Oh, that's a fixed point of an iterative (or recursive) function. That's a beautiful subject, deep and marvelous, and with real world applications. There's a certain temperature and pressure at which water can exist in three phases at once; solid, liquid, and gaseous. That's the triple point of water, and it is, you've guessed it, a fixed point. Nature generally does that one thing it knows how, and then it just keeps doing that to whatever comes out, so we should expect to find lots of fixed points. But it's not always just one fixed point. Sometimes it's a pair, where f(x) = y, and f(y) = x; or it's three points, or four, five... You've probably seen the Mandelbrot Set, and you may have wondered; "yes, it's beautiful, but what is it really?" It's a map, and it's a map of all the fixed points! Since Kaprekar's nice little game has a fixed point, there is a way to map it into the Mandelbrot Set, I don't know how to do it, and I won't be bothered to find out, but in some general sense, appropriately transmogrified, 6174 lives in there. And if you were to play this game yourself, you may note that on the way to the fixed point, some numbers pop up more than others, and some may even seem inevitable. And if you were to count the number of ways that you can get to a certain value, that number is a measure of the entropy of that value; that, in a very real sense, is what entropy is. Note that there is nothing random here: everything is deterministic. And yet there is entropy. And if you were to take the (in)famous Riemann zeta-function, which I am not going to write down here, and then replace the ones in the denominator with these entropy numbers, you'll get a Dirichlet L-function, and those tell of deep relations between physics, geometry, and number theory; and the subject of those relations is called Langland's Program. But by now I'm getting out of my depth, and maybe no longer know what I'm talking about. Like I said, it's a nice little game, and yet, if you play it right, it is like a sacred key that opens the doors to the universe, that you will finally see it. Thank you.
Revt.Dr.Kaprwker was my Guru... from 1978 to 1981 I had learnt maths of 5 th , 6th,7th standard from this Genius...at that time He lived in Abhinav Bharat ( Swatantravir Sawarkars Resd. ) in Nashik, Maharashtra...he took an annual fees Rs.1 from us ... when he died... India was unaware..BBC gave the bad new...while writing this my eyes are full of tears...The Man Of Simplicity... Prof.Raj Sinnarkar, Nashik
@@vectoracademy3992 genious never run for money that was time when so genious were teaching without money. And remember our country's nature and our nature is to ignor and criticise their talent it is unfortunate practice in our country.
I’m so unlucky person guys, This legend was living in Deolali Gaon and I live almost 5-7 km away from his residence but I never heard about him. I think this is not just my fault it’s also foult of my education system or society! Some buddy like this legend with great knowledge in maths but i never heard about him in my hole life. This is really so unfortunate thing for me! Thanks to Prime Newton who introduced us about this legend! Love from India sir! 🇮🇳 ❤
@@PrimeNewtons he used to stay in a very small room. When he died, all his room was full of mathematical equations, even puzzles were solved on bus tickets also. Very much fond of speed. Always prefers to sit at window seat and just do calculations wrt speeds. I am proud that I came from his town, and had seen him almost everyday.
I love how you are so excited to show off something you've discovered and haven't even read everything about it. That's the true spirit of a mathematician!
I'm an engineer. I wish I had a teacher like you in my maths class. It was a struggle but I eventually got there. To all those people struggling with maths do not be afraid to ask for help. Keep up the good work brother.
I learnt about Kaprekar constant and Kaprekar Routine back in 1983 when I was learning BASIC programming. It was such a joy to see the steps in the routine and finally reaching Kaprekar constant! Thank you for refreshing those memories. And I do love the enthusiasm and passion with which you communicate.
THANK YOU for sharing in such a nice and engaging way! Too good! I attended a short lecture by Mr. Kaprekar in Mumbai in 1983 (or '84), where he illustrated amazing properties of palindromic numbers. I was 15 then, and was one of the award recipients from Mr. Kaprekar in that very event, for finishing among the top 3 in a secondary-school level mathematics competition held at Mumbai's Nehru Planetarium. I not only enjoyed your video but also reminisced such a great event in my life!
Your enthusiam is infectious! Your motto is fabulous - Never Stop Learning. I'm 63 and people think I'm nuts because I am alway learning something new. These things I learn will probably never be used - except of the wonder of I my learning it. AND THAT IS MORE THAN GOOD ENOUGH. I am thrilled to be a new subcriber to your channel.
Afro Bro. More than that Kaprekar's constant your romantic pronounciation and english flavour are exciting. I loved your teaching sir. My salute and by touching your feet and spirit. Thanks from India.
I ran a python program to figure out how many times it takes for all numbers between 1000 and 9999 and it's true that 7 is the max. 9004, 9005, 9006 are examples of consecutive numbers with all 7 times. Same with 9015, 9016, and 9017.
@@vadimbe9783 Nice, that's correct for n >= 1000, but for n >= 1 (padded with leading zeros, e.g. by f'{n:04d}'), you will get 2184 numbers. Anyways the longest list of numbers is received for a target of 3 iterations (2124 for n >= 1000 and 2400 for n >= 1).
As an Indian I am so proud to see this video....In our Indian school our teachers haven't taught us about this constant, it's my first time to know this.....Thank you so much for the information. I'm subscribing you for this brilliant information😊
@@bittertruth27the day u say indian. Hindi states ppl mean north indian overlook u. Cuz according to north India India means Hindi. Even they also don't know Hindi is not their language. They lost their language due to Hindi.
@@bittertruth27Comment is referring to state not caste. Kapreker was from Maharashtra which is part of India. If you identify from any state it doesn't mean you are less patriotic to India, I think being Maharashtrian already accounts for being Indian (Foreigners might not get it). People from various states live in Maharashtra, they always tell other people about their state for them Being Indian Is Obvious. It is Not Chak de India. Kapreker being a Maharashtrian is more relatable and personal to a Maharashtrian. WHY ANYONE SHOULD THROW AWAY THEIR STATE IDENTITY AND IDENTIFY AS AN INDIAN?? Why not identify as both? A Indian from the state of Maharashtra.
Gentleman the way and excitement with which you teach... am very sure the students who love mathematics must be madly in love with the subject... God bless young man... _May all teachers of the world become like you..._ 🙏
I gotta love this man - so much enthusiasm, so much involvement, he makes science fun! And this is the greatest service one can do to science. Thank you so much for this presentation, truly brilliant!
Where are you going after you die? What happens next? Have you ever thought about that? Repent today and give your life to Jesus Christ to obtain eternal salvation. Tomorrow may be too late my brethen😢. Hebrews 9:27 says "And as it is appointed unto man once to die, but after that the judgement
@@JesusPlsSaveMeThat is not how one is saved. We must believe that the Lord Jesus Christ died for our sins, was buried and rose from the dead. The Bible never tells us to give our lives to Jesus.
@@ppal64nothing, he was just born into human family. And you are committing like old fart aunty the moment you see the word Marathi. People are proud to be their mother tongue so keep your fingers to where they belong, yes, that’s your butts..
Actually marathi is a mother tongue language in India... Marathi speaking people are scattered throughout india but mainly marathi speaking state is Maharashtra...... And Mr. Kaprekar is a marathi man that is his mother tongue was marathi!!!!!
That is quite amazing. I never heard about that constant before. And: You have a very clear and beautiful hand writing. I think one of the best on RUclips!
Thank you for another wonderful and blessed video. I had not heard of Kaprekar's constant until now, so thank you for helping me to never stop learning. I think a proof (improved from an earlier version by reducing the number of cases) of the result that a unique invariant under the Max-Min process exists and equals 6174 goes like this. Label the 4 digits of the number N chosen as a4, a3, a2 and a1 in order from largest to smallest (the all equal case is excluded). Then the max and min numbers that can be formed from its digits are Max = 1000a4 + 100a3 + 10a2 + a1 Min = 1000a1 + 100a2 + 10a3 + a4. Hence Max-Min = 1000(a4-a1) + 100(a3-a2)+ 10(a2-a3) + (a1-a4). The latter two coefficients are non-positive, so "carry" to rewrite this expression as Max-Min = 1000(a4-a1) + 100(a3-a2-1) + 10(a2--a3+10-1) + (a1-a4+10). Hence Max-Min = abcd where a = a4-a1 b = a3-a2-1 c = a2-a3+9 d = a1-a4+10 denoted equations (1) unless a3=a2, in which case b above is negative and hence invalid and must be replaced with a3-a2-1+10 by carrying from a, which reduces to a4-a1-1. Note that c and d are always non-negative because both a2-a3 and a1-a4 are at least -9. a4, a3, a2 & a1 are now redefined as the 4 digits of the number abcd in order from largest to smallest, and the Max-Min process repeated. Note that they cannot all be equal again because a=d requires a4-a1=5, b=c requires a3-a2=5 and hence a=b requires 5=5-1=4, contradiction. And in the case where b is replaced by carrying from a, a=d requires a4--a1=11/2, contradiction. Note that when a3>a2 so that b above is non-negative and hence does not need to be replaced by carrying from a a+d=10 b+c=8 a>b because a4-a1 is at least as large as a3-a2, and c > or = d-1 for the same reason denoted conditions (2). Because it is invariant under the Max-Min process, the only possibilities for Kaprekar's constant are as follows. First note the possibility that its a3=a2 is excluded because if that was so: b = a3-a2-1+10 = 9, from which it follows that its a4=9 a = a4-a1-1 = 8-a1 c = a2-a3-1+10 = 9, from which it follows that its a3=9 and hence also that its a2=9 d = a1-a4+10 = a1+1 Hence the constant could only be: a1-999 where a1 = 8-a1 hence a1 = 4 and a1+1=9, contradiction. or 999-a1 where 9=8-a1 hence a1=-1, contradiction. Hence its a3>a2. Next note the inequality c > or = d -1 in conditions (2) is in fact strict because c = d-1 iff a4-a1 = a3-a2 iff a4=a3 and a2=a1 iff the constant has the form a3a3a2a2. But this and the second of equations (1) then mean a3 = b = a3-a2-1 hence a2=-1, contradiction. Hence c > d-1 hence c > or = d. This strengthened inequality, together with the inequality a>b in conditions (2), then mean the only possibilities for the constant are: Case 1: a4a3a2a1 Case 2: a4a2a3a1 Case 3: a4a1a3a2 Case 4: a3a2a4a1 Case 5: a3a1a4a2 Case 6: a2a1a4a3 where a4, a3, a2 & a1 satisfy equations (1). All cases lead to a contradiction except Case 5, which yields the result 6174. Case 1: a3=b=a3~a2-1 hence a2 = -1, contradiction. Cases 2 & 4: a1=d=a1-a4+10 hence a4=10, contradiction. Case 3: a4=a=a4-a1 hence a1=0 a1=b=a3-a2-1 hence a3=a2+1 a3=c=a2-a3+9 hence 2a3=a2+9 Hence 2a2 +2 = a2+9 hence a2=7 a2=d=a1-a4+10 hence a4=3 < a2, contradiction. Case 6: a2=a=a4-a1 hence a4=a1+a2 a1=b=a3-a2-1 hence a1+a2=a3-1 a4=c=a2-a3+9 hence a3+a4=a2+9 a3=d=a1-a4+10 hence a4=a1-a3+10 Substituting for a4 from the first equation into the last two equations gives a3+a1+a2=a2+9 hence a1+a3=9 a1+a2=a1-a3+10 hence a2+a3=10 Hence the second equation gives 9-a3+10-a3= a3-1 hence 3a3 = 20 hence a3=20/3, contradiction. Case 5: a3=a=a4-a1 hence a4=a1+a3 a1=b=a3-a2-1 hence a1+a2=a3-1 a4=c=a2-a3+9 hence a3=a2-a4+9 a2=d=a1-a4+10 hence a4=a1-a2+10 Substituting for a4 from the first equation into the last two equations gives a3=a2-a1-a3+9 hence 2a3=a2-a1+9 a1+a3=a1-a2+10 hence a3=10-a2 Hence 20-2a2=a2-a1+9 hence 3a2=11+a1 Hence the second equation gives a1+a2=9-a2 hence 2a2 = 9- a1 Hence 27-3a1 = 22+2a1 hence 5a1=5 Hence a1=1, a2=4, a3=6, a4=7. Hence Kaprekar's constant = a3a1a4a2 = 6174. Again, thank you for your lovely videos and i look forward to seeing the next one. God bless you ❤
I wish you had been my maths teacher ....your enthusiasm would have made the subject I loathed most exciting and interesting.... You are so likeable ....
Thanks for enlighting this very interesting subject. I made a few tests on an excel sheet with 2, 3 and 5 digit numbers by curiousity and got the interesting following results ! I hope you like my inquiry and study, tell me if you like it ! >For 2 digit numbers, the Kaprekar iteration process doesn't converge, but gives a cyclic set of following numbers after 7 iterations max: 9, 81, 63, 27, 45 and 9 again and so on. >For 5 digit numbers, the Kaprekar iteration process doesn't converge, but has got a different constant value (depending on the intial value) repeated every 4 iterations. The periodic behaviour seems to appear after 8 interations max. >And last, with 3 digit numbers, the Kaprekar iteration process converges towards the value 495 and seems to after 7 interations max. To finish with my post. There is a trivial result to be mentioned: The Kaprekar iteration process gives 0 as result for all intial numer having the same value for all digits. For example, 2222 gives 2222 as "max" and "min" so the there difference is 0. Greetings and keep up the good work !
He didn't have any degrees or anything. He was just curious about numbers and that's it! As an Indian I am deeply heartbroken that I haven't heard a single thing about him in my 16 years of studying maths. A real shame on my people for teaching system for teaching me who looted us , ruled us , and didn't even considered to talk about D.R.Kapreker. 😢
There is another famous Indian mathematician who was largely self-taught: Ramanujan. He did amazing things with repeating fractions and much much more. He went to England to study further but sadly died very young. Perhaps you could talk about something he solved in one of your videos?
Thank you so much for creating these videos! I LOVE math so much I decided to never stop learning it after college. You are so good with your teaching, and your enthusiasm is contagious. Keep up the good work! 👏👏
randomly came across this video n was happy to see another maths teacher with such enthusiasm- the first one was my 5th grade math teacher!! hugs from India!!
Finally, a math teacher who can write legibly! Too many math people don't write very well, so it's often hard to tell exactly what it is that they've written. 👍
...especially when greek letters suddenly come out to play but you don't know their names, or have trouble remembering them all and the upper and lower cases, grrr!
Given any 4-digit number abcd where a>b>c>d, the first subtraction yields the expression 900(a-d) + 90(a+b-c-d) + 9(a-d). This produces a number of the form 9(efe) or 9(efg) where e=g+1. The second of these cases occurs if a+b is 10 or more than c+d, because of the carry-over term. The above expression is used again with the new digits obtained from the first subtraction, and in general, it indicates that the total number of subtractions to get to 6174 is a function of the differences between the digits of the starting number. Studying the relatively straightforward algebra behind the process can take the magic away!
According to Wolfeam & Wikipedia Consider an n-digit number k. Square it and add the right n digits to the left n or n-1 digits. If the resultant sum is k, then k is called a Kaprekar number. For example, 9 is a Kaprekar number since 9^2=81 8+1=9, and 297 is a Kaprekar number since 297^2=88209 88+209=297.
Would be interesting to see how that works in binairy. Maximum and minimum only depend on the total number of ones and total number of zeros. I wonder if it can be generalized for any base and any size analytically. And I have to figure out how it works in balanced ternary!
@@jahbiniThe biggest and smallest numbers generally won't be the same in other bases. They may not even have the same number of digits. There might be other constants in other bases though. Nice idea!
I came to know about Kaprekar's constant couple of years back. Thank you for covering this art of mathematics. Also, if I remember correctly, Numberphile made a video about most iterations required for a number to reach 6174 or like that.
I was born in the year l96l. Fun fact: even if you turn this number upside down it reads the same. 😄 By the way, great content. Glad to know about Mr Kaprekar. Never heard about him though I live 4 hours drive from where he lived.
Mathematics scares me, but your calmness is so inviting that it has actually sparked a curiosity in me about math. Teaching is an art, and you, sir, are an artist.
That was enjoyable and refreshing. Finally I found someone to guide me through the world of numbers. And thanks for making this easy to understand and pleasurable to watch.
Amazing! Thank you for introducing me to this concept. Truly fascinating. I used to hate Maths at school but am absolutely in awe of it now. Thanks for the video and stay happy.
Coolest mathematics teacher.... Thank your sir for teaching something new... I belong to the state of kaprekar sir and now i m wondering why this theory was not in our school syllabus... This is the first time i am getting to hear about kaprekar constant... Thank you sir for teaching something new..
Wow that black board and clean writing on it. You reminded me my school day teacher. Coincidentally he was teaching us Maths. I felt nostalgic for your writing and was amazed for this constant.
Thanks for the knowledge. It inspired me to write an algorithm to test every number from 0001 to 9999. There were lots of numbers that took 7 turns to arrive at the constant. Many took just one turn like 0026, 0062, 0136 and none took more than 7 turns. Of course numbers with four identical digits were not tested. Just for the record the amount of numbers taking one to seven turns are 383 | 576 | 2400 | 1272 | 1518 | 1656 | 2184. Meaning most numbers took 3 turns to arrive at the constant.
I never seen a video for maths, but today youtube suggested this video and i watched. its really a mind blowing fact and good to know. thanks to the creator of this video
I am from India. I am not a big fan of Mathematics. However seeing the video title I got curious. It is simply amazing... (Because being an Indian whenever we think of Maths we only recollect Ramanujam a great wizard of Mathematics).
Actually, studying the numbers XY00 with X≥Y (with the help of Excel…), I found that 4100, 5100, 5200, 6100, 8500, 9400, 9500 and 9600 were the numbers of this kind needing 7 steps. Funny enough, 6200 goes directly to 6174 (in 1 step), and is the only one of the kind. Thanks for your interesting videos !
i made a model in excel using the formula "=VALUE(CONCAT(SORT(MID(A1,SEQUENCE(LEN(A1)),1),,1)))" and "=VALUE(CONCAT(SORT(MID(A1,SEQUENCE(LEN(A1)),1),,-1)))"i run it from 1000 to 9999 , it was fun to see which numbers to reach 7 times
I wonder if you can create an X, Y graph of the results that shows the number of steps it takes to get there - perhaps showing points of different colours according to the number of steps to get to 6174 - EG 1 step = red, 2 = orange, 3 yellow etc. I wonder what that would look like?
From the second step onwards, both the resulting number and the numbers generated in each rearrangement (ascending and descending) all have a digit sum of 18. This indicates that the process has entered a stable cycle where all results maintain a digit sum of 18, eventually leading to 6174.
I'm not a mathematician or student, but what a wonderful and informative video. It certainly sparks your brain into thinking differently. Thank you good sir!
Amazing!!!! I am not a mathematician but I thoroughly enjoyed your presentation, Sir! Also, God bless the soul of that simple, but great man, Pandit Kaprekar! ❤❤❤🙏🙏🙏🪷🪷🪷🙇♂️
From one who loves unique math problems, this is instructive and fun. Works with 495,too, just use any 3 digit number. Thanks for introducing me to something new. I am passing it on. One 7th grade relative likes it and said he’ll show it to his math teacher tomorrow.
You have great charm and enthusiasm in your way of demonstrating and presenting. I always hated mathematics and assumed I was just plain thick - until doing a course as an adult with a great teacher and realised I was enjoying it. It wasn't me it was just careless and cold instruction - the subject itself was fine. Great and interesting stuff here - I'll be checking out more of your videos.
Oh my god. Indeed he is great mathematician. He used to stay in the same city where i live. It is Donbivli in Thane district near Mumbai. We didn't know about him initially but few years back on railway platofrm wall painitng under the title of great human beigns from our city I saw his photo. Really got shocked that he was such a great mathematician and from the same city❤🎉
I love it!! My father mentioned this to me years ago, but I obviously didn`t pay attention. I have since become much more interested in numbers and mathematics. Thanks so much for putting this up.
Try to draw a directed graph (vertex/edge kind, not the plot) with all 1000 numbers as vertices and arrows pointing to the next vertex. Maybe start with two digit numbers to try the idea out first.
Oh, that's a fixed point of an iterative (or recursive) function. That's a beautiful subject, deep and marvelous, and with real world applications. There's a certain temperature and pressure at which water can exist in three phases at once; solid, liquid, and gaseous. That's the triple point of water, and it is, you've guessed it, a fixed point.
Nature generally does that one thing it knows how, and then it just keeps doing that to whatever comes out, so we should expect to find lots of fixed points. But it's not always just one fixed point. Sometimes it's a pair, where f(x) = y, and f(y) = x; or it's three points, or four, five...
You've probably seen the Mandelbrot Set, and you may have wondered; "yes, it's beautiful, but what is it really?"
It's a map, and it's a map of all the fixed points! Since Kaprekar's nice little game has a fixed point, there is a way to map it into the Mandelbrot Set, I don't know how to do it, and I won't be bothered to find out, but in some general sense, appropriately transmogrified, 6174 lives in there.
And if you were to play this game yourself, you may note that on the way to the fixed point, some numbers pop up more than others, and some may even seem inevitable. And if you were to count the number of ways that you can get to a certain value, that number is a measure of the entropy of that value; that, in a very real sense, is what entropy is. Note that there is nothing random here: everything is deterministic. And yet there is entropy. And if you were to take the (in)famous Riemann zeta-function, which I am not going to write down here, and then replace the ones in the denominator with these entropy numbers, you'll get a Dirichlet L-function, and those tell of deep relations between physics, geometry, and number theory; and the subject of those relations is called Langland's Program. But by now I'm getting out of my depth, and maybe no longer know what I'm talking about.
Like I said, it's a nice little game, and yet, if you play it right, it is like a sacred key that opens the doors to the universe, that you will finally see it.
Thank you.
Beautiful narrative.
How could all of this be a mistake ? Such intelligent design ? Thank you for your detailed comment o stranger
@@realitycheck816 certainly not coincidence some of us believe the lord is all
your explanation is interesting😊
what's the real-world application of the Kaprekar constant?
Revt.Dr.Kaprwker was my Guru... from 1978 to 1981 I had learnt maths of 5 th , 6th,7th standard from this Genius...at that time He lived in Abhinav Bharat ( Swatantravir Sawarkars Resd. ) in Nashik, Maharashtra...he took an annual fees Rs.1 from us ... when he died... India was unaware..BBC gave the bad new...while writing this my eyes are full of tears...The Man Of Simplicity...
Prof.Raj Sinnarkar, Nashik
Reverend? Was he Christian?
No @@invictus143
When Amitabh Bachan dies India will be aware
@@invictus143 suppose he was and suppose he wasn't ? How would you're views change ? Seems like you're a bigot
@@vectoracademy3992 genious never run for money that was time when so genious were teaching without money. And remember our country's nature and our nature is to ignor and criticise their talent it is unfortunate practice in our country.
As an Indian I never knew there is something like this. Thanks a loads
Indian knows a lot about the number, that;s why many of them working at the 07-11....LOL
Oh wow.
Expert @@yrot1123
@@yrot1123 thoughtful comment... feeling proud about your "wit", no doubt
Hahhaha indians are the highest earning individuals in usa
Well what can I say? You Indians are the best.
I am fortunate to listen kaprekar sir in my school days
That guy in thumbnail seems like wearing some Marathi culture clothing!?
@@mattdamon2084 Because he was!
Really? Where?
@@mattdamon2084he is😂😊
I have learned something new today and it was fun learning.
I’m so unlucky person guys,
This legend was living in Deolali Gaon and I live almost 5-7 km away from his residence but I never heard about him. I think this is not just my fault it’s also foult of my education system or society!
Some buddy like this legend with great knowledge in maths but i never heard about him in my hole life.
This is really so unfortunate thing for me!
Thanks to Prime Newton who introduced us about this legend!
Love from India sir! 🇮🇳 ❤
😂😂😂
Hole life , lmao.
@@anomalous5048 😛
I guess its your fault I was thaught about this in university
@@anomalous5048 how many languages you know perfectly sir?
English isn't our primary language, please don't make fun of others.
I have seen him in person. We used to stay @100 m from each other. Simple man, used to walk around with his umbrella. I was school going boy then.
That makes you a part of the history of mathematics.
@@PrimeNewtons he used to stay in a very small room. When he died, all his room was full of mathematical equations, even puzzles were solved on bus tickets also. Very much fond of speed. Always prefers to sit at window seat and just do calculations wrt speeds. I am proud that I came from his town, and had seen him almost everyday.
@@kulkashishhow lucky!!
Where he lives in Pune?? Address pls
I thought you were going to say, "I lived 6174 ft. from his house" 😂!
I love this presenter’s enthusiasm and ease of explanation. Good style and good substance.
I love how you are so excited to show off something you've discovered and haven't even read everything about it. That's the true spirit of a mathematician!
depriciation?
Sarcasm 😂
Fantastic
I'm an engineer. I wish I had a teacher like you in my maths class. It was a struggle but I eventually got there. To all those people struggling with maths do not be afraid to ask for help. Keep up the good work brother.
I agree. He is amazing. From old retired PhD engineer.
Greetings. A great advice. God's blessings in all that you do.
I learnt about Kaprekar constant and Kaprekar Routine back in 1983 when I was learning BASIC programming. It was such a joy to see the steps in the routine and finally reaching Kaprekar constant!
Thank you for refreshing those memories. And I do love the enthusiasm and passion with which you communicate.
THANK YOU for sharing in such a nice and engaging way! Too good! I attended a short lecture by Mr. Kaprekar in Mumbai in 1983 (or '84), where he illustrated amazing properties of palindromic numbers. I was 15 then, and was one of the award recipients from Mr. Kaprekar in that very event, for finishing among the top 3 in a secondary-school level mathematics competition held at Mumbai's Nehru Planetarium. I not only enjoyed your video but also reminisced such a great event in my life!
Fascinating! I haven’t been in a math class for 50 years. I can’t wait to share this with my grandchildren. What a fun math concept! ❤
Please do!
"Recreational Mathematician" sounds so much better than "retired math teacher". Thank you for this wonderful spin.
Very cool👍
Jesus is the way, the truth and the life. Turn to him and repent from your sins today!
I love your contagious playful enthusiastic energy! In Peace and Friendship,
Pierre Pagé
Thank you, Pierre.
Not only have you broadened my mind, you have taught me a new way to do 9s.
I love the energy and joy you bring into the subject.
Your enthusiam is infectious! Your motto is fabulous - Never Stop Learning. I'm 63 and people think I'm nuts because I am alway learning something new. These things I learn will probably never be used - except of the wonder of I my learning it. AND THAT IS MORE THAN GOOD ENOUGH. I am thrilled to be a new subcriber to your channel.
NICE TO WATCH YOUR CHANNEL.
I am honored.
Afro Bro. More than that Kaprekar's constant your romantic pronounciation and english flavour are exciting. I loved your teaching sir. My salute and by touching your feet and spirit. Thanks from India.
You are a true AH, should not bring race in every breath
True @@nripensaha2210
@@nripensaha2210Funny you! Why shouldn't him bring admiration to race of the teacher's spoken accent.
What a pathetic comment in the honour of kaprekar.......hats off to you .........Mahan log aapke jaise he india me isiliye india is having more gutter
I'm so lucky to be an maratha and he kaprekar even taught me in my school days I remember this thing thank you so much to remind me ❤
Maths is full of wonder but you also bring joy, and that’s how the love of something so important can spread. Thank you!
Your classes are so therapeutic and I have never seen anyone who draw a perfect straight line on board like you do..bravo..🎉
I can't seem to be able to draw one with a straitedge! 😅
I ran a python program to figure out how many times it takes for all numbers between 1000 and 9999 and it's true that 7 is the max. 9004, 9005, 9006 are examples of consecutive numbers with all 7 times. Same with 9015, 9016, and 9017.
I also wrote a Python program - looks like there are 1980 four digit numbers which require 7 iterations, the smallest being 1004 and the largest 9985
Javascript programmer here. But would you mind sharing your git? I am very interested in learning more maths myself snd contributing to it.
@@juangalton999 Try it yourself. It's easy. Just some looping.
@@vadimbe9783 the smallest is 0014. And 3 consecutive numbers with all 7 steps:
0014 -> 4086 -> 8172 -> 7443 -> 3996 -> 6264 -> 4176 -> 6174 (7 steps)
0015 -> 5085 -> 7992 -> 7173 -> 6354 -> 3087 -> 8352 -> 6174 (7 steps)
0016 -> 6084 -> 8172 -> 7443 -> 3996 -> 6264 -> 4176 -> 6174 (7 steps)
@@vadimbe9783 Nice, that's correct for n >= 1000, but for n >= 1 (padded with leading zeros, e.g. by f'{n:04d}'), you will get 2184 numbers. Anyways the longest list of numbers is received for a target of 3 iterations (2124 for n >= 1000 and 2400 for n >= 1).
As an Indian I am so proud to see this video....In our Indian school our teachers haven't taught us about this constant, it's my first time to know this.....Thank you so much for the information. I'm subscribing you for this brilliant information😊
❤
Delighted.
I have never heard of Kaprekars constant.
Thanks.
This number is in you, in time and more, I invite you to see Universum 6174
Wow! I am a Maharashtrian and had never heard of this math genius. Thank you for re-introducing this to the world.
@@bittertruth27the day u say indian. Hindi states ppl mean north indian overlook u. Cuz according to north India India means Hindi. Even they also don't know Hindi is not their language. They lost their language due to Hindi.
@@sankkham true...👍👍..
@@bittertruth27Comment is referring to state not caste. Kapreker was from Maharashtra which is part of India. If you identify from any state it doesn't mean you are less patriotic to India, I think being Maharashtrian already accounts for being Indian (Foreigners might not get it). People from various states live in Maharashtra, they always tell other people about their state for them Being Indian Is Obvious. It is Not Chak de India. Kapreker being a Maharashtrian is more relatable and personal to a Maharashtrian. WHY ANYONE SHOULD THROW AWAY THEIR STATE IDENTITY AND IDENTIFY AS AN INDIAN?? Why not identify as both? A Indian from the state of Maharashtra.
@@bittertruth27 bro he is just proud to be maharashtrian. It doesn't he will not be indian that is foolish in my are way of thinking.
I am in Pune. Had never heard of him!
Thank you.
I showed the whole family.
Messed it up first,but got it right 2nd time.
Collective amazement!
Gentleman the way and excitement with which you teach... am very sure the students who love mathematics must be madly in love with the subject...
God bless young man... _May all teachers of the world become like you..._
🙏
I gotta love this man - so much enthusiasm, so much involvement, he makes science fun! And this is the greatest service one can do to science. Thank you so much for this presentation, truly brilliant!
This is the perfect way to teach some maths to children, bringing these magical features about numbers. Excellent work.
Where are you going after you die?
What happens next? Have you ever thought about that?
Repent today and give your life to Jesus Christ to obtain eternal salvation. Tomorrow may be too late my brethen😢.
Hebrews 9:27 says "And as it is appointed unto man once to die, but after that the judgement
@@JesusPlsSaveMe Idiot, when we die we go to the cemetery
@@JesusPlsSaveMeThat is not how one is saved. We must believe that the Lord Jesus Christ died for our sins, was buried and rose from the dead. The Bible never tells us to give our lives to Jesus.
Proud to be Marathi, and Marathi scientists. Thank you for bringing this up 😊
🙌
What did you do to be Marathi?
Yeah this is new disease these days in indians that do nothing and feel proud...
@@ppal64nothing, he was just born into human family. And you are committing like old fart aunty the moment you see the word Marathi. People are proud to be their mother tongue so keep your fingers to where they belong, yes, that’s your butts..
Actually marathi is a mother tongue language in India... Marathi speaking people are scattered throughout india but mainly marathi speaking state is Maharashtra...... And Mr. Kaprekar is a marathi man that is his mother tongue was marathi!!!!!
That is quite amazing. I never heard about that constant before.
And: You have a very clear and beautiful hand writing. I think one of the best on RUclips!
Humans are essentially curious by nature. You help to bring out the curiosity and wonder in all of your videos. Bravo, sir! Take a bow!
Thank you for another wonderful and blessed video.
I had not heard of Kaprekar's constant until now, so thank you for helping me to never stop learning.
I think a proof (improved from an earlier version by reducing the number of cases) of the result that a unique invariant under the Max-Min process exists and equals 6174 goes like this.
Label the 4 digits of the number N chosen as a4, a3, a2 and a1 in order from largest to smallest (the all equal case is excluded).
Then the max and min numbers that can be formed from its digits are
Max = 1000a4 + 100a3 + 10a2 + a1
Min = 1000a1 + 100a2 + 10a3 + a4.
Hence
Max-Min = 1000(a4-a1) + 100(a3-a2)+ 10(a2-a3) + (a1-a4).
The latter two coefficients are non-positive, so "carry" to rewrite this expression as
Max-Min = 1000(a4-a1) + 100(a3-a2-1) + 10(a2--a3+10-1) + (a1-a4+10).
Hence Max-Min = abcd where
a = a4-a1
b = a3-a2-1
c = a2-a3+9
d = a1-a4+10
denoted equations (1)
unless a3=a2, in which case b above is negative and hence invalid and must be replaced with a3-a2-1+10 by carrying from a, which reduces to a4-a1-1. Note that c and d are always non-negative because both a2-a3 and a1-a4 are at least -9.
a4, a3, a2 & a1 are now redefined as the 4 digits of the number abcd in order from largest to smallest, and the Max-Min process repeated. Note that they cannot all be equal again because a=d requires a4-a1=5, b=c requires a3-a2=5 and hence a=b requires 5=5-1=4, contradiction. And in the case where b is replaced by carrying from a, a=d requires a4--a1=11/2, contradiction.
Note that when a3>a2 so that b above is non-negative and hence does not need to be replaced by carrying from a
a+d=10
b+c=8
a>b because a4-a1 is at least as large as a3-a2, and
c > or = d-1 for the same reason
denoted conditions (2).
Because it is invariant under the Max-Min process, the only possibilities for Kaprekar's constant are as follows.
First note the possibility that its a3=a2 is excluded because if that was so:
b = a3-a2-1+10 = 9, from which it follows that its a4=9
a = a4-a1-1 = 8-a1
c = a2-a3-1+10 = 9, from which it follows that its a3=9 and hence also that its a2=9
d = a1-a4+10 = a1+1
Hence the constant could only be:
a1-999 where a1 = 8-a1 hence a1 = 4 and a1+1=9, contradiction. or
999-a1 where 9=8-a1 hence a1=-1, contradiction.
Hence its a3>a2.
Next note the inequality c > or = d -1 in conditions (2) is in fact strict because c = d-1 iff a4-a1 = a3-a2 iff a4=a3 and a2=a1 iff the constant has the form a3a3a2a2. But this and the second of equations (1) then mean a3 = b = a3-a2-1 hence a2=-1, contradiction.
Hence c > d-1 hence c > or = d.
This strengthened inequality, together with the inequality a>b in conditions (2), then mean the only possibilities for the constant are:
Case 1: a4a3a2a1
Case 2: a4a2a3a1
Case 3: a4a1a3a2
Case 4: a3a2a4a1
Case 5: a3a1a4a2
Case 6: a2a1a4a3
where a4, a3, a2 & a1 satisfy equations (1).
All cases lead to a contradiction except Case 5, which yields the result 6174.
Case 1: a3=b=a3~a2-1 hence a2 = -1, contradiction.
Cases 2 & 4: a1=d=a1-a4+10 hence a4=10, contradiction.
Case 3: a4=a=a4-a1 hence a1=0
a1=b=a3-a2-1 hence a3=a2+1
a3=c=a2-a3+9 hence 2a3=a2+9
Hence 2a2 +2 = a2+9 hence a2=7
a2=d=a1-a4+10 hence a4=3 < a2, contradiction.
Case 6: a2=a=a4-a1 hence a4=a1+a2
a1=b=a3-a2-1 hence a1+a2=a3-1
a4=c=a2-a3+9 hence a3+a4=a2+9
a3=d=a1-a4+10 hence a4=a1-a3+10
Substituting for a4 from the first equation into the last two equations gives
a3+a1+a2=a2+9 hence a1+a3=9
a1+a2=a1-a3+10 hence a2+a3=10
Hence the second equation gives 9-a3+10-a3= a3-1 hence 3a3 = 20 hence a3=20/3, contradiction.
Case 5: a3=a=a4-a1 hence a4=a1+a3
a1=b=a3-a2-1 hence a1+a2=a3-1
a4=c=a2-a3+9 hence a3=a2-a4+9
a2=d=a1-a4+10 hence a4=a1-a2+10
Substituting for a4 from the first equation into the last two equations gives
a3=a2-a1-a3+9 hence 2a3=a2-a1+9
a1+a3=a1-a2+10 hence a3=10-a2
Hence 20-2a2=a2-a1+9 hence 3a2=11+a1
Hence the second equation gives a1+a2=9-a2 hence 2a2 = 9- a1
Hence 27-3a1 = 22+2a1 hence 5a1=5
Hence a1=1, a2=4, a3=6, a4=7.
Hence Kaprekar's constant = a3a1a4a2 = 6174.
Again, thank you for your lovely videos and i look forward to seeing the next one. God bless you ❤
Man, you are good
Very elaborate working. Is this the way mathematicians really work .Oh my God.
Mind boggling! True logical, mathematical mind . Hats off for proving 6174 is constant other way 😊
My brain cannot handle your comment
Felicitări pentru democrație. ❤
I wish you had been my maths teacher ....your enthusiasm would have made the subject I loathed most exciting and interesting.... You are so likeable ....
Thanks for enlighting this very interesting subject.
I made a few tests on an excel sheet with 2, 3 and 5 digit numbers by curiousity and got the interesting following results !
I hope you like my inquiry and study, tell me if you like it !
>For 2 digit numbers, the Kaprekar iteration process doesn't converge, but gives a cyclic set of following numbers after 7 iterations max:
9, 81, 63, 27, 45 and 9 again and so on.
>For 5 digit numbers, the Kaprekar iteration process doesn't converge, but has got a different constant value (depending on the intial value) repeated every 4 iterations.
The periodic behaviour seems to appear after 8 interations max.
>And last, with 3 digit numbers, the Kaprekar iteration process converges towards the value 495 and seems to after 7 interations max.
To finish with my post. There is a trivial result to be mentioned: The Kaprekar iteration process gives 0 as result for all intial numer having the same value for all digits.
For example, 2222 gives 2222 as "max" and "min" so the there difference is 0.
Greetings and keep up the good work !
He didn't have any degrees or anything. He was just curious about numbers and that's it! As an Indian I am deeply heartbroken that I haven't heard a single thing about him in my 16 years of studying maths. A real shame on my people for teaching system for teaching me who looted us , ruled us , and didn't even considered to talk about D.R.Kapreker. 😢
A wonderful maths game i never new this gentle man who invented this constant is from my country. And you are a good teacher
I am so excited, it is wonderful. Express how a great mathematician searched and concluded.
There is another famous Indian mathematician who was largely self-taught: Ramanujan. He did amazing things with repeating fractions and much much more. He went to England to study further but sadly died very young. Perhaps you could talk about something he solved in one of your videos?
Boy : tons of stuff,,,,, even not yet proven!
@@jceepf will never be proven by anyone, for that we will need to search for another ramanujan from india.
Yeah i would love a video on Ramanujan 's paradox
Really
But not about the infinite sum of natural numbers. Everybody know it’s bullshit still speaking about it only because it was from Ramanujan
Thank you so much for creating these videos! I LOVE math so much I decided to never stop learning it after college. You are so good with your teaching, and your enthusiasm is contagious. Keep up the good work! 👏👏
You are a very inspirational teacher. Your enthusiasm for your subject shines through.
randomly came across this video n was happy to see another maths teacher with such enthusiasm- the first one was my 5th grade math teacher!! hugs from India!!
Finally, a math teacher who can write legibly! Too many math people don't write very well, so it's often hard to tell exactly what it is that they've written. 👍
...especially when greek letters suddenly come out to play but you don't know their names, or have trouble remembering them all and the upper and lower cases, grrr!
Started with 2438. It took 8 steps =4176. Thanks for sharing.
@@duku3535
1. 8432-2348=6084
2. 8640-0468=8172
3. 8721-1278=7443
4. 7443-3447=3996
5. 9963-3699=6264
6. 6642-2466=4176
7. 7641-1467=6174
Took me 7 steps, not 8.
Given any 4-digit number abcd where a>b>c>d, the first subtraction yields the expression 900(a-d) + 90(a+b-c-d) + 9(a-d). This produces a number of the form 9(efe) or 9(efg) where e=g+1. The second of these cases occurs if a+b is 10 or more than c+d, because of the carry-over term. The above expression is used again with the new digits obtained from the first subtraction, and in general, it indicates that the total number of subtractions to get to 6174 is a function of the differences between the digits of the starting number. Studying the relatively straightforward algebra behind the process can take the magic away!
According to Wolfeam & Wikipedia
Consider an n-digit number k. Square it and add the right n digits to the left n or n-1 digits. If the resultant sum is k, then k is called a Kaprekar number. For example, 9 is a Kaprekar number since
9^2=81 8+1=9,
and 297 is a Kaprekar number since
297^2=88209 88+209=297.
Sir, I love your dramatic pauses, and your smile when teaching the subject. It is so infectious and fun to watch. 👍🏼🙏🏼
Man, you are the best. I love your lessons.
I’ve spent the las forty years of my life hating math with a passion, yet this video has done the impossible, it’s gotten me interested in math!
I love it! Now I want to write a program hack to search the n-digit numbers for their endpoints and rates of convergence..... Thanks for the tip!
And can it work in different bases? and ...
Would be interesting to see how that works in binairy. Maximum and minimum only depend on the total number of ones and total number of zeros. I wonder if it can be generalized for any base and any size analytically. And I have to figure out how it works in balanced ternary!
@@jahbiniThe biggest and smallest numbers generally won't be the same in other bases. They may not even have the same number of digits.
There might be other constants in other bases though.
Nice idea!
I thank this teacher. His language and explanation are fantastic.
Amazing 6174
A very interesting result.I tried it on some random four digit number , and after a few steps I reached Kaprekar's constant.
Charming presentation. And fun. Thanks from an old engineer. My dad used to joke "You learn something new every day if you're not careful!"
I came to know about Kaprekar's constant couple of years back.
Thank you for covering this art of mathematics.
Also, if I remember correctly, Numberphile made a video about most iterations required for a number to reach 6174 or like that.
Brilliant communicator explaining something very, very interesting, and always with a smile on his face. Thank you very much Sir!
I was born in the year l96l.
Fun fact: even if you turn this number upside down it reads the same. 😄 By the way, great content. Glad to know about Mr Kaprekar. Never heard about him though I live 4 hours drive from where he lived.
Mathematics scares me, but your calmness is so inviting that it has actually sparked a curiosity in me about math. Teaching is an art, and you, sir, are an artist.
Being from Maharashtra, i don't know about him and his contribution, thankyou for bringing this 🙏Jai Maharashtra
Politicians are more important than genuine genius people like these...
That was enjoyable and refreshing.
Finally I found someone to guide me through the world of numbers.
And thanks for making this easy to understand and pleasurable to watch.
Interesting that. I've personally never heard of Kapreka's Constant so thanks for introducing me to this phenomenon.
Amazing! Thank you for introducing me to this concept. Truly fascinating. I used to hate Maths at school but am absolutely in awe of it now. Thanks for the video and stay happy.
Excellent lesson professor. I had no idea about this mathematical ‘wonder’.👏👏👏👏
Coolest mathematics teacher.... Thank your sir for teaching something new... I belong to the state of kaprekar sir and now i m wondering why this theory was not in our school syllabus... This is the first time i am getting to hear about kaprekar constant...
Thank you sir for teaching something new..
Wow that black board and clean writing on it. You reminded me my school day teacher. Coincidentally he was teaching us Maths. I felt nostalgic for your writing and was amazed for this constant.
Thanks for the knowledge. It inspired me to write an algorithm to test every number from 0001 to 9999.
There were lots of numbers that took 7 turns to arrive at the constant. Many took just one turn like 0026, 0062, 0136 and none took more than 7 turns. Of course numbers with four identical digits were not tested. Just for the record the amount of numbers taking one to seven turns are 383 | 576 | 2400 | 1272 | 1518 | 1656 | 2184. Meaning most numbers took 3 turns to arrive at the constant.
A recreational genius.
I never seen a video for maths, but today youtube suggested this video and i watched. its really a mind blowing fact and good to know. thanks to the creator of this video
I am from India. I am not a big fan of Mathematics. However seeing the video title I got curious. It is simply amazing...
(Because being an Indian whenever we think of Maths we only recollect Ramanujam a great wizard of Mathematics).
Beautiful!! ❤❤❤Makes me proud as an Indian and I haven't even heard about this!!
Actually, studying the numbers XY00 with X≥Y (with the help of Excel…), I found that 4100, 5100, 5200, 6100, 8500, 9400, 9500 and 9600 were the numbers of this kind needing 7 steps.
Funny enough, 6200 goes directly to 6174 (in 1 step), and is the only one of the kind.
Thanks for your interesting videos !
Very happy that you made a video on it Kaprekar was a school teacher but his love for mathematics is worth commendable
I am from Maharashtra, I am Marathi and yet I had never heard about him before.. thanks for introducing him, the constant theory is interesting
Magnificent, thank you! Teaching is not only to show the way but to open the imagination to discovery!
Thank you Professor!
Congrats on 200K! 🥳🥳🎉
I like the way you speak. Very calm and pleasant to listen to.
i made a model in excel using the formula "=VALUE(CONCAT(SORT(MID(A1,SEQUENCE(LEN(A1)),1),,1)))" and "=VALUE(CONCAT(SORT(MID(A1,SEQUENCE(LEN(A1)),1),,-1)))"i run it from 1000 to 9999 , it was fun to see which numbers to reach 7 times
good sir
I wonder if you can create an X, Y graph of the results that shows the number of steps it takes to get there - perhaps showing points of different colours according to the number of steps to get to 6174 - EG 1 step = red, 2 = orange, 3 yellow etc. I wonder what that would look like?
Thoroughly enjoyed this. Beautifully clear writing. Appreciate that!!
From the second step onwards, both the resulting number and the numbers generated in each rearrangement (ascending and descending) all have a digit sum of 18. This indicates that the process has entered a stable cycle where all results maintain a digit sum of 18, eventually leading to 6174.
I didnt notice that it does add up to 18=9 . The magic nine.
Happy to see someone admire the beauty of mathematics. Love from India, brother.
I never heard this number. Thanks for your energetic teachings.
I'm not a mathematician or student, but what a wonderful and informative video. It certainly sparks your brain into thinking differently. Thank you good sir!
Absolutely delighted to find inspiration that, would fuel my liking for numbers.
For the first time in my life I came across such a wonderful number. Thanks for video... 🙏🙏🙏
just found this young man, very clear explanation! Cool.
Amazing!!!! I am not a mathematician but I thoroughly enjoyed your presentation, Sir! Also, God bless the soul of that simple, but great man, Pandit Kaprekar! ❤❤❤🙏🙏🙏🪷🪷🪷🙇♂️
From one who loves unique math problems, this is instructive and fun. Works with 495,too, just use any 3 digit number.
Thanks for introducing me to something new. I am passing it on. One 7th grade relative likes it and said he’ll show it to his math teacher tomorrow.
You have great charm and enthusiasm in your way of demonstrating and presenting. I always hated mathematics and assumed I was just plain thick - until doing a course as an adult with a great teacher and realised I was enjoying it. It wasn't me it was just careless and cold instruction - the subject itself was fine. Great and interesting stuff here - I'll be checking out more of your videos.
That was amazing - and I’m a bit numberphobic. Your enthusiasm is contagious
Hi dear im from Kaprekars district but i heard from you about him thanks keep your hardwork continue...
It's simplicity of Vedic Mathematics where one plays with Number...
And becomes Ganitanand or Recreational Mathematician
"I didn't used to write my 9s like that"
Love that
Very refreshing delivery
Thank you
O wooooow....!!!!
I am from Mumbai, India but I wasn't aware of this fact .
Great to know about a great Maharashtrian mathematician.
Oh my god. Indeed he is great mathematician. He used to stay in the same city where i live. It is Donbivli in Thane district near Mumbai. We didn't know about him initially but few years back on railway platofrm wall painitng under the title of great human beigns from our city I saw his photo. Really got shocked that he was such a great mathematician and from the same city❤🎉
Learnt something new today. Thanks for the video
I love it!! My father mentioned this to me years ago, but I obviously didn`t pay attention. I have since become much more interested in numbers and mathematics. Thanks so much for putting this up.
Try to draw a directed graph (vertex/edge kind, not the plot) with all 1000 numbers as vertices and arrows pointing to the next vertex. Maybe start with two digit numbers to try the idea out first.
I bought this number 6174 and struck the lottery.
Now I'm a millionaire and enjoying my life. 😊
The number that will take the whole 7 steps is 3148
Thank You Prime Newton for this new information (for me), amazing. I did not know since many many years.
There is a Fibonacci set 2,3, 5 and 8. Wondering if this is another coincidence.
That's interesting 🤔
This Kapreker’s Mathematical Constant of 6174 is freaky! I never knew about this. Thank you for sharing it!