2023-10-31 Q1(m=6)

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

⟨6th 5th 4th 3rd 2nd 1st 0th⟩

✅Match

6th → a, 5th → b, a+b=5

Jump(0,6) = 0

⟨⋯ Perm(4,5,6) ⋯⟩

⟨⋯ 2 ⋯ ? 1 ⋯ (?−2)⟩ (?≠3)

⛔Avoid

⟨? ⋯ 1 (?+2) ⋯ 0 ⋯⟩ (?≠1)

#125034_v2.1

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

Proof of 2023-10-31 Q1(m=6)
═══════════════════════════

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

Let a := [6th] and b := [5th]. ✅【6th → a, 5th → b, a+b=5】 implies that there are three possibilities of {a,b}:

(i) {5,0}
(ii) {4,1}
(iii) {3,2}

We claim that case (i) and (ii) are not possible.

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

Suppose otherwise that case (i) holds. If (a,b) = (5,0), then ✅【⟨⋯ Perm(4,5,6) ⋯⟩】 cannot be matched; else if (a,b) = (0,5), then ✅【Jump(0,6) = 0】 cannot be matched. Hence, it is a contradiction.

Else, suppose case (ii) holds. If (a,b) =(4,1), then ✅【⟨⋯ Perm(4,5,6) ⋯⟩】 cannot be matched; else if (a,b) = (1,4), then ✅【⟨⋯ 2 ⋯ ? 1 ⋯ (?−2)⟩ (?≠3)】 cannot be matched. Therefore, it is also a contradiction.

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

It follows that case (iii) holds. In addition, noting that ✅【Jump(0,6) = 0】 and ✅【⟨⋯ Perm(4,5,6) ⋯⟩】 imply 0,4,5,6 are adjacent to each other, we have two situations:

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

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

where {a,b} = {3,2} and the "/" are occupied by 0,4,5,6.

We need to match ✅【⟨⋯ 2 ⋯ ? 1 ⋯ (?−2)⟩ (?≠3)】. A fortiori, 1 != [0th]. It follows that (1) is impossible and we have [4th] = 1 as our first step:

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

--- Idle ---
┌───┬───┬───┬───┬───┬───┬───┐
│   │ 2 │ 6 │ 3 │ 0 │ 4 │ 5 │
└───┴───┴───┴───┴───┴───┴───┘

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

Next, we consider whether (a,b) = (3,2) or (2,3). If the latter holds, then the "?" in ✅【⟨⋯ 2 ⋯ ? 1 ⋯ (?−2)⟩ (?≠3)】 is 3 and [0th] = 1, which is a contradiction. Hence, we have (a,b) = (3,2).

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

--- Idle ---
┌───┬───┬───┬───┬───┬───┬───┐
│   │   │ 6 │   │ 0 │ 4 │ 5 │
└───┴───┴───┴───┴───┴───┴───┘

Accordingly, the "?" in ✅【⟨⋯ 2 ⋯ ? 1 ⋯ (?−2)⟩ (?≠3)】 is 2, so [0th] = 0. Using ✅【Jump(0,6) = 0】 too, we get

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

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

Finally, ⛔「⟨? ⋯ 1 (?+2) ⋯ 0 ⋯⟩ (?≠1)」 requires that 5 is not next to 1. So, we reach

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

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

Q.E.D.

#125034_v2.1