2023-12-20 Q1(m=6)

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

⟨6th 5th 4th 3rd 2nd 1st 0th⟩

✅Match

⟦4,5⟧ ∋ 2,6

⟨⋯ Perm(0,3,4) ⋯⟩

2nd → a, 1st → b, a+b=7

4th → 3|4

⛔Avoid

⟨         ²ⁿᵈa   ⁰ᵗʰb ⟩, (ab)₁₀ ≤ 60

5th → 0|1|3|5

#125034_v2.2

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

Proof of 2023-12-20 Q1(m=6)
═══════════════════════════

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

Plainly, to avoid ⛔「⟨         ²ⁿᵈa   ⁰ᵗʰb ⟩, (ab)₁₀ ≤ 60」, we need 6 = [2nd]. Then, to match ✅「2nd → a, 1st → b, a+b=7」, we need 1 = [1st].

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

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

Next, we determine the value of [0th]. By ✅「⟨⋯ Perm(0,3,4) ⋯⟩」, 0,3,4 are adjacent, so they cannot be placed at 0th. On the other hand, by ✅「⟦4,5⟧ ∋ 2,6」, 2 is between 4,5. A fortiori 2 is not in corners. Therefore, [0th] = 5.

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

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

Since 0,3,4 are adjacent by ✅「⟨⋯ Perm(0,3,4) ⋯⟩」, we have 2 = [6th] or [3rd]:

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

As 2 is not in corners by ✅「⟦4,5⟧ ∋ 2,6」, the latter case holds. It then follows from ⛔「5th → 0|1|3|5」 that [5th] = 4:

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

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

Finally, to match ✅「4th → 3|4」, we finish by

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

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

Q.E.D.

#125034_v2.2