know that the quotient is 8, we compute 18*8=144, which needs 15 bits (two
single-digit multiplications of 5.64 bits and increment of 332 bits); and then
157-144=13 for the remainder, which needs 15 bits in general (there are about 50 possibilities for single-digit subtractions, so addition and subtraction coincide in complexity). At each step, the subtraction of the duplex to the remainder*10 takes 15 bits. Then, we have 15+15+15=45 bits. In total, 7*45=315 bits for the whole root computation.
3) We add 300+315=615, which gives slightly more than 100 bits per second if
the square root is solved in 6 seconds. This is vastly superior compared to the
other disciplines. For instance, the world record for the multiplication of two
five-digit numbers, which requires 25*5.64+24*15.00=501 bits of processing, is
5.94 seconds on average (84 bits per second). Moreover, the computation of square
roots has a challenge compared to addition, multiplication or calendar dates: for each division, if the remainder minus the duplex is negative, the quotient has
to be decremented by one. Thus, there is a guess at each step that might need
readjustment. This slows down the computation severely, compared to the other disciplines. Therefore, the processing speed for solving a square root in 6 seconds is quite greater than 100 bits per seconds.
I read/heard from some competitors that some tables may be memorized. I
understand that it is possible to know the squares of numbers 0-1000 by heart, but
this might not be a great advantage in practice (because of the larger size of the numbers involved, memory demand, etc.).
A plausible option to solve square roots is the Newton-Raphson method. As it shows quadratic convergence rate, roughly speaking one could get twice of digits accuracy per iteration. So an initial guess that is correct to four significant digits may give the sought result of eight significant digits in only one iteration. This iteration requires the square of the four-digit initial guess, the
remainder and its division by twice the initial guess (the derivative). Too much to do in just six seconds.
What are your thoughts? 🙂 Do you have any idea on the optimal strategies for square roots?
Hi Daniel. I totally agree with all these rough estimates. Despite that the human brain is analogue and not digital; such kinds of brute quantifications happen very often in neural networks. I wish there was more research done on this, elsewhere in literature. Unlike the electric binary switches in CPU, we human competitors also require fast ‘eye-hand coordination’ in order to either type, utter or write the correct result. And we often calibrate intricate adjustments to make even more shortcuts. Example, in calendars, some competitors like me, only use just 1 modular addition instead of 2. This obviously cuts the total bits by 4, for each date, (22 instead of 26) but requires a more extensive memory load of 500 code values for each 1 of those standard competition dates (1600-2099). Also, I believe that the fastest mental calculation processing occurs in Flash Anzan (=’mental soroban’). Besides to what we have seen in competitions from Mrs. Lee (S. Korea) and others, there is also a Youtube video of a Japanese class adding 15 3-digits in 1.85 seconds. ( youtu.be/7ktpme4xcoQ ) And assuming you’re right with this estimate of 7.3 bits per single addition, that’s 3 x 15 x 7.3 = 328.5 bits per 1.85”, or nearly 178 bits per second. Finally, since besides calculation you are also interested in memory, it would be also interesting to estimate the brain speed of top memorizers, like those who memorize binary digits , decimals or a full deck of cards in less than 17”. I know they are not ‘calculating per se’ ; but their information processing speed is very impressive, nonetheless. Regards, Nodas
Nodas I believe you’re correct about flash anzan being the fastest of all the mental calculation disciplines – the actual processing speed is slightly less as they only needed to do 13 additions in the 1.85 seconds (the 1st was not an addition – merely reading the number, and the last can be done after the 1.85 seconds are over) and it may be possible to be lazy with the most significant digit as the typical answer will usually be around 8000 (not e.g. 2000 or 11,000), this speed is still well over 100 bits per second!
I do wonder what the difference is. Maybe it’s possible to improve our processing speed with intensive training on flash anzan, or maybe the “60 bits per second” approximate limit is greater when dealing with bigger data (such as 3- and 4-digit numbers) rather than single digits (as in calendar dates). It could also be that in e.g. 10×10 addition, the focus of eye needs to physically move across the screen, but in flash anzan it can be kept in one place on the screen.
For memory, it would be difficult to calculate correctly. 226 bits are required to describe the order of a pack of cards (log_2(52!)), and a natural description would require 52 * ceiling(log_2(52)) = 416 bits. Even in 17 seconds (Alex Mullen) this is not an impressive speed (less than 25 bits per second) but this ignores the fact that the memory athlete has to look around memory palaces and invent stories – so their actual processing speed is significantly higher!
Hello! It would be nice to have a discussion about square roots. This discipline has been always mysterious to me, in terms of the world record timings. I use
the duplex method (see
https://forum.artofmemory.com/t/duplex-method-for-square-roots/32913 or from page 335 in The Mental Calculator’s Handbook), which is a small variation of the method explained in this blog (https://worldmentalcalculation.com/mental-square-roots-algorithm/); I use the double of the first digit solution as the divisor and not work with decimal numbers. My average time for each
six-digit radicand square root, with result of eight significant digits, is 20
seconds. I find astonishing the computation of such square root in 6 seconds, as
the current world record requires. Is this possible with the duplex method? Maybe an analysis of the number of bits per second could give an insight into the methods used by the world record holders.
I show my sloppy computations on the bits (I am not confident on these
computations, so maybe there are mistakes):
1) The duplex of a number is the largest cross when squaring such number by
cross multiplication. For example, the duplex of 5134 is 2x(5×4+1×3)=46, the
duplex of 513 is 2x(5×3+1^2)=32, etc. For an n-digit number, there are like n/2
single-digit multiplications and n/2 two-digit sums. Each single-digit multiplication
needs 5.64 bits of processing, and each two-digit sum needs 15.00 bits of
processing (see the mathematical derivation in the section “multiplication” above). Then n/2*(5.64+15.00)=10.32*n bits of processing. As the square root up to eight significant digits implies seven duplexes of one to seven digits, we have 10.32*(
1+2+3+4+5+6+7)=288.96 bits. If we take into account the double in the duplex
definition (double of a two or three-digit number), we have approximately 300 bits of processing for all duplexes.
2) On the other hand, the divisor is twice the first significant digit of the result. There are seven divisions in the root computation (usually a 2 or 3-digit number divided by a small 2-digit number). Suppose the division 157/18. If we know that the quotient is 8, we compute 18*8=144, which needs 15 bits (two
single-digit multiplications of 5.64 bits and increment of 332 bits); and then
157-144=13 for the remainder, which needs 15 bits in general (there are about 50 possibilities for single-digit subtractions, so addition and subtraction coincide in complexity). At each step, the subtraction of the duplex to the remainder*10 takes 15 bits. Then, we have 15+15+15=45 bits. In total, 7*45=315 bits for the whole root computation.
3) We add 300+315=615, which gives slightly more than 100 bits per second if
the square root is solved in 6 seconds. This is vastly superior compared to the
other disciplines. For instance, the world record for the multiplication of two
five-digit numbers, which requires 25*5.64+24*15.00=501 bits of processing, is
5.94 seconds on average (84 bits per second). Moreover, the computation of square
roots has a challenge compared to addition, multiplication or calendar dates: for each division, if the remainder minus the duplex is negative, the quotient has
to be decremented by one. Thus, there is a guess at each step that might need
readjustment. This slows down the computation severely, compared to the other disciplines. Therefore, the processing speed for solving a square root in 6 seconds is quite greater than 100 bits per seconds.
I read/heard from some competitors that some tables may be memorized. I
understand that it is possible to know the squares of numbers 0-1000 by heart, but
this might not be a great advantage in practice (because of the larger size of the numbers involved, memory demand, etc.).
A plausible option to solve square roots is the Newton-Raphson method. As it shows quadratic convergence rate, roughly speaking one could get twice of digits accuracy per iteration. So an initial guess that is correct to four significant digits may give the sought result of eight significant digits in only one iteration. This iteration requires the square of the four-digit initial guess, the
remainder and its division by twice the initial guess (the derivative). Too much to do in just six seconds.
What are your thoughts? 🙂 Do you have any idea on the optimal strategies for square roots?
Hi Daniel. I totally agree with all these rough estimates. Despite that the human brain is analogue and not digital; such kinds of brute quantifications happen very often in neural networks. I wish there was more research done on this, elsewhere in literature. Unlike the electric binary switches in CPU, we human competitors also require fast ‘eye-hand coordination’ in order to either type, utter or write the correct result. And we often calibrate intricate adjustments to make even more shortcuts. Example, in calendars, some competitors like me, only use just 1 modular addition instead of 2. This obviously cuts the total bits by 4, for each date, (22 instead of 26) but requires a more extensive memory load of 500 code values for each 1 of those standard competition dates (1600-2099). Also, I believe that the fastest mental calculation processing occurs in Flash Anzan (=’mental soroban’). Besides to what we have seen in competitions from Mrs. Lee (S. Korea) and others, there is also a Youtube video of a Japanese class adding 15 3-digits in 1.85 seconds. ( youtu.be/7ktpme4xcoQ ) And assuming you’re right with this estimate of 7.3 bits per single addition, that’s 3 x 15 x 7.3 = 328.5 bits per 1.85”, or nearly 178 bits per second. Finally, since besides calculation you are also interested in memory, it would be also interesting to estimate the brain speed of top memorizers, like those who memorize binary digits , decimals or a full deck of cards in less than 17”. I know they are not ‘calculating per se’ ; but their information processing speed is very impressive, nonetheless. Regards, Nodas
Nodas I believe you’re correct about flash anzan being the fastest of all the mental calculation disciplines – the actual processing speed is slightly less as they only needed to do 13 additions in the 1.85 seconds (the 1st was not an addition – merely reading the number, and the last can be done after the 1.85 seconds are over) and it may be possible to be lazy with the most significant digit as the typical answer will usually be around 8000 (not e.g. 2000 or 11,000), this speed is still well over 100 bits per second!
I do wonder what the difference is. Maybe it’s possible to improve our processing speed with intensive training on flash anzan, or maybe the “60 bits per second” approximate limit is greater when dealing with bigger data (such as 3- and 4-digit numbers) rather than single digits (as in calendar dates). It could also be that in e.g. 10×10 addition, the focus of eye needs to physically move across the screen, but in flash anzan it can be kept in one place on the screen.
For memory, it would be difficult to calculate correctly. 226 bits are required to describe the order of a pack of cards (log_2(52!)), and a natural description would require 52 * ceiling(log_2(52)) = 416 bits. Even in 17 seconds (Alex Mullen) this is not an impressive speed (less than 25 bits per second) but this ignores the fact that the memory athlete has to look around memory palaces and invent stories – so their actual processing speed is significantly higher!