This post is split into three parts because of Atmos's character limit. This is part 2.

Part 1: https://projectatmos.space/profile/1p8WCZnqqG6N3ZOsJxBgUTo/p1SJCxFjA6yCA52Hz

Part 3: https://projectatmos.space/profile/1p8WCZnqqG6N3ZOsJxBgUTo/p1mZOwFRzbxYBjidY

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The describers are always in ascending order, because I chose to pick the lowest number resulting from rearrangements and excluded all other rearrangements.

The describers need to add up to the length of the number, because you describe all digits exactly once, just grouped together by which digit it is.

To have the sum of the describers be the same as the length of the number, you could for example just have every describer be a 2. Your number would then be `2?2?2?2?…`

. This obviously does not work, because only the 2 occurs multiple times and the described digits occur once (unless it's also a 2), so you would need a 1 per described digit, which isn't there. And you would need a pretty high number to describe how many 2s there are. So this pattern only works in the case of `22`

.

If you want to deviate from the `2,2,2,…`

pattern, you need to always increase one describer while decreasing another, that way the sum stays the same. Since a describer 0 can't happen (a number with `09`

in it already includes at least one 9, right there), that means that you have to change a describer 2 to a describer 1 for every other describer you increase by 1. And if you increase a describer from e.g. 2 to 4, you need to change two 2s to 1s and so on. This quickly leads to a lot of describer 1s. Keep this "trade a 1 for a 3, two 1s for a 4, etc." mechanic in mind, I'll use it a lot from now on.

The most common describer (which is itself described by the last describer, because that's the biggest) can only be 1 or 2, because any bigger describer would need at least an equal amount of 1s as describers, which would make 1 the most common describer again. (2 actually can't be the most common describer either if the number has at least 10 digits, but the reason for that is complicated and not needed for this proof.)

Now let's look at the last two describers: They can't be `2,2`

, because the describers are sorted in ascending order, so all other describers would be either 1 or 2. Only 2s doesn't work, as shown above, and substituting some 2s for 1s wouldn't work, because the sum of all describers would be less than the length of the number. For the same reason the last two describers can also not be `1,2`

or `1,1`

.

The last two describers can only be `2,3`

if either the 1 or the 2 occurs exactly twice as describer (because 3 is the last describer and 1 or 2 must be the most common describer) and the other (2 or 1) occurs at most once as describer (because it's the second most common describer, no describer above 3 occurs, because 3 is the last describer). This already limits the number in length, because if 1 only occurs at most twice and 2 occurs at most twice, then the remaining describers can be at most two 3s or one 4, otherwise the sum would be higher than the length. So there are at most 6 describers and at most 12 digits. The 12 digit numbers can easily be tested with a program, which I did, the solution is `22, 12143133, 14212332, 1415223133`

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Public

1 year ago

(edited 1 year ago)

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