math mystery - the number 495

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these 2 numbers - 495 and 6174 and known as "Kaprekar's Constants" named for the famous Indian mathematician of that name

Good luck to anybody trying to figure this one out

this will work for ALL 3 digit numbers except when they are the same number such as 000 or 111 or 222 etc.

1. Think of a 3-digit number where not all the digits are the same.
2. Arrange the digits in descending order and ascending order from left to right. This will give you two 3-digit numbers — one the smallest number that you an form from the digits, and the other the largest.
3. Subtract the smaller number from the larger number.
4. Go back to step 2

If you repeat this process over and over again, you will end up with 495!
You don’t believe it? Let’s have 3 examples:

Example 1
Our number is 592.
Arranging it in descending and ascending number, we have 952 and 259.
Subtracting 259 from 592 we get 693.
Repeating steps 2-3, we have 963 -369 = 594
Repeating steps 2-3 again, we have 954 – 459 = 495
Notice: If the difference contains the digits 9, 5, and 4, you will end up with 495 in each repetition.


Example 2
Our number is 466
Arranging in ascending and descending we have 664 and 466.
Subtracting 466 from 664 we have 198.
Repeating steps 2-3, we have 981 – 189 = 792.
Repeating steps 2-3, we have 972 – 279 = 693. And this is the same as the number 3 in Example 1 above, which will end up in 495.


Let’s have 990.
Arranging the digits in ascending and descending, we have 990 and 099 or simply 99.
Subtracting, we have 990 – 99 = 891.
Repeating steps 2-3, we have 981 – 189 =792. And this is the same as difference in Example 2 step 4. This means that it will also end up in 495!

The number 495 is the Kaprekar constant for three digit numbers. Yes, if you choose any 3-digit number, you will end up in 495 (try other numbers). The more famous Kaprekar constant is 6174 for 4-digit numbers.


here is a vid which might make it easier to follow the mystery of these 2 numbers:






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I guess it's kind of interesting. Obviously, props to Kaprekar for figuring it out; although, I can't see how this would ever be useful except for some kind of prop bet.

The way it works out is that you can't avoid odd numbers and because the middle digit is always going to get changed to a nine for the next number. The reason that it's always going to get changed to a 9 is because the middle digits from whatever starting number are going to be the same, but you will always have to, "Carry a ten," to get the terminal number to subtract because the terminal bottom number (ascending order), by necessity, will be greater than the terminal number when you do it in descending order.

Anyway, the trick is that the numbers are being forced into descending order on top and ascending order on the bottom, which limits the possibilities for what the digits for the next number can be...and also means the new number in the middle position will be forced to be a nine.

The four digit one is some variation of what I said above.

Anyway, when you know THAT it works, the why it works isn't hard to figure out. It's nevertheless really impressive that Kaprekar figured out THAT it works.

I thought this was going to be a thread about craps probabilities!

It makes sense.
495 is the Federal designation for the Long Island Expressway, which starts at the mid-Town tunnel in NYC.,aka The Center of The Universe.

Quote: lilredrooster

this will work for ALL 3 digit numbers except when they are the same number such as 000 or 111 or 222 etc.

1. Think of a 3-digit number where not all the digits are the same.
2. Arrange the digits in descending order and ascending order from left to right. This will give you two 3-digit numbers — one the smallest number that you an form from the digits, and the other the largest.
3. Subtract the smaller number from the larger number.
4. Go back to step 2

If you repeat this process over and over again, you will end up with 495!

(1) The number is 100a + 10b + c, where a, b, and c are the digits
(2) This is the same as limiting the number in (1) to a <= b <= c; the two numbers are 100a + 10b + c and 100c + 10b + a
(3) (100c + 10b + a) - (100a + 10b + c) = 100 (c - a) + (a - c) = 99 (c - a)
Since the digits are all different, c > a
There are 9 such numbers: 099, 198, 297, 396, 495, 594, 693, 792, 891, and 990
With 297 and 792: 792 - 297 = 495
With 396 and 693: 693 - 396 = 297, which becomes 495
With 198 and 891: 891 - 198 = 693, which becomes 297, which becomes 495
With 099 and 990: 990 - 99 = 891, which becomes 693, which becomes 297, which becomes 495

I already knew ThatDonGuy is much smarter than I am, so I remain satisfied with my response.

Thanks for the excellent explanation!

Quote: billryan

It makes sense.
495 is the Federal designation for the Long Island Expressway, which starts at the mid-Town tunnel in NYC.,aka The Center of The Universe.

But yet., the 495 area code is undefined.

Quote: DRich

But yet., the 495 area code is undefined.

(918) 495-6174 goes to one of the numbers for the College of Nursing at Oral Roberts. I wouldn't normally list an actual phone number, but that's a business anyway and is publicly listed on their website.

Quote: Mission146

(918) 495-6174 goes to one of the numbers for the College of Nursing at Oral Roberts. I wouldn't normally list an actual phone number, but that's a business anyway and is publicly listed on their website.

Don't mess with them. Oral knows people in high places.