1) I already had this affinity for numbers (at an early age) and this certain talent, which one can mention, but which also seems to be present in this way (by quotation or in the text). I was often given arithmetic problems that were absolutely not age-appropriate (like 7 times 17, at pre-school age), but which I was still able to solve in principle (even if, as described, I made a mistake). There is a lot of “evidence” for this talent. For example, shortly after my first day at school, we had a competition in “arithmetic” (that was the name of the school subject), which I won hands down (I was the first to finish ALL the tasks, but the tasks never had the level of 7 times 17). It seemed so easy to me and I couldn’t understand the congratulations or admiration of my classmates, because it was really so easy…!
But this was not the story at all, it was just part of the back story. You grow up with it, you are asked, you get your image, however, you also cultivate it, almost inevitably. But at some point in my life I had this encounter: there was a chess player at my level who became more and more of a good friend. You immediately noticed that he was special, but we also liked each other from the first meeting. At some point you realise why this might be the case?! Exactly: one has similar dispositions – and senses this without immediately figuring out what it would be. At some point I discovered that this man excelled in mental arithmetic. One came to this question, to that problem, one calculated quickly, he was faster, I was faster, not significant. Only once I asked him, when for a curious reason he gave me a random, high number, whether he could not please break it down into its prime factors? And promptly it happened: he calculated the prime factors in no time at all, audibly to me, and told me them one after the other. This was so incredibly powerful that I came home later and, instead of sleeping at night, at least tried to reconstruct this calculation. And I succeeded! I was extremely proud of it. Only: it took me a whole hour, he managed it in one or two minutes.
After that, of course, he was my “idol”. We talked about the different forms of this talent, because he also recognised mine, of course. He said that he used to square four-digit numbers in his head. The bet was: you have to be through in less than two minutes – and it has to be right, of course. Now I started to practise this too. And indeed I managed to solve these tasks within two minutes (one practice method was to do it on the motorway, for example, at any number plate; I calculated the number, had a pen and paper in my pocket, wrote down the result, with hands on the wheel, but just in calmer seconds, and checked the result at the next stop; it could also be more than one task by then). Only we then competed against each other ONCE. And he finished in barely a minute, whereas I needed the full two. Say: there are these limits everywhere. It didn’t depress me one bit, rather thrilled me that there was demonstrably someone better than me at that and I was proud to know that man and could tell that story anytime and happily. Do you understand roughly? I think so, the only question would be how to incorporate that into the text?
2) Now that I have told this story, I would have to add the next one right away, which would have almost the same character, but we would thus already come to five in total.
So I had another acquaintance (by the way, I had a completely surprising dream about him last night). Unfortunately, we don’t have any contact any more, because he was my reference person regarding Subbuteo and he continues not only to play this game but also to really “promote” it, which we did together at that time, but he was perhaps disappointed that I didn’t continue to play the game (we played many tournaments together and thought and exchanged a lot about improving the rules; the world didn’t quite understand, but ours were better, just not to be enforced; similar to my mission in “real football”). In my case, however, it was both family reasons and my back problems, which were quite frequent when playing at the plate.
Anyway, it was clear that this boy also had (at least) a special talent. We were pretty much in agreement and pretty much best friends. At some point we did an intelligence test together, online, on the computer. He basically just used his intelligence to be supremely funny. He just couldn’t take it all that seriously, didn’t want to mess with anyone or argue with anyone, so he became funny instead, in which his talent became obvious – but perhaps even went unnoticed by others in the laughter, and above all it never came across as arrogant. He understood (better), the joke made it clear – no one could be angry with him.
In this intelligence test, his attitude was similar: it’s funny, but not important. One could, much more than that, but what for? Laughter is healthy.
But suddenly there was this one task. I calculated for a moment, thought about it, it was already difficult, but I thought I had it. My solution was wrong, of course, as I immediately noticed from the way he calculated. Now I didn’t correct my thinking, but listened to him, fascinated. At last he was a bit challenged. At last a problem of this kind appealed to him. Finally one could witness what he was REALLY capable of. And he couldn’t or didn’t want to hide it. Now the maths was done, here was a serious problem worth doing. So I listened and waited. At some point he said: “Wait a minute, I’ll be right back…” and then he spat out the – naturally correct – solution. The same tenor: there are limits, I’m excited about it, I’m proud to have this boy in my circle of acquaintances.
3) A third story, of which I would now have six, I’ll try to make a little shorter: we once did an intelligence test against Garry Kasparov, a number of Berlin’s top chess players. There was an article about it in Der Spiegel. Garry won, of course. It was an intelligence-chess endurance test. So chess in between, then intelligence again, chess again, for one day. But at some point in the test, there were practically pure arithmetic problems (a little bit in words, perhaps, but still “simple” for me). We were given TWO minutes for these tasks. I finished AFTER ONE minute. I turned the page – nothing, no more tasks. I looked around – everyone was calculating feverishly. I had nothing to do. Testing was ridiculous. The inventors of the test had thought: no one can do ALL of them anyway, let’s see who gets how far and who gets how much right. For me, however, there should have been twice as many tasks to discover the special talent.
By the way, this also happened once before in primary school. There were teacher trainees in our class and they gave us each a task sheet to solve. I was finished after a few minutes. As I sat idly in my seat, one of the students came up to me and asked me if I was getting nowhere and if he should help me? I showed him my paper and said, “No, I’m done.”
By the way, every time I did a maths test, I handed it in after half the time at the latest.
So in that sense: I could do something, but there were also limits that you find in completely different places.
4) I solved the Rubik’s Cube, just for myself, at the end of the 70s. I got it, I screwed around with it for a day, I discovered more and more, I put it together for the first time on the evening of the first day, but I couldn’t repeat this, but the next morning, when I picked it up again, I had the final solution, which I could repeat at any time. I don’t know how many people in the world have actually solved it this way, but as an example, my uncle was an absolute puzzle freak (all kinds of puzzles I mean), but he visited me even before I solved it myself with an article from Der Spiegel in which a solution was printed (with pictures and turning instructions). I couldn’t believe that he only wanted to understand it and not solve it himself (an example of what he asked you to do: when I had finished a puzzle that wasn’t even that easy for my age, he asked me to solve it BACKwards; that is, without any clue as to the colours or the overall picture; I did it; he always made the highest demands on himself, too, with the highest degree of difficulty – and was practically always successful). I didn’t look at the article at all, nor did I want to be changed by him, but set myself to work on it, with said result.
Later, when other types of freaks wanted to show me what faster solutions were available, I wasn’t interested at all. I had my solution – and it was good and always worked, reliably, with a certain number of turns (you could say: maximum number, because faster was of course always possible, depending on the current constellation). However, I did it every morning on the way to school, on the bike. It was in my pocket, I could take it out of my pocket if the traffic situation allowed it and quickly do a few necessary turns, because I didn’t have to look at the cube (only once briefly in between). The goal was to have it ready by the time I got to school – and I did it (here it was a certain disadvantage that the way to school was quite short).
5) Curiously, I was able to speak backwards even before school. The reason why I know this so well is this: I used to sing “ächte, ächte tchalegsua” (however this should be written) at a child’s birthday party. My parents asked me what I was warbling about? I said truthfully that it was “Ätsche, Ätsche ausgelacht”, spoken backwards. But I didn’t know the letters yet.
So everyone recognised that I always “knew” that for all the words and immediately, but you couldn’t check it yourself so quickly. No matter how long or complicated the words were: I answered spontaneously, even with sentences (which, however, I always turned around word by word; that is, I didn’t recite the sentence from the back). To get “proof” that I always did it right, a neighbour who was also a hobbyist came. He brought his tape recorder. Words that were intentionally difficult were recited to me in rapid succession. The tape recorded. Afterwards he put the tape in upside down and we could check all the words. Although the intonation was wrong and the words “echoed” but did not sound out, I had not made a single mistake. The only thing I remember is this one word that was put in front of me with the intention of “fooling” the child of five or six (which of course was not really the intention). It was the street at the next crossroads. The Grunewaldstraße. I said this, without the slightest delay (I just didn’t have to think, I could do it just like that), all right.
I gave up this game for various reasons – and in fact later completely forgot it. Once a woman appeared on television who sang an entire song backwards. This was recorded exactly as I had done it for the test. A tape recorder, she sang everything (which sounded totally off-key and stupid), the tape was played from behind – and everything fitted perfectly. Of course I said – and everyone agreed – that in principle it would be no problem to rehearse something like that and that anyone could do it under the circumstances, but it still frustrated me that someone with a much higher requirement was allowed to perform on television.
Another reason was that I gradually realised that, for example, you didn’t have to or weren’t allowed to pronounce a “Z” (also and especially in the word) like “Z”, but rather like “ts”, i.e., backwards, as “st”. There were a few more examples of this (of course, I always pronounced “sch” as “sch” and not as one would have had to read it, since, as in the example of “tchalegsua”, I couldn’t even write it yet). So I always wavered about how to do it “correctly” (especially since the sentences meant that you had to start the sentence with the last word anyway, which I didn’t do). And finally gave it up altogether. There was no “right”. No one ever needed to ask me anymore.
6) Since you alluded to the scientific recognition that I did not receive: at the beginning of the millennium, I once sent a formula to the university in Leipzig, to a certain Mr. Poppe (on his way to a doctorate). That was the place to which one should send one’s discoveries, I was told at some point (I simply didn’t bother myself; also a little out of frustration or the conviction that no one would recognise it anyway, no matter how good). I even spoke to Mr Poppe personally (a few times) afterwards. Apparently he didn’t find it all that uninteresting?! However, I remember him saying early on: “Mr. Paulsen, Mr. Paulsen, listen, there IS nothing new. It’s all been there, you haven’t made a new discovery.” Without him really knowing it yet. Now I had sent him a quite simple formula, actually for four reasons: a) I had discovered it myself, b) it was inconceivable that no one should have come up with something like this before, if you like, c) please someone look up for me who it was and when, d) would then be: may the mathematician who is looking at my discovery please be able to judge on the basis of this that I might have others besides this formula and that I work scientifically and can think?
So Mr. Poppe researched for a few days. He found what he was looking for “of course”. The good Mr Zermelo had already discovered and published this in 1929. I would still have said that Mr. Zermelo was missing a practical application, which I not only found but basically established and made demonstrably usable, but here you go: he was right so far.
Now I assured him that I had only shown him this for the aforementioned reasons and that it was not the formula I actually wanted to show him. Accordingly, I left him with another, much deeper and better formula, with considerable practical use and which represented a serious advance for mathematics (and especially statistics or probability theory, which, however, in my view, determines our lives, insofar as it is almost a philosophical answer to many questions that occupy mankind) (of course I have it ready at any time and could add its usefulness and reliability at any time; especially since even then it remains quite simple, despite its depth).
Now he went into seclusion for a few days. Presumably, in search of the predecessor. When he did not find it, he changed strategy: he looked for the mistake, or at least SOME mistake, in the concept. In fact, according to him, he found one. So he sent me the “error”, which, however, only referred to a side aspect that I had well considered, but which was known to me and which a) could easily be remedied but b) basically had nothing to do with the validity of the formula itself (and its value). In this respect, I got back to him and told him that I could easily fix this (ridiculous and insignificant) “error”. I did so the next day and sent him this part. Now there was no longer any doubt: I had something to show that was good and right and had not been discovered before. With the following consequence: it was the last exchange that ever took place with Mr Poppe.
I don’t want to offend him and he had said several times that he was in the middle of his doctoral thesis and that he could only do this on the side, so in this respect a variety of alternative reasons were possible for the silence. However, it is also possible that he could not accept this from a non-scientist, that his original statement was rock solid and irrefutable (“there is nothing new and you have nothing, unless it is wrong…”) or, as a last possibility, that he thanked me for my kindness in presenting him with a new method that could advance mathematics – and we will soon be presented with it by HIM. Of course, all this was said with plenty of imagination and a lot of tongue-in-cheek.
Nevertheless, what remains of this story is that I get no recognition, although I certainly deserve some. For me, it remains that finding these formulas was very easy, insofar as “nothing great” was created by me either way.
7) A small example from my school days: in the ninth grade we were asked to draw a line with the length of the square root of 13. The teacher’s idea was clear: 4 + 9 = 13, 2 ^2 = 4 and 3 ^2 = 9. So: you draw a right-angled triangle in which one cathetus would have the length 2, the other the length 3 (you start with one of the two, make a right angle, draw the other at this angle, the hypotenuse would have the length root 13 (the sum of the squares of the cathets is equal to the square of the hypotenuse; 4 + 9 = 13; hypotenuse would have the length root 13). All right, so far simple. Only: would any of the classmates or the teacher also have found a solution if you had asked for root 11 or root 17? Apart from the possibility of asking for any other number, which would not have to be odd at all?
Little Pauli had a general solution. It is as follows: divide the number from which the root is sought by 2. Add 0.5 to this result and subtract 0.5 from it. These two numbers are entered into the right-angled triangle: the larger is the hypotenuse, the smaller is one of the two cathets. The other cathetus would have the length of the root in question.
Applied to the simple example root 13 of mine, which I also presented and explained like this in the paper:
13 divided by 2 = 6.5
6.5 + 0.5 = 7
6.5 – 0.5 = 6
The little miracle of mathematics now shows itself :
7^2 = 49
6^2 = 36
49 – 36 = 13
So this means: you draw the hypotenuse with length 7, one of the cathets with length 6. The other cathetus has length root 13. The difference to the required solution: mine is ALWAYS valid and for every number x, from which you want to draw the root.
The peculiarities of the story: the teacher gave me 0 points, in the paper I got a 3. I did NOT complain, but continued to laugh at her, as I had done before (my neighbours said: whenever I intervened in the lesson, she got red marks on her neck, indicating nervousness; because: she knew she had made a mistake at that moment). I didn’t go to the headmaster to challenge the result either.
Another small peculiarity, of course, was that I had come up with it all by myself. So I always played a bit with numbers and “discovered” such things, knowing full well that I wouldn’t have this knowledge exclusively.
Another little thing: when I showed my father this “discovery”, he explained to me calmly and amicably that it wasn’t such a great discovery (he didn’t belittle me or anything, he knew what to make of me, didn’t praise me too much, but didn’t blame me either; he just explained calmly). He explained it to me graphically: if you draw a square with 55, you can expand it very simply to a 66 square. To do this, you add five small squares at the bottom, for example one row down, so that it is effectively 5*6. Then you add these five squares to the right side (for example; the left side would also work). Now you just have to add the one, small, missing square.
So very easy to recognise graphically. In this respect, my insight was “disenchanted”, but that didn’t surprise me and didn’t stop me from trying and researching further.
I would have several more examples, only briefly mentioned:
8) the perfect typing game. I have explained this in videos (you can probably find them under this name).
9) the tennis formula, which could also be applied to chess (this is better than the Elo formula).
10) That an odds represents a probability, in the reciprocal value is a kind of “open secret”. But how to turn the “fair odds” determined in this way into a “pay odds” is still a mystery to many. For this I have developed a formula. This has a lot of amazing properties that bookmakers neither know nor understand – but basically all work according to it. Just to test you (or any respondent) once: what would be your pay rate if the probability, calculated by you this way, for an event is 1%?
11) I was once given the task of memorising pi to the first 30 digits, within two minutes. I succeeded. I could recite it without mistakes.
12) A game I like to play to this day is: “Tell me your date of birth, with the year of birth. Give me two minutes and I’ll tell you the day of the week.” (Why always two minutes? I don’t know, but that’s how it was).
