2024-02-06 Q1(m=6)

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

⟨6th 5th 4th 3rd 2nd 1st 0th⟩

✅Match

⟨ ⁶ᵗʰa       ²ⁿᵈb     ⟩, min⟦a,b⟧ = 1

⟨? 6 ⋯ (?+4) ⋯ 5 ⋯⟩ (?≠2)

Jump(2,4) = 0

⟦1,2⟧ ∋ 3,4,6

⛔Avoid

3rd → 3|4

#125034_v2.3

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

Proof of 2024-02-06 Q1(m=6)
═══════════════════════════

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

Plainly, our first step follows from ✅「⟨? 6 ⋯ (?+4) ⋯ 5 ⋯⟩ (?≠2)」:

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

--- Idle ---
┌───┬───┬───┬───┬───┬───┬───┐
│ 1 │ 2 │   │ 3 │ 0 │ 4 │ 5 │
└───┴───┴───┴───┴───┴───┴───┘

Next, in view of ✅「⟦1,2⟧ ∋ 3,4,6」, we need to place 1 or 2 at 6th, otherwise 6 would not be contained in ⟦1,2⟧. Since [6th] != 2 by ✅「⟨? 6 ⋯ (?+4) ⋯ 5 ⋯⟩ (?≠2)」, we have [6th] = 1.

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

--- Idle ---
┌───┬───┬───┬───┬───┬───┬───┐
│   │ 2 │   │ 3 │ 0 │ 4 │ 5 │
└───┴───┴───┴───┴───┴───┴───┘

Then, it follows from ✅「⟦1,2⟧ ∋ 3,4,6」 and ✅「Jump(2,4) = 0」 that we have to match the following pattern:

(1) ⟨16 ⋯ 3 ⋯ 42 ⋯⟩.

Accordingly, there are only four ways to place 2,4:

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

Case (4) holds actually, for:

● If (2) holds, then we could not match ✅「⟦1,2⟧ ∋ 3,4,6」 wherever 3 is placed;

● If (3) holds, then we match ⛔「3rd → 3|4」;

● If (5) holds, then we could not match ✅「⟨ ⁶ᵗʰa       ²ⁿᵈb     ⟩, min⟦a,b⟧ = 1」 wherever 0 is placed.

Hence, we get

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

--- Idle ---
┌───┬───┬───┬───┬───┬───┬───┐
│   │   │   │ 3 │ 0 │   │ 5 │
└───┴───┴───┴───┴───┴───┴───┘

Next, to match ✅「⟨ ⁶ᵗʰa       ²ⁿᵈb     ⟩, min⟦a,b⟧ = 1」, we have 0 != [4th] and 0 != [3rd]. Therefore, 0 = [0th]:

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

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

Finally, to avoid ⛔「3rd → 3|4」, we finish by

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

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

Q.E.D.

#125034_v2.3