The Lenny Conundrum round 228? Help me!?

Question

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The Altadorian Archives are the repository of all Altadorian knowledge, science, and history. Thousands of books and scrolls populate its shelves. Amongst these books is the huge, multi-volume Encyclopedia Altadoria, volumes 1 through 9950.
The archivist misplaced his shopping list inside one of the books. He has a handy way to remember which volume he placed it in, though. The volume number was a four-digit number, in which all four digits were unique. If you take the largest possible four-digit number that can be made by rearranging the four digits, and you subtract from that the smallest possible four-digit number that can be made by rearranging the four digits, the number you get is volume number, but with the digits in reverse order.
In what volume of the Encyclopedia Altadoria did he leave his shopping list?
You don't have to give me the answer, I would just like some help

Answer 1

I don't see how that last person thought they were trying to help... the examples that he gave didn't have the criteria for an answer and was just random stuff.

Here's how I look at it:
We get a 4 digit number with digits with decreasing value: for example 7531.
Then we reverse it (for the smallest possible) and take it away from the larger one:
The result of which has to have all of the digits that the original number did... no more, no less, no difference in digits

Another way of looking at it is this:
wxyz - zyxw = any combination of w, x, y or z

We can automatically say that 7, 8 and 9 can't be the LOWEST. 8 and 9 can't be the second lowest. 9 can't be the third lowest.
so can't be (if in ascending value): 7xxx, 8xxx, 9xxx, x8xx, x9xx, xx9x
2, 1 and 0 can't be the HIGHEST. 2 and 1 can't be the second highest. 2 can't be the third highest.
so can't be(if in ascending value): xxx0, xxx1, xxx2, xx1x, xx2x, x2xx

The 2 numbers in the middle cannot be within 1 of each other otherwise you'll end up with a 0 in there because of the fact the first 2 numbers of the largest one will also be the last 2 numbers of the smallest one, that means that they have to be bigger meaning that you need to "borrow" from column 2+3

Now any number that has a 0 in it means that the first digit of the smallest number possible has to be a zero. Meaning that the highest will stay the same. Which tells us that the result of any other part of it cannot equal the highest.

Hope that helped!

Answer 2

subtracting the smallest 4 digit non repeating number from the largest possible number using the same unique 4 digits results in several fixed answers like.

9081, 4086, 3087, 5175, 6174 etc... try a few 4 digit numbers for yourself.

Now notice the pattern the first and last digit add up to 10 and the middle 2 add up to 8.

Hope that helps.

Answer 3

Let the 4 digits be a, b, c, d

..abcd
- dcba
---------
= pqrs
---------

Since the total (pqrs) must equal 9950 you can figur eout the following:
s = d - a + 10
r = c - b + 9
q = b - c - 1
p = a - d

s + p = 10
r + q = 8

2 of the digits add up to 8. They can be
0+8
1+7
2+6
3+5

The other 2 add up to 10. They can be
1+9
2+8
3+7
4+6

So there are 16 possible cases. Since the 4 digits are unique (non matching) we can eliminate a few more cases, thus are left with 10 possible cases.
9810, 8730, 8721, 7641, 9621, 7632, 9531, 6543, 8640, 8532

Try them one by one.
0,8,1,9

9810 - 0189 = 9621 (wrong; the answer must contain 0,8,1,9)

0,8,3,7
8730 - 0378 = 8352 (wrong; the answer must contain 0,8,3,7)

...

Finally we found the unique solution which is
1,7,4,6

7641 - 1467 = 6174 (Right; answer contains 1,7,4,6)

Now reverse the 6174...
The shopping list is in volume 4716

Answer 4

I think there are just loads and loads of possibilities, but somehow you have to choose one. I think there's probably a clue in the question like there sometimes is (e.g Lenny Conundrum Round 225) but so far I haven't found one yet.

The bit where it said "The volume number was a four-digit number, in which all four digits were unique", UNIQUE might be the clue.

Answer 5

how approximately some clues the main important unique variety is 9876 the smallest unique variety is ____ (unusual,even,unusual,even) subtract and you get ____(all even unique numbers) this in opposite is ____