2024-10-08 Q1(m=6)

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

⟨6th 5th 4th 3rd 2nd 1st 0th⟩

✅Match

⟨⋯ 6 ⋯ 0 ⋯⟩

⟨⋯ Perm(0,1,4) ⋯⟩

Jump(4,5) = 1

5th → a, 1st → b, ab=0

⛔Avoid

Jump(2,3) ≥ 2

min ⊢3⊣ ≥ 2

⟨⋯ 2 ⋯ 6 ⋯⟩

#125034_v2.6

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

Proof of 2024-10-08 Q1(m=6)
═══════════════════════════

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

We begin with an observation. To avoid ⛔「min ⊢3⊣ ≥ 2」, we need

(1) 3 is adjacent to 1 or 0.

We start the proof. By ✅「5th → a, 1st → b, ab=0」, we have 0 = [5th] | [1st].

(2) We claim that 0 = [1st].

------------------------------

For, suppose on the contrary 0 = [5th]. By ✅「⟨⋯ 6 ⋯ 0 ⋯⟩」, we have

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

Then, in view of ✅「⟨⋯ Perm(0,1,4) ⋯⟩」 and ✅「Jump(4,5) = 1」, two cases follow:

    ┌───┬───┬───┬───┬───┬───┬───┐
    │6th│5th│ 4▲│ 3▲│ 2▲│ 1▲│0th│
    ╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡
(3) │ 6 │ 0 │ 1 │ 4 │   │ 5 │   │
    ├───┼───┼───┼───┼───┼───┼───┤
(4) │ 6 │ 0 │ 4 │ 1 │ 5 │   │   │
    └───┴───┴───┴───┴───┴───┴───┘

Both cases give contradiction, however, as we have to match (1).

------------------------------

We have verified our claim in (2). Accordingly, we get

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

--- Idle ---
┌───┬───┬───┬───┬───┬───┬───┐
│ 1 │ 2 │ 6 │ 3 │   │ 4 │ 5 │
└───┴───┴───┴───┴───┴───┴───┘

Next, by ✅「⟨⋯ Perm(0,1,4) ⋯⟩」, there are two possibilities: 

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

where "▬" are occupied by 1,4.

If case (5) holds, then (1) implies

┌───┬───┬───┬───┬───┬───┬───┐
│6th│5th│4th│3rd│ 2▲│1st│ 0▲│
╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡
│   │   │   │   │ 1 │ 0 │ 4 │
└───┴───┴───┴───┴───┴───┴───┘

But then we cannot match ✅「Jump(4,5) = 1」. Therefore, case (5) does not hold and case (6) holds instead. Then, in view of (1), there are three cases:

    ┌───┬───┬───┬───┬───┬───┬───┐
    │6th│5th│ 4▲│ 3▲│ 2▲│1st│ 0▲│
    ╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡
(7) │   │   │   │ 4 │ 1 │ 0 │ 3 │
    ├───┼───┼───┼───┼───┼───┼───┤
(8) │   │   │   │ 1 │ 4 │ 0 │ 3 │
    ├───┼───┼───┼───┼───┼───┼───┤
(9) │   │   │ 3 │ 1 │ 4 │ 0 │   │
    └───┴───┴───┴───┴───┴───┴───┘

Note that if case (7) or (8) holds, then we cannot avoid ⛔「Jump(2,3) ≥ 2」. Therefore, case (9) holds actually. We get

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

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

Then, by ✅「Jump(4,5) = 1」, we have 

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

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

Finally, in view of ⛔「⟨⋯ 2 ⋯ 6 ⋯⟩」, we finish by

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

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

Q.E.D.

#125034_v2.6