2023-12-14 Q1(m=6)

Rearrange the digits in ⟨1263045⟩ to meet the rules below.

⟨6th 5th 4th 3rd 2nd 1st 0th⟩

✅Match

⟦2,6⟧ ∋ 1,3

⟨? ⋯ 0 (?−1) ⋯ 4 ⋯⟩ (?≠1)

⟨⋯ Perm(1,3,4) ⋯⟩

2nd → a, 1st → b, a+b=9

⛔Avoid

3rd → 0|2|4|6

#125034_v2.2

       ┌───┬───┬───┬───┬───┬───┬───┐
       │6th│5th│4th│3rd│2nd│1st│0th│▒
       ╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡▒
Step 1 │   │ 0 │   │   │   │   │   │▒
       ├───┼───┼───┼───┼───┼───┼───┤▒
Step 2 │ 2 │ 0 │   │   │   │   │   │▒
       ├───┼───┼───┼───┼───┼───┼───┤▒
Step 3 │ 2 │ 0 │ 1 │   │   │   │   │▒
       ├───┼───┼───┼───┼───┼───┼───┤▒
Step 4 │ 2 │ 0 │ 1 │ 3 │   │   │   │▒
       ├───┼───┼───┼───┼───┼───┼───┤▒
Step 5 │ 2 │ 0 │ 1 │ 3 │ 4 │   │   │▒
       ├───┼───┼───┼───┼───┼───┼───┤▒
Step 6 │ 2 │ 0 │ 1 │ 3 │ 4 │ 5 │   │▒
       ├───┼───┼───┼───┼───┼───┼───┤▒
Step 7 │ 2 │ 0 │ 1 │ 3 │ 4 │ 5 │ 6 │▒
       └───┴───┴───┴───┴───┴───┴───┘▒
        ▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒

Proof of 2023-12-14 Q1(m=6)
═══════════════════════════

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

To match ✅「2nd → a, 1st → b, a+b=9」, we have {a,b} = {3,6} or {4,5}, where a:=[2nd] and b:=[1st]. Combining this with ✅「⟨⋯ Perm(1,3,4) ⋯⟩」, we have:

    ┌───┬───┬───┬───┬───┬───┬───┐
    │6th│5th│ 4▲│ 3▲│ 2▲│*1 │0th│
    ╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡
(1) │   │   │ X │ Y │ Z │ b │   │
    └───┴───┴───┴───┴───┴───┴───┘

where {X,Y,Z} = {1,3,4}, and b ∈ {5,6}.

Observe that ✅「⟨? ⋯ 0 (?−1) ⋯ 4 ⋯⟩ (?≠1)」 and ✅「⟨⋯ Perm(1,3,4) ⋯⟩」 imply the following pattern:

(2) ⟨? ⋯ 0 ⋯ Perm(1,3,4) ⋯⟩ (?=6|5|2)

To match (1) and (2), we need [5th] = 0:

┌───┬───┬───┬───┬───┬───┬───┐
│6th│ 5■│4th│3rd│2nd│1st│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡
│   │ 0 │ X │ Y │ Z │ b │   │
└───┴───┴───┴───┴───┴───┴───┘

--- Idle ---
┌───┬───┬───┬───┬───┬───┬───┐
│   │ 2 │ 6 │   │   │   │ 5 │
└───┴───┴───┴───┴───┴───┴───┘

We proceed to determine the value of [6th]. To match ✅「⟦2,6⟧ ∋ 1,3」, we need [6th] = 2|6, for otherwise 1,3 would not be between 2 and 6. In view of ✅「⟨? ⋯ 0 (?−1) ⋯ 4 ⋯⟩ (?≠1)」, it is not possible that [6th] = 6, for X != 5. 

Hence, [6th] = 2, and then the preceding pattern gives X = 1 as well.

┌───┬───┬───┬───┬───┬───┬───┐
│ 6■│5th│ 4■│3rd│2nd│1st│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡
│ 2 │ 0 │ 1 │ Y │ Z │ b │   │
└───┴───┴───┴───┴───┴───┴───┘

--- Idle ---
┌───┬───┬───┬───┬───┬───┬───┐
│   │   │ 6 │   │   │   │ 5 │
└───┴───┴───┴───┴───┴───┴───┘

Next, by ⛔「3rd → 0|2|4|6」, we get Y = 3 and Z = 4. So, our first five steps are:

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

--- Idle ---
┌───┬───┬───┬───┬───┬───┬───┐
│   │   │ 6 │   │   │   │ 5 │
└───┴───┴───┴───┴───┴───┴───┘

We finish by matching ✅「2nd → a, 1st → b, a+b=9」:

       ┌───┬───┬───┬───┬───┬───┬───┐
       │6th│5th│4th│3rd│2nd│ 1■│ 0■│▒
       ╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡▒
       │ 2 │ 0 │ 1 │ 3 │ 4 │   │   │▒
       ├───┼───┼───┼───┼───┼───┼───┤▒
Step 6 │ 2 │ 0 │ 1 │ 3 │ 4 │ 5 │   │▒
       ├───┼───┼───┼───┼───┼───┼───┤▒
Step 7 │ 2 │ 0 │ 1 │ 3 │ 4 │ 5 │ 6 │▒
       └───┴───┴───┴───┴───┴───┴───┘▒
        ▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒

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

Q.E.D.

#125034_v2.2