2024-11-26 Q1(m=6)

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

⟨6th 5th 4th 3rd 2nd 1st 0th⟩

✅Match

Jump(3,5) ≤ 1

⟨? ⋯ 2 ⋯ (?+4) ⋯⟩ (?≠2)

⟦0,6⟧ ∋ 4,5

⟦2,3⟧ ∋ 1,5,6

⛔Avoid

min {p5, p3, p1} = 1

#125034_v2.7

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

Proof of 2024-11-26 Q1(m=6)
═══════════════════════════

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

By ✅「⟨? ⋯ 2 ⋯ (?+4) ⋯⟩ (?≠2)」, we have [6th] = 0|1. As 1 is not at corners by ✅「⟦2,3⟧ ∋ 1,5,6」, we get [6th] = 0:

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

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

Then, to match ✅「⟦2,3⟧ ∋ 1,5,6」, there are three possible ways to place 2 and 3:

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

Case (3) holds actually, for otherwise we have

    ┌───┬───┬───┬───┬───┬───┬───┐
    │6th│5th│4th│3rd│2nd│1st│0th│
    ╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡
(4) │ 0 │ ▲ │156│156│156│ ▲ │ 4 │
    ├───┼───┼───┼───┼───┼───┼───┤
(5) │ 0 │ 4 │ ▲ │156│156│156│ ▲ │
    └───┴───┴───┴───┴───┴───┴───┘

Note that (4) does not match ✅「⟦0,6⟧ ∋ 4,5」 and (5) does not match ✅「⟨? ⋯ 2 ⋯ (?+4) ⋯⟩ (?≠2)」. So they are contradictions.

Combining (3) with ✅「⟨? ⋯ 2 ⋯ (?+4) ⋯⟩ (?≠2)」, we get

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

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

Then, to match ✅「Jump(3,5) ≤ 1」, we have 5 = [2nd] | [1st]. We would fail to match ✅「⟦0,6⟧ ∋ 4,5」 if 5 = [1st]. Therefore, 5 = [2nd]:

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

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

Then, it follows from ✅「⟦0,6⟧ ∋ 4,5」 that [1st] = 6:

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

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

Finally, in view of ⛔「min {p5, p3, p1} = 1」, we finish by

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

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

Q.E.D.

#125034_v2.7