In graduate school back in the 1980s, our class was dumbfounded by a blind classmate who nonchalantly performed an astounding act of mathematical gymnastics.
Since Martin gave both the matrix and its inverse, my guess is that, rather than calculating the inverse from scratch, Emilio simply multiplied the two matrices together to see if it produced the identity. Thus 1x(-24) + 2x20 +3x(-5) = 1 as it should, but 0x18 + 1x15 + 4x4 is not 1. But replace the 15 by -15 and it is 1. Still very impressive, as he has to do this for all 9 entries, and to notice that a simple change of sign will do it.
Joel Kupperman was seven years old and on Quiz Kids radio show, and could multiply and number by 99 in his head. When asked how he did it, he said "it's a secret twick" (the lad became semi-famous in part for his lisp). I believe it was multiplying by 100 then subtracting the original number.
(Which I had to get a calculator out to confirm. Not on the level of the matrices question, but it shows that some people just cut the Gordian knot by mentally processing something in a completely different manner.)
In the 1700s, in elementary school, Carl Friedrich Gauss was asked to calculate the sum of integers from 1 to 100 and quickly said 5,050. The teacher was stunned. What he had done was 1+100=101, 2+99=101, 3+98=101, etc. He realized that there were 50 such pairs, so all he had to do was calculate 50x101.
Lots of material to work with. I have another piece in the queue about an astounding singer. Wonderful just to hear, but then the backstory ... ... ... !
For those interested in these fascinating matters of mind that Grayboyes so interestingly details, I highly recommend "The Matter With Things," by Iain McGilchrist. It's only a bit over 1,500 pages, so not for the faint of mind.
One possibility is that Emilio didn't calculate the entire matrix in his head, but "spot-checked" it to see if basic things worked out like "is this the correct sign?" which is a simpler mathematical operation. When he saw a value where the sign was flipped, he could calculate that value quickly enough.
We have a spectrum person in the family. Her first unusual feat was holding the second piece in from the right corner of a 1000 piece puzzle till the end. How did she spot that piece of puzzle on puzzles she worked on. She scored a 5 on AP statistics when she was 14. It’s as if when something is exciting or interesting to her, the sky’s the limit. It’s like Temple Grandin said or implied. There’s extra circuitry that can produce usually outcomes.
Sounds fascinating! Could you explain what you mean by, "holding the second piece in from the right corner of a 1000 piece puzzle till the end"? I think I understand what she did, but I'm not certain.
I’m incorrect on the size of the puzzle. It was when she was 4 ish and the puzzles were more like 100 pieces. She would hold the piece in her hand and place it last. Just an odd thing to see her spot the piece and then hold it till the end.
Now my head hurts. I do regret that I was never exposed to calculus in school. I did have a very rushed semester of physics which I can't say enhanced my knowledge or intellectual capabilities. But calculus, no. I do cherish having had a pre-dumbed down education, so that I can rightfully continue to call myself Math In Head Girl ... but not at this level.
Welllll ... none of us could have done this either. Or even begun to grasp how he did it. I should have added to the story that he wasn't some robot-math-oddball. He was just a charming, easy-going, likeable guy who had obviously compensated magnificently for his lack of sight.
Interesting story, although..I dunno, I can do the inverse of a 2x2 in my head pretty easily, and checking that one 2x2 is the inverse of another is even easier.
I'm vaguely reminded of a story I heard about Fermi once, someone I surmise to be one of the dozen or so smartest people ever to have lived. Supposedly Fermi rarely read the literature, preferring to work stuff out for himself, and supposedly this led before he was sufficiently famous to visitors having strange experiences. They would go in to talk to Fermi, and start jabbering about this and that current topic, and find that Fermi seemed strangely ignorant. He didn't understand all kinds of basic aspects of the topic, and at first the visitor would think huh this guy is slow -- or if they knew about him, the Great Fermi is not all he's cracked up to be, must coast on his lab chops -- Fermi was first known as a skilled and intuitive experimentalist[1] before he became famous as a theorist[2].
But then...as the conversation progressed, Fermi would start accelerating, as he worked out on the fly, in his head, all the basic aspects that other people would know from reading the literature, until after a while Fermi would seem no longer dull, and all caught up.
I have imagined that often, then, Fermi would continue accelerating, so that he would soon work out the state of the art and then surge ahead, posing questions and discussing insights that would leave his visitor (if he was a mere mortal) baffled and feeling in his turn stupid. So that, overall, talking to Fermi would consist of Phase A, where Fermi appeared dumb, then a brief glorious Phase B, where Fermi and you were conversing on the same level, and then a humiliating Phase C, where you were struggling to keep up and wondering if you were boring the great man.
-------------------
[1] Although his most famous miss was failing to look for much lighter fragments when he studied the response of uranium atoms when bombarded with neutrons. At the time, everyone assumed any fragments would be lighter than U by only 4-12 or so, consistent with alpha decay. Nobody thought to look for something only half the mass of U. Fermi was the first to technically observe nuclear fission, but he didn't recognize it. He was in good company, though, the great Marie Curie also technically saw it, but didn't recognize it.
[2] Supposedly one of his most consequential theoretical insights happened shortly after fission was discovered, on the eve of war. Everyone understood the military implications immediately, and theorists around the world whipped out their envelopes and scribbled down an estimate of the critical mass. Apparently Heisenberg in Germany and Fermi (then in the United States) both made the same initial and incorrect estimate, which showed a critical mass of hundreds of kg, completely impractical. But Fermi redid his calculation, for whatever strange reason, and came up with the correct order of magnitude of a few kg. It's likely the difference was a significant reason why the US pursued the atomic bomb with strenuous effor, but the German atomic bomb effort was anemic and ultimately fruitless. After the war I believe Heisenberg asserted that he did not pursue the goal with energy because he did not want Hitler to have the weapon, but the Farm Hall transcriptions reveal that in fact he was surprised at the success of the American effort -- he had not suspected the critical mass was as small as it turned out to be -- and and it may be he favored the other explanation from wounded pride.
I used to tell my students that sometimes enormous world events can turn on the ability to do reasonably accurate back of the envelope estimates. In this case, had Heisenberg done his estimate correctly, world history might've been very different indeed. (Although it would also have required the early discovery of Pu, since wartime Germany utterly lacked the industrial capacity to enrich U sufficiently. But Germany had excellent radiochemists, so that isn't so much of a stretch.)
And this harks back to my Enrico Fermi story above—his insistence that young physicists have the capacity to do back-of-the-envelope statistics, plus the knowledge that they OUGHT to use such calculations.
Having enjoyed your incredibly erudite comments for a while, I have no doubt that you can invert matrices in your head, Carl. Some people can also understand Nadia’s gymnastics.
Great Fermi stories. At least once, he threatened to veto a doctoral student’s dissertation defense because the student didn’t know how to answer the question, “How many taxicabs are there in Chicago?” He thought any mathematically inclined student should be able to come up with a reasonably good guess. The student didn’t understand that Fermi didn’t want the right answer—just a good, well-reasoned guess.
I can believe the Heisenberg story. He’s quite the morally ambiguous character. Perhaps you know the story of Moe Berg https://www.smithsonianmag.com/history/the-baseball-player-turned-spy-who-went-undercover-to-assassinate-the-nazis-top-nuclear-scientist-180982813/). “The traditional narrative, as outlined in Robert Rodat’s 2018 biopic, The Catcher Was a Spy, holds that Berg listened to Heisenberg’s talk about a totally unrelated topic in physics, gleaned that the scientist was either anti-Nazi or unbelievably behind in the race to harness nuclear energy, and felt it was unnecessary to execute him. As Rodat told the New York Times in 2018, Berg “sensed when a runner was going to steal, and even though Heisenberg was trying to hide it, Berg knew he was despondent because Germany didn’t have the bomb and was going to lose the war.”
I did not know the story of Berg! And this is of interest to me, since (as a child of the Cold War, and trained in physics) I have a morbid side-interest in the history of nuclear weapons. Thanks very much for this pointer. If you know of a good book on the subject, I would be quite interested.
Yes, I've tended to generally agree with Fermi on this. Usually it's a major aspect of my participation in oral exams: yes, I know you know your research well, almost certainly (if you are any good) better than the committee. What I want to know is: can you think on your feet, estimate, rule stuff broadly in or out, reason without access to the books, literature, conventional wisdom? This has so much (I think) to do with whether you can be a genuine contributor to science. Having been on the other side of that (my committee apparently felt the same way, so karma has its way) I know it can be excruciating, but it's necessary, I think. And those who come through it and learn to hone the skill are more confident and productive ever afterward.
I wish we taught it more in high school and college to everyone! We live in an era of unprecedented access to information, the Conventional Wisdom, to the thoughts of others, often others far more experienced and knowledgeable than ourselves. But I feel like an unfortunate concomitant of that is a slow atrophy of the skills of thinking for yourself, and critical evaluation of what others say.
The "Wikipedia Generation," as I call them when feeling unusually liverish, seems to believe the search for truth can be reduced to the search for the authoritative citation. This is deeply unfortunate. The highly educated and informed are hardly immune to error, intellectual blindness, canalization, fossilization of thought. A healthy intellectual culture *must* prize every last individual's ability to reason for himself, to estimate, to guage the plausibility of arguments via his own thought and experience, and to encourage the tendency to accept uncritically almost no opinion, however accredited and accomplished or venerated the opinionator. (You can tell I worship at the temple of Empiricism ha ha.)
There are those who suggest as compensation a general increases in distrust of the expert opinion, but that's not what I mean at all. I mean we should try much harder to teach our young people the reasoning skills they need to be skilled skeptics, people who can pull apart chains of reasoning, detect assumptions both explicity and implicit, recognize fallacies (formal and informal), and yes make Fermi estimates of what they don't know directly.
I remember reading on Quora once that a person the author knew could invert 3x3 matrices in his head. I decided to give it a try. For the first few, I made mistakes. But after a few tries, in about 60 seconds per matrix, I could invert a 3x3 matrix only in my head. What Emilio did in this story is significantly more impressive. I was able to see the matrix I was inverting, and I was able to calculate each element in the inverted matrix and save it. I couldn’t imagine having to keep track of that many numbers in short term memory.
A friend of mine told a math professor that his equation was wrong. Apparently the full paper was to be published the next year in a prestigious journal. He was so proud of the work that he was giving the class a sneak preview.
He was about to demolish her and walked over to his class notes quite ostentatiously. His glasses were strategically dipped down his nose — never a good sign.
Though everyone expected him to start in on her, he sheepishly returned to the chalk board and mumbled “Sorry, typographical error.”
Most people were surprised by this incident. But I knew my friend. She could spot things were wrong even though she had no knowledge of the math material. It was a gift that simply couldn’t be explained.
Since Martin gave both the matrix and its inverse, my guess is that, rather than calculating the inverse from scratch, Emilio simply multiplied the two matrices together to see if it produced the identity. Thus 1x(-24) + 2x20 +3x(-5) = 1 as it should, but 0x18 + 1x15 + 4x4 is not 1. But replace the 15 by -15 and it is 1. Still very impressive, as he has to do this for all 9 entries, and to notice that a simple change of sign will do it.
As I told another commenter: "Any explanation offered is possible. But no explanation offered is comprehensible." :)
Joel Kupperman was seven years old and on Quiz Kids radio show, and could multiply and number by 99 in his head. When asked how he did it, he said "it's a secret twick" (the lad became semi-famous in part for his lisp). I believe it was multiplying by 100 then subtracting the original number.
(Which I had to get a calculator out to confirm. Not on the level of the matrices question, but it shows that some people just cut the Gordian knot by mentally processing something in a completely different manner.)
In the 1700s, in elementary school, Carl Friedrich Gauss was asked to calculate the sum of integers from 1 to 100 and quickly said 5,050. The teacher was stunned. What he had done was 1+100=101, 2+99=101, 3+98=101, etc. He realized that there were 50 such pairs, so all he had to do was calculate 50x101.
Wow! Great story.
I love your enthusiasm and celebration of our humanity - thank you!❤️🥳✨
Lots of material to work with. I have another piece in the queue about an astounding singer. Wonderful just to hear, but then the backstory ... ... ... !
I am fascinated by the layers of human that unlock to liberate a pure voice - can’t wait to read!!
Very soon!
For those interested in these fascinating matters of mind that Grayboyes so interestingly details, I highly recommend "The Matter With Things," by Iain McGilchrist. It's only a bit over 1,500 pages, so not for the faint of mind.
I won't be tackling 1,500 pages of ANYTHING! But I might scan it.
thank you for this article. it so reminds of me of what the psalmist says in Psalm 139, '.. "for I am fearfully and wonderfully made"
Indeed!
One possibility is that Emilio didn't calculate the entire matrix in his head, but "spot-checked" it to see if basic things worked out like "is this the correct sign?" which is a simpler mathematical operation. When he saw a value where the sign was flipped, he could calculate that value quickly enough.
Any explanation offered is possible. But no explanation offered is comprehensible. :)
We have a spectrum person in the family. Her first unusual feat was holding the second piece in from the right corner of a 1000 piece puzzle till the end. How did she spot that piece of puzzle on puzzles she worked on. She scored a 5 on AP statistics when she was 14. It’s as if when something is exciting or interesting to her, the sky’s the limit. It’s like Temple Grandin said or implied. There’s extra circuitry that can produce usually outcomes.
Sounds fascinating! Could you explain what you mean by, "holding the second piece in from the right corner of a 1000 piece puzzle till the end"? I think I understand what she did, but I'm not certain.
I’m incorrect on the size of the puzzle. It was when she was 4 ish and the puzzles were more like 100 pieces. She would hold the piece in her hand and place it last. Just an odd thing to see her spot the piece and then hold it till the end.
Cool story!
It IS magical when we see something so wonderful we (understand that we) understand less than before. Thank you for this essay.
And thank you!
As one who used to be able invert matrices, your story i stupefying. Fascinating, as always, Robert.
Isn’t it? Thanks.
Now my head hurts. I do regret that I was never exposed to calculus in school. I did have a very rushed semester of physics which I can't say enhanced my knowledge or intellectual capabilities. But calculus, no. I do cherish having had a pre-dumbed down education, so that I can rightfully continue to call myself Math In Head Girl ... but not at this level.
Welllll ... none of us could have done this either. Or even begun to grasp how he did it. I should have added to the story that he wasn't some robot-math-oddball. He was just a charming, easy-going, likeable guy who had obviously compensated magnificently for his lack of sight.
Interesting story, although..I dunno, I can do the inverse of a 2x2 in my head pretty easily, and checking that one 2x2 is the inverse of another is even easier.
I'm vaguely reminded of a story I heard about Fermi once, someone I surmise to be one of the dozen or so smartest people ever to have lived. Supposedly Fermi rarely read the literature, preferring to work stuff out for himself, and supposedly this led before he was sufficiently famous to visitors having strange experiences. They would go in to talk to Fermi, and start jabbering about this and that current topic, and find that Fermi seemed strangely ignorant. He didn't understand all kinds of basic aspects of the topic, and at first the visitor would think huh this guy is slow -- or if they knew about him, the Great Fermi is not all he's cracked up to be, must coast on his lab chops -- Fermi was first known as a skilled and intuitive experimentalist[1] before he became famous as a theorist[2].
But then...as the conversation progressed, Fermi would start accelerating, as he worked out on the fly, in his head, all the basic aspects that other people would know from reading the literature, until after a while Fermi would seem no longer dull, and all caught up.
I have imagined that often, then, Fermi would continue accelerating, so that he would soon work out the state of the art and then surge ahead, posing questions and discussing insights that would leave his visitor (if he was a mere mortal) baffled and feeling in his turn stupid. So that, overall, talking to Fermi would consist of Phase A, where Fermi appeared dumb, then a brief glorious Phase B, where Fermi and you were conversing on the same level, and then a humiliating Phase C, where you were struggling to keep up and wondering if you were boring the great man.
-------------------
[1] Although his most famous miss was failing to look for much lighter fragments when he studied the response of uranium atoms when bombarded with neutrons. At the time, everyone assumed any fragments would be lighter than U by only 4-12 or so, consistent with alpha decay. Nobody thought to look for something only half the mass of U. Fermi was the first to technically observe nuclear fission, but he didn't recognize it. He was in good company, though, the great Marie Curie also technically saw it, but didn't recognize it.
[2] Supposedly one of his most consequential theoretical insights happened shortly after fission was discovered, on the eve of war. Everyone understood the military implications immediately, and theorists around the world whipped out their envelopes and scribbled down an estimate of the critical mass. Apparently Heisenberg in Germany and Fermi (then in the United States) both made the same initial and incorrect estimate, which showed a critical mass of hundreds of kg, completely impractical. But Fermi redid his calculation, for whatever strange reason, and came up with the correct order of magnitude of a few kg. It's likely the difference was a significant reason why the US pursued the atomic bomb with strenuous effor, but the German atomic bomb effort was anemic and ultimately fruitless. After the war I believe Heisenberg asserted that he did not pursue the goal with energy because he did not want Hitler to have the weapon, but the Farm Hall transcriptions reveal that in fact he was surprised at the success of the American effort -- he had not suspected the critical mass was as small as it turned out to be -- and and it may be he favored the other explanation from wounded pride.
I used to tell my students that sometimes enormous world events can turn on the ability to do reasonably accurate back of the envelope estimates. In this case, had Heisenberg done his estimate correctly, world history might've been very different indeed. (Although it would also have required the early discovery of Pu, since wartime Germany utterly lacked the industrial capacity to enrich U sufficiently. But Germany had excellent radiochemists, so that isn't so much of a stretch.)
And this harks back to my Enrico Fermi story above—his insistence that young physicists have the capacity to do back-of-the-envelope statistics, plus the knowledge that they OUGHT to use such calculations.
Having enjoyed your incredibly erudite comments for a while, I have no doubt that you can invert matrices in your head, Carl. Some people can also understand Nadia’s gymnastics.
Great Fermi stories. At least once, he threatened to veto a doctoral student’s dissertation defense because the student didn’t know how to answer the question, “How many taxicabs are there in Chicago?” He thought any mathematically inclined student should be able to come up with a reasonably good guess. The student didn’t understand that Fermi didn’t want the right answer—just a good, well-reasoned guess.
I can believe the Heisenberg story. He’s quite the morally ambiguous character. Perhaps you know the story of Moe Berg https://www.smithsonianmag.com/history/the-baseball-player-turned-spy-who-went-undercover-to-assassinate-the-nazis-top-nuclear-scientist-180982813/). “The traditional narrative, as outlined in Robert Rodat’s 2018 biopic, The Catcher Was a Spy, holds that Berg listened to Heisenberg’s talk about a totally unrelated topic in physics, gleaned that the scientist was either anti-Nazi or unbelievably behind in the race to harness nuclear energy, and felt it was unnecessary to execute him. As Rodat told the New York Times in 2018, Berg “sensed when a runner was going to steal, and even though Heisenberg was trying to hide it, Berg knew he was despondent because Germany didn’t have the bomb and was going to lose the war.”
I did not know the story of Berg! And this is of interest to me, since (as a child of the Cold War, and trained in physics) I have a morbid side-interest in the history of nuclear weapons. Thanks very much for this pointer. If you know of a good book on the subject, I would be quite interested.
Yes, I've tended to generally agree with Fermi on this. Usually it's a major aspect of my participation in oral exams: yes, I know you know your research well, almost certainly (if you are any good) better than the committee. What I want to know is: can you think on your feet, estimate, rule stuff broadly in or out, reason without access to the books, literature, conventional wisdom? This has so much (I think) to do with whether you can be a genuine contributor to science. Having been on the other side of that (my committee apparently felt the same way, so karma has its way) I know it can be excruciating, but it's necessary, I think. And those who come through it and learn to hone the skill are more confident and productive ever afterward.
I wish we taught it more in high school and college to everyone! We live in an era of unprecedented access to information, the Conventional Wisdom, to the thoughts of others, often others far more experienced and knowledgeable than ourselves. But I feel like an unfortunate concomitant of that is a slow atrophy of the skills of thinking for yourself, and critical evaluation of what others say.
The "Wikipedia Generation," as I call them when feeling unusually liverish, seems to believe the search for truth can be reduced to the search for the authoritative citation. This is deeply unfortunate. The highly educated and informed are hardly immune to error, intellectual blindness, canalization, fossilization of thought. A healthy intellectual culture *must* prize every last individual's ability to reason for himself, to estimate, to guage the plausibility of arguments via his own thought and experience, and to encourage the tendency to accept uncritically almost no opinion, however accredited and accomplished or venerated the opinionator. (You can tell I worship at the temple of Empiricism ha ha.)
There are those who suggest as compensation a general increases in distrust of the expert opinion, but that's not what I mean at all. I mean we should try much harder to teach our young people the reasoning skills they need to be skilled skeptics, people who can pull apart chains of reasoning, detect assumptions both explicity and implicit, recognize fallacies (formal and informal), and yes make Fermi estimates of what they don't know directly.
I'm sure Emilio was better acquainted with his wife's beauty than anyone who could only see her.
And, as we all knew, that was the real meaning of his friend’s joke.
I remember reading on Quora once that a person the author knew could invert 3x3 matrices in his head. I decided to give it a try. For the first few, I made mistakes. But after a few tries, in about 60 seconds per matrix, I could invert a 3x3 matrix only in my head. What Emilio did in this story is significantly more impressive. I was able to see the matrix I was inverting, and I was able to calculate each element in the inverted matrix and save it. I couldn’t imagine having to keep track of that many numbers in short term memory.
I’m impressed by both of you.
I have difficulty remembering the two-factor authentication code when it’s more than six digits long. :)
A friend of mine told a math professor that his equation was wrong. Apparently the full paper was to be published the next year in a prestigious journal. He was so proud of the work that he was giving the class a sneak preview.
He was about to demolish her and walked over to his class notes quite ostentatiously. His glasses were strategically dipped down his nose — never a good sign.
Though everyone expected him to start in on her, he sheepishly returned to the chalk board and mumbled “Sorry, typographical error.”
Most people were surprised by this incident. But I knew my friend. She could spot things were wrong even though she had no knowledge of the math material. It was a gift that simply couldn’t be explained.