2023-11-22 WR

Rearrange the digits in ⟨125034⟩ to meet the rules below.

⟨5th 4th 3rd 2nd 1st 0th⟩

✅Match

⟨⋯ Perm(3,4,5) ⋯⟩

⟦1,4⟧ ∋ 0,3

3rd → a, 0th → b, ab=0

Jump(2,4) = 2

----- Information -----

🔲 「⟨⋯ Perm(3,4,5) ⋯⟩」

3,4,5 are adjacent.

144 permutations match this pattern.

Examples: ⟨345201⟩, ⟨345102⟩, ⟨102453⟩.

🔲 「⟦1,4⟧ ∋ 0,3」

The closed interval given by 1 and 4 contains 0, 3.

120 permutations match this pattern.

Examples: ⟨150342⟩, ⟨153042⟩, ⟨453012⟩.

🔲 「3rd → a, 0th → b, ab=0」

240 permutations match this pattern.

Examples: ⟨350142⟩, ⟨450321⟩, ⟨320541⟩.

🔲 「Jump(2,4) = 2」

There are 2 numbers between 2 and 4.

144 permutations match this pattern.

Examples: ⟨523041⟩, ⟨215430⟩, ⟨230451⟩.

#125034_v2.2

       ┌───┬───┬───┬───┬───┬───┐
       │5th│4th│3rd│2nd│1st│0th│▒
       ╞═══╪═══╪═══╪═══╪═══╪═══╡▒
Step 1 │   │   │ 0 │   │   │   │▒
       ├───┼───┼───┼───┼───┼───┤▒
Step 2 │   │ 2 │ 0 │   │   │   │▒
       ├───┼───┼───┼───┼───┼───┤▒
Step 3 │ 1 │ 2 │ 0 │   │   │   │▒
       ├───┼───┼───┼───┼───┼───┤▒
Step 4 │ 1 │ 2 │ 0 │   │ 4 │   │▒
       ├───┼───┼───┼───┼───┼───┤▒
Step 5 │ 1 │ 2 │ 0 │ 3 │ 4 │   │▒
       ├───┼───┼───┼───┼───┼───┤▒
Step 6 │ 1 │ 2 │ 0 │ 3 │ 4 │ 5 │▒
       └───┴───┴───┴───┴───┴───┘▒
        ▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒

Proof of 2023-11-22 WR
══════════════════════

Notation: if nth -> a, then we write [nth] = a.

By ✅「3rd → a, 0th → b, ab=0」, we have 0 = [3rd] or [0th]. Since ✅「⟦1,4⟧ ∋ 0,3」 implies that 0 is not in corners, we have 0 = [3rd].

       ┌───┬───┬───┬───┬───┬───┐
       │5th│4th│ 3■│2nd│1st│0th│▒
       ╞═══╪═══╪═══╪═══╪═══╪═══╡▒
Step 1 │   │   │ 0 │   │   │   │▒
       └───┴───┴───┴───┴───┴───┘▒
        ▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒

--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ 1 │ 2 │ 5 │   │ 3 │ 4 │
└───┴───┴───┴───┴───┴───┘

To match ✅「⟨⋯ Perm(3,4,5) ⋯⟩」, we need 3,4,5 to be adjacent, so we have

    ┌───┬───┬───┬───┬───┬───┐
    │5th│4th│3rd│2nd│1st│0th│
    ╞═══╪═══╪═══╪═══╪═══╪═══╡
(1) │ ▬ │ ▬ │ 0 │ ▲ │ ▲ │ ▲ │
    └───┴───┴───┴───┴───┴───┘

where "▬" are occupied by 1,2 and "▲" are occupied by 3,4,5.

(2) In particular, we have 2 = [5th] or [4th]. If 2 = [5th], then it follows from ✅「Jump(2,4) = 2」 that 4 = [2nd]:

┌───┬───┬───┬───┬───┬───┐
│ 5■│4th│3rd│ 2■│1st│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╡
│ 2 │   │ 0 │   │   │   │
├───┼───┼───┼───┼───┼───┤
│ 2 │   │ 0 │ 4 │   │   │
└───┴───┴───┴───┴───┴───┘

But then we would never match ✅「⟦1,4⟧ ∋ 0,3」, which is a contradiction. Hence, back to (2), we have 2 = [4th] as our next step.

       ┌───┬───┬───┬───┬───┬───┐
       │5th│ 4■│3rd│2nd│1st│0th│▒
       ╞═══╪═══╪═══╪═══╪═══╪═══╡▒
       │   │   │ 0 │   │   │   │▒
       ├───┼───┼───┼───┼───┼───┤▒
Step 2 │   │ 2 │ 0 │   │   │   │▒
       └───┴───┴───┴───┴───┴───┘▒
        ▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒

--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ 1 │   │ 5 │   │ 3 │ 4 │
└───┴───┴───┴───┴───┴───┘

Using (1) and ✅「Jump(2,4) = 2」, we find the positions of 1 and 4 as well:

       ┌───┬───┬───┬───┬───┬───┐
       │ 5■│4th│3rd│2nd│ 1■│0th│▒
       ╞═══╪═══╪═══╪═══╪═══╪═══╡▒
       │   │ 2 │ 0 │   │   │   │▒
       ├───┼───┼───┼───┼───┼───┤▒
Step 3 │ 1 │ 2 │ 0 │   │   │   │▒
       ├───┼───┼───┼───┼───┼───┤▒
Step 4 │ 1 │ 2 │ 0 │   │ 4 │   │▒
       └───┴───┴───┴───┴───┴───┘▒
        ▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒

--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│   │   │ 5 │   │ 3 │   │
└───┴───┴───┴───┴───┴───┘

Lastly, to match ✅「⟦1,4⟧ ∋ 0,3」, 3 has to be at the left of 4. We finish by

       ┌───┬───┬───┬───┬───┬───┐
       │5th│4th│3rd│ 2■│1st│ 0■│▒
       ╞═══╪═══╪═══╪═══╪═══╪═══╡▒
       │ 1 │ 2 │ 0 │   │ 4 │   │▒
       ├───┼───┼───┼───┼───┼───┤▒
Step 5 │ 1 │ 2 │ 0 │ 3 │ 4 │   │▒
       ├───┼───┼───┼───┼───┼───┤▒
Step 6 │ 1 │ 2 │ 0 │ 3 │ 4 │ 5 │▒
       └───┴───┴───┴───┴───┴───┘▒
        ▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒

--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│   │   │   │   │   │   │
└───┴───┴───┴───┴───┴───┘

Q.E.D.

#125034_v2.2