2023-12-12 Q1(m=6)

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

⟨6th 5th 4th 3rd 2nd 1st 0th⟩

✅Match

⟨⋯ 5 ⋯ ? ⋯ 6 (?−1)⟩ (?≠6)

⟨⋯ Perm(0,4,6) ⋯⟩

⟨? 3 ⋯ (?−4) ⋯ 2 ⋯⟩

⛔Avoid

4th|3rd|2nd → 2

⟨⋯ 4 ⋯ 0 ⋯ 6 ⋯⟩

#125034_v2.2

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

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

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

Plainly, ✅「⟨⋯ 5 ⋯ ? ⋯ 6 (?−1)⟩ (?≠6)」 implies [1st] = 6, and ✅「⟨? 3 ⋯ (?−4) ⋯ 2 ⋯⟩」 implies [5th] = 3.

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

--- Idle ---
┌───┬───┬───┬───┬───┬───┬───┐
│ 1 │ 2 │   │   │ 0 │ 4 │ 5 │
└───┴───┴───┴───┴───┴───┴───┘

Next, by ⛔「4th|3rd|2nd → 2」, we have 2 = [6th] or [0th]. The latter holds because ✅「⟨? 3 ⋯ (?−4) ⋯ 2 ⋯⟩」 implies 2 is not in the left corner.

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

--- Idle ---
┌───┬───┬───┬───┬───┬───┬───┐
│ 1 │   │   │   │ 0 │ 4 │ 5 │
└───┴───┴───┴───┴───┴───┴───┘

As ✅「⟨⋯ 5 ⋯ ? ⋯ 6 (?−1)⟩ (?≠6)」 becomes the pattern

⟨⋯ 5 ⋯ 3 ⋯ 62⟩,

we see that 5 = [6th].

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

--- Idle ---
┌───┬───┬───┬───┬───┬───┬───┐
│ 1 │   │   │   │ 0 │ 4 │   │
└───┴───┴───┴───┴───┴───┴───┘

By ✅「⟨⋯ Perm(0,4,6) ⋯⟩」, 0,4,6 are adjacent, so we have to place 1 at 4th.

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

--- Idle ---
┌───┬───┬───┬───┬───┬───┬───┐
│   │   │   │   │ 0 │ 4 │   │
└───┴───┴───┴───┴───┴───┴───┘

Finally, to avoid ⛔「⟨⋯ 4 ⋯ 0 ⋯ 6 ⋯⟩」, we reach

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

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

Q.E.D.

#125034_v2.2