Anyone explain the mystery of the number 2178?

Question

Alright. I've been trying to figure this out for years. I'm hoping to gain some insight into this unique number.
Let me set it up.
Take a number, any number (4 digits is good, because so is the number I'm going to be working with, but it works with any).
I'll use 2349 (picked at random).
Now, because the number is positive, we'll subtract. We are going to subtract the digits in opposite order. So:
2349-9432 = -7053
Now that the number is negative, we will ADD the digits in opposite order. We'll continue to do this until we notice a pattern. Bear with me.
-7053 + 3507 = -3546
-3546 + 6453 = 2907
2907 - 7092 = -4185 (etc, let me just give you the answers now that you know the process)
1629, -7632, -5265, 360, 279, -495, 99, 0
That's my point. When this process is used, nearly every number imaginable goes to 0. Try it yourself!

EXCEPT 2178 and it's derivatives:
2178, -6534, -2178(!!), 6534, 2178(!!)

Do you see what I mean? Does that make sense?
Can anyone fill me in on why it's unique?

@ Rich Z: My point is that it DOES go to zero for MOST numbers, my question is why isn't that the case for 2178?

@Kyle: What doesn't the number 8723 work? According to your theory, it fits a working number doesn't it?
8 = 7+1 Check
2 = 3 - 1 Check
|7 - 2| = |8 - 3| => |5| = |5| Check
But when you go through it it fails immediately:
8723 - 3278 = 5445 -5445 = 0

@ McFate: I did my reverse digits differently than you did, I think
When I had a zero at the end of a number, when I did the next thing, I left it off. Example:
6510 - 0156 (I treated the 0 as an irrelevant placeholder and did 6510 - 156).
I believe you wrapped the zero around?
QUOTE:
"v=360, subtracting rv=630
v=-270, adding rv=720"
I'm sure that's the reason we're coming to different conclusions about how many cycle and how many don't. Though I'm not sure the ramifications of that difference.
I'd be VERY interested to see your program though!

Also @ McFate: Here's a big difference. You restrained yourself to 4 digits when you started with 4 digits and I didn't. At one of your examples, you got to
QUOTE : "v=999, subtracting rv=9990"
At which point I considered it going to zero:
MY WORK: 999 - 999 = 0
I started at a 4 digit number, but did not require it to stay 4 digits, if my return was only 3 or 2.

One more @ McFate: WOW. Just wow. I do have just one more question. If they don't cycle, but don't go to zero, what happens to them?

Sorry, I misread what you had said. I understand.
That's really cool. Thank you for that answer.
Now all I need to do is figure out WHY! Haha. I told you 2178 was special. It and 6534 are the only true cyclical numbers. I think your answer plus a combonation of Kyle's answer might be what we're looking for.

Answer 1

Okay, switching to your algorithm, though one modification is that I take absolute value and then always subtract the reversed digits (same net effect but don't have to worry about negative numbers being treated differently).

(1) all one-digit numbers go to zero immediately (1 - 1 = 0, 2 - 2 = 0, etc.)

(2) all two-digit numbers go to zero because they immediately become a multiple of 9. The number 10A+B becomes (10A+B - 10B+A -> 9A-9B, which is 9(A-B)). All two-digit multiples of 9 reduce to a single digit (9) eventually, which per above immediately becomes 0. (90 -> 81 -> 63 -> 27 -> 45 -> 9 -> 0; any reversal of those two digits starts in the same place, e.g., 36 starts where 63 is listed).

(3) All 3-digit numbers go to zero since they immediately become multiples of 99
(100A + 10B + C) - (100C + 10B + A) =
99A - 99C = 99(A-C)
This works the same way as the 9's but with an extra digit in the middle: (990 -> 891 -> 693 -> 297 -> 495 -> 99 -> 0; any reversal of those three digits starts in the same place, e.g., 396 starts where 693 is listed).

(4) Four-digit numbers go from (1000A + 100B + 10C + D) to (999A + 90B - 90C - 999D), or a multiple of 90(B-C) and a multiple of 999(A-D). Since they're not a single multiple of a difference of digits, there are a lot more cases to consider.

However, it does funnel 9000 starting numbers (1000 to 9999 inclusive) through only about 171 (9*19) numbers at the second step -- [0 to 9 times 999] plus [-9 to 9 times 90].

The reason 2178 always shows up eventually for any number that doesn't reduce to zero, is that of those 171 numbers, only 12 of them cause a cycle, and all those cycles end in 2178 - 6534 - 2178 - 6534. So, really 6534 and 2178 are the only true cycle among four-digit numbers, and there are other numbers which feed into that cycle.

6*999 + 6*90 <-> 2*999 + 2*90

Here are the 12:
[1089] 1089-9801; 8712-2178; 6534-4356; 2178-8712; [6534]
[1188] 1188-8811; 7623-3267; 4356-6534; 2178-8712; 6534-4356; [2178]
[2178] 2178-8712; 6534-4356; [2178]
[3267] 3267-7623; 4356-6534; 2178-8712; 6534-4356; [2178]
[3366] 3366-6633; 3267-7623; 4356-6534; 2178-8712; 6534-4356; [2178]
[4356] 4356-6534; 2178-8712; 6534-4356; [2178]
[4455] 4455-5544; 1089-9801; 8712-2178; 6534-4356; 2178-8712; [6534]
[5544] 5544-4455; 1089-9801; 8712-2178; 6534-4356; 2178-8712; [6534]
[6534] 6534-4356; 2178-8712; [6534]
[6633] 6633-3366; 3267-7623; 4356-6534; 2178-8712; 6534-4356; [2178]
[7623] 7623-3267; 4356-6534; 2178-8712; 6534-4356; [2178]
[8712] 8712-2178; 6534-4356; 2178-8712; [6534]

A fair number of four-digit numbers DON'T go to zero -- 637 of them, in fact. Here are all the ones which cycle just between 1,000 and 2,000:

[1012] 1012-2101; 1089-9801; 8712-2178; 6534-4356; 2178-8712; [6534]
[1023] 1023-3201; 2178-8712; 6534-4356; [2178]
[1034] 1034-4301; 3267-7623; 4356-6534; 2178-8712; 6534-4356; [2178]
[1045] 1045-5401; 4356-6534; 2178-8712; 6534-4356; [2178]
[1067] 1067-7601; 6534-4356; 2178-8712; [6534]
[1078] 1078-8701; 7623-3267; 4356-6534; 2178-8712; 6534-4356; [2178]
[1089] 1089-9801; 8712-2178; 6534-4356; 2178-8712; [6534]
[1100] 1100-11; 1089-9801; 8712-2178; 6534-4356; 2178-8712; [6534]
[1122] 1122-2211; 1089-9801; 8712-2178; 6534-4356; 2178-8712; [6534]
[1133] 1133-3311; 2178-8712; 6534-4356; [2178]
[1144] 1144-4411; 3267-7623; 4356-6534; 2178-8712; 6534-4356; [2178]
[1155] 1155-5511; 4356-6534; 2178-8712; 6534-4356; [2178]
[1177] 1177-7711; 6534-4356; 2178-8712; [6534]
[1188] 1188-8811; 7623-3267; 4356-6534; 2178-8712; 6534-4356; [2178]
[1199] 1199-9911; 8712-2178; 6534-4356; 2178-8712; [6534]
[1210] 1210-121; 1089-9801; 8712-2178; 6534-4356; 2178-8712; [6534]
[1232] 1232-2321; 1089-9801; 8712-2178; 6534-4356; 2178-8712; [6534]
[1243] 1243-3421; 2178-8712; 6534-4356; [2178]
[1254] 1254-4521; 3267-7623; 4356-6534; 2178-8712; 6534-4356; [2178]
[1265] 1265-5621; 4356-6534; 2178-8712; 6534-4356; [2178]
[1287] 1287-7821; 6534-4356; 2178-8712; [6534]
[1298] 1298-8921; 7623-3267; 4356-6534; 2178-8712; 6534-4356; [2178]
[1320] 1320-231; 1089-9801; 8712-2178; 6534-4356; 2178-8712; [6534]
[1342] 1342-2431; 1089-9801; 8712-2178; 6534-4356; 2178-8712; [6534]
[1353] 1353-3531; 2178-8712; 6534-4356; [2178]
[1364] 1364-4631; 3267-7623; 4356-6534; 2178-8712; 6534-4356; [2178]
[1375] 1375-5731; 4356-6534; 2178-8712; 6534-4356; [2178]
[1397] 1397-7931; 6534-4356; 2178-8712; [6534]
[1408] 1408-8041; 6633-3366; 3267-7623; 4356-6534; 2178-8712; 6534-4356; [2178]
[1430] 1430-341; 1089-9801; 8712-2178; 6534-4356; 2178-8712; [6534]
[1452] 1452-2541; 1089-9801; 8712-2178; 6534-4356; 2178-8712; [6534]
[1463] 1463-3641; 2178-8712; 6534-4356; [2178]
[1474] 1474-4741; 3267-7623; 4356-6534; 2178-8712; 6534-4356; [2178]
[1485] 1485-5841; 4356-6534; 2178-8712; 6534-4356; [2178]
[1507] 1507-7051; 5544-4455; 1089-9801; 8712-2178; 6534-4356; 2178-8712; [6534]
[1518] 1518-8151; 6633-3366; 3267-7623; 4356-6534; 2178-8712; 6534-4356; [2178]
[1540] 1540-451; 1089-9801; 8712-2178; 6534-4356; 2178-8712; [6534]
[1562] 1562-2651; 1089-9801; 8712-2178; 6534-4356; 2178-8712; [6534]
[1573] 1573-3751; 2178-8712; 6534-4356; [2178]
[1584] 1584-4851; 3267-7623; 4356-6534; 2178-8712; 6534-4356; [2178]
[1595] 1595-5951; 4356-6534; 2178-8712; 6534-4356; [2178]
[1606] 1606-6061; 4455-5544; 1089-9801; 8712-2178; 6534-4356; 2178-8712; [6534]
[1617] 1617-7161; 5544-4455; 1089-9801; 8712-2178; 6534-4356; 2178-8712; [6534]
[1628] 1628-8261; 6633-3366; 3267-7623; 4356-6534; 2178-8712; 6534-4356; [2178]
[1650] 1650-561; 1089-9801; 8712-2178; 6534-4356; 2178-8712; [6534]
[1672] 1672-2761; 1089-9801; 8712-2178; 6534-4356; 2178-8712; [6534]
[1683] 1683-3861; 2178-8712; 6534-4356; [2178]
[1694] 1694-4961; 3267-7623; 4356-6534; 2178-8712; 6534-4356; [2178]
[1705] 1705-5071; 3366-6633; 3267-7623; 4356-6534; 2178-8712; 6534-4356; [2178]
[1716] 1716-6171; 4455-5544; 1089-9801; 8712-2178; 6534-4356; 2178-8712; [6534]
[1727] 1727-7271; 5544-4455; 1089-9801; 8712-2178; 6534-4356; 2178-8712; [6534]
[1738] 1738-8371; 6633-3366; 3267-7623; 4356-6534; 2178-8712; 6534-4356; [2178]
[1760] 1760-671; 1089-9801; 8712-2178; 6534-4356; 2178-8712; [6534]
[1782] 1782-2871; 1089-9801; 8712-2178; 6534-4356; 2178-8712; [6534]
[1793] 1793-3971; 2178-8712; 6534-4356; [2178]
[1815] 1815-5181; 3366-6633; 3267-7623; 4356-6534; 2178-8712; 6534-4356; [2178]
[1826] 1826-6281; 4455-5544; 1089-9801; 8712-2178; 6534-4356; 2178-8712; [6534]
[1837] 1837-7381; 5544-4455; 1089-9801; 8712-2178; 6534-4356; 2178-8712; [6534]
[1848] 1848-8481; 6633-3366; 3267-7623; 4356-6534; 2178-8712; 6534-4356; [2178]
[1870] 1870-781; 1089-9801; 8712-2178; 6534-4356; 2178-8712; [6534]
[1892] 1892-2981; 1089-9801; 8712-2178; 6534-4356; 2178-8712; [6534]
[1903] 1903-3091; 1188-8811; 7623-3267; 4356-6534; 2178-8712; 6534-4356; [2178]
[1925] 1925-5291; 3366-6633; 3267-7623; 4356-6534; 2178-8712; 6534-4356; [2178]
[1936] 1936-6391; 4455-5544; 1089-9801; 8712-2178; 6534-4356; 2178-8712; [6534]
[1947] 1947-7491; 5544-4455; 1089-9801; 8712-2178; 6534-4356; 2178-8712; [6534]
[1958] 1958-8591; 6633-3366; 3267-7623; 4356-6534; 2178-8712; 6534-4356; [2178]
[1980] 1980-891; 1089-9801; 8712-2178; 6534-4356; 2178-8712; [6534]

Answer 2

2178 - 8712 = -6534
-6534 + 4356 = -2178
-2178 + 8712 = 6534
6534 - 4356 = 2178

It's because the first two digits are consecutive, decreasing and the second two are consecutive, increasing and the difference between the 1st and 4th digits and 2nd and 3rd digits are the same (6). So you will just keep cycling.

All of your "derivatives" follow that pattern (ignoring sign)

4356 follows that pattern too
4356 - 6534 = -2178

So does
6578
6578 - 8756 = -2178

9812 will work too
9812 - 2189 = 7623
7623 - 3267 = 4356
4356 - 6534 = -2178

So conclusion...
If this is your number ABCD
And these properties are true:
A = B + 1
C = D - 1
|B - C| = |A - D|
Then the pattern will follow.

edit:
You are right. I'll think about it.
Although I think this is a great question, I do want to point out that I have at least proven that 2178 and it's derivatives is not unique.

Answer 3

God does not exist because God can do and creat anything or everything God created life for what reason then. We cant help solve or destroy God so all we can do it pleasure God right And if life is for God pleasure anything goes = chaos world Why did God create Life? The only reason is nothing else except pleasure otherwise what can you think of. You say we need to worship God to get into heaven but why worship something which created something you had no choice or didnt ask for. Did I ask to be born? No So why should I be thankfull for life. Yes I am thankfull for life because I know what it feels like to be alive but if I were not alive in the first place I couldnt not even exist The only mystery is that its 2008 and people still frow away a percentage of their free will into religion. Life is an element to this Univese The Universe is built on negatives, neutral, positives infinity, probailities, and spontaneous hence you can never write the unverse in the book and knowlegde about it is infinity it was not created nor will it end. Example (Your life) Positive- Your born Neutral- You live Negative- You die Spontaneous- Your life appear from nowhere Probabilities- Events in your life and its outcome Infinity-The atoms which make up 100% of you have and will always be here

Answer 4

I don't have a perfect explanation, but check this out:

2178/2=1089.

1089*1=1089
1089*2=2178
1089*3=3267
1089*4=4356
1089*5=5445
1089*6=6534
1089*7=7623
1089*8=8712
1089*9=9801

If you notice 1089 is 9801 backwards. 2718 is 8712 backwards, etc.

Answer 5

There are a lot of interesting number problems like that.
I don't know the answer to that one but for your interest check out this

http://www.durangobill.com/Ramanujan.html

Read "The Man Who Knew Infinity"

It is unique because of what you showed it to do?...It may not be unique otherwise?...just a guess

Also get books by David Wells, a dyed in the wool number nut and a great guy to boot, you gotta love this guy if the only number you like is the sqrt of your age...LOL

Answer 6

It might have to do with the ubiquitous number 22, and 100 minus 22 is 78.
21 is off by one from 22.
Just a thought...

Answer 7

That use of add/subtract in reverse sequence is equivalent to subtracting it from itself and that will always get you to zero.

Answer 8

I get what u mean but i dont understand the problem.sorry. I would try a google it.