MAGICAL Indian Math Discovery - Numbers 495 and 6174 (Kaprekar Constants)

The Indian mathematician Dattaraya Ramchandra Kaprekar discovered the number 6174 is reached after repeatedly subtracting the smallest number from the largest number that can be formed from any four digits not all the same. The number 495 is similarly reached for three digit numbers.

Kaprekar routine

Mysterious Number 6174

Magic Number 495

Number 6174 and 495

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[chrisg]

What's the connection between 495 and 6174? Add 495 to its reverse, 594, and you get that other famously cool number 1089. Do the same to 6174 and you get 10890. But then 1089 + 9801 = 10890 too.

[karthikm]

If you are getting 495 in the end, its mathematically proven that death is the end because 4=d, 9=I and 5=e. You 495(die).

Great

[anshumanmisra]

Any of these mathematical mysteries will always have 9 as the fulcrum. Add the constants here and (4+9+5=18 or 1+8 = 9) or (6+1+7+4) and the result is 9. Magical 9.

[gausspapei]

Clearly this is an amazing result but it would be more interesting to make the proof of this and show the reason of this result..

[ymblcza]

For 5-digit numbers, there are 3 possible loops:

[74943, 62964, 71973, 83952]
[63954, 61974, 82962, 75933]
[53955, 59994]


For 6 digit numers, there are 1 loop and 2 'magic' numbers:

[851742, 750843, 840852, 860832, 862632, 642654, 420876]
[631764]
[549945]

The 851742 reminds me of 1/7

[themoagoddess]

something curious I noticed about this:

if you do the same sort of thing with two digit numbers, you get nine times the difference between the larger and smaller digit (ie. 64 and 46 become 18, 33 becomes 0, 90 and 9 become 81, etc)

and when I tried to do it with five digit numbers, I cycled back to my original set of digits(though I only tested this twice) here's the numbers I got:

41976(start)
82962
75933
63954
61974(this one bugged me considerably given Kaprekar's four digit constant)
82962(end)

82963(start)
74943
62964
71973
83952
74943(end)

[yuvalbg]

The real magic number is 73.
In the 73rd episode of The Big Bang Theory Sheldon explains that:
73 is the 21st prime number,
its mirror 37 is the 12th prime number which is also the mirror of 21, the multiplication of 7 and 3 is also 21, and in binary code 73 is a palindrome

[robertosamaniegoeconmusic]

So cool to see one of your older videos for the first time!

[stvp]

I wonder how mathematicians came up with the idea to try this

[ffggddss]

Note that for 3-digit numbers, the first step necessarily gives 99n, with n being one of 1 through 9, inclusive.
So maybe the "homing in" on 495 (which is one of those 9) isn't quite so remarkable. Still, that's pretty interesting.

For the 4-digit numbers, you'll always get some multiple of 999 (from the 2 outer digits) plus some multiple of 90 (from the 2 inner digits).
999m + 90n, where m and n are each between 1 and 9, inclusive.
That makes 9x9 = 81 possible results each time.
[CORRECTION: n can also be 0, because the two inner digits can be equal. This raises the possibilities to 9x10 = 90.]

The fact that the process, in both cases, always ends on a value that repeats forever, rather than a cycle of two or more that repeat, is remarkable; the fact that it's always the same final number, is even moreso.

For 2-digit numbers, it will just cycle around the first 5 odd multiples of 9, in the order: (81, 63, 27, 45, 09).

What does it do for 5-digit and longer numbers?

Fred

[gonza]

My question:
How do the numbers change in different bases

[sidhollander]

Doesn't it always work if you reverse the digits and subtract the smaller from the larger?

[ericveneto]

I assume the catch is: NO repeat digits?

[redthorne]

Weird, if you do 5 digit numbers, it kind of cycles endlessly though a series instead of sticking to one. Still, 9's everywhere though.

Also weird, 495*9 = 4455, and 6174*9 = 55566. If you try to reverse the process and do things like 666677, or 344, you get repeating decimals instead of whole numbers. I thought that had something to do with it, but I tried (same pattern, repeating digits with two digits one higher on the end, divisible by 9) and it cycles through about 7 or 8 numbers infinitely.

Well, screw it, I didn't need sleep tonight anyway.

Also, useless fact, 6174/495 is 686/55. Also, math is ultra weird. If you imagine 686 as a pattern of a number on each end, and the middle filled with a number two higher, then 55 is 5's with 0 7s between. 686 is 6's with 1 8 in between.

Here's why it's weird. 686/55 is 12.47272727 repeating.
If one less is 4's on both ends with "-1" 6 in the middle, just make it 4 (because you have to remove a number). 4/55 is 0.0727272727 repeating. 55/55 is obviously 1, 686/55 is that weird number, but hey...looks familiar. What if we do (686/55)-(4/55)? 682/55 is 12.4. Okay, keep moving up. 7997/55 is 145.4?

From here we have two possibilities. 80008 would be next? or is it 8(10)(10)(10)8? If it's the second, we have to assume it's 91108 (as if we're carrying the one, right?) Well...
80008/55 = 1454.690909 repeating (wait, that's almost exactly [7997/55]*10), 91108/55 = 1656.509091...
MORE...
911119/55 = 16565.8 (that's awfully similar to the second one above times ten...) The second version would be 1022219 (with carrying), so 1022219/55 = 18585.8...
One more...
10222230/55 (ends become 10, last 10 carries one over)? 185858.7273 (Oh...that's...also familiar...), and finally, (with full carrying) = 206060.5455

So there's this odd pattern where carrying with one is almost exactly 1/10 of the non-carrying version after it, each time raising from 1656... to 1858... to 2060...?

At this point it's not even math, it's just nonsense...but I love it. :D

[Abhisruta]

This is greatly appreciated discovery in India...

[stereolifestyle]

Why in every magical math facts this number "9" is always includes

[jaimeduncan]

I was expecting more discusión about this results, like : is it a property of decimal system or do this numbers exists for all the bases ? Does this work for any. Inner of digits ? (Clearly the numbers will change) etc

[avirup]

If we take any 2 digit no say 34 and subtract it with its reverse the ans we get in the first step will always same after 4 iterations with the preceding answers.

[allah___maadarchod]

3:15 it says to exclude 3 digit numbers where all numbers are same e.g. 111, 333 etc.
but if you take into consideration the numbers 100, 101, 110, 211, 221 then you can beat 495 magic like a boss with another magic number 99 😎
similarly you can beat 6174 with 999.

[TheRealKitWalker]

This happens with 45 as well. I too discovered this long ago 😊