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