2024-01-09 Q1(m=6)

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

⟨6th 5th 4th 3rd 2nd 1st 0th⟩

✅Match

⟨⋯ a ⋯ 3 ⋯⟩, a = 4|6

⟨? 6 ⋯ (?−3) ⋯ 0 ⋯⟩ (?≠6)

1st → 0|1|2|3|6

⟨     ⁴ᵗʰa     ¹ˢᵗb ⁰ᵗʰc ⟩, (abc)₁₀ ≤ 461

⟦2,6⟧ ∋ 0

⛔Avoid

1st → 1|2|3

⟨⋯ 1 ⋯ ? 5 ⋯ (?−1)⟩ (?≠5,6)

6th → a, 3rd → b, ab=0+5n

#125034_v2.2

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

Proof of 2024-01-09 Q1(m=6)
═══════════════════════════

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

Plainly, by ✅「⟨? 6 ⋯ (?−3) ⋯ 0 ⋯⟩ (?≠6)」, we have [5th]  =  6:

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

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

We proceed to determine the value of [6th]. By ✅「⟨? 6 ⋯ (?−3) ⋯ 0 ⋯⟩ (?≠6)」 and ✅「⟨⋯ a ⋯ 3 ⋯⟩, a = 4|6」, we have [6th] = 5|4. To avoid ⛔「6th → a, 3rd → b, ab=0+5n」, [6th] cannot be 5. Therefore, [6th] = 4.

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

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

Next, by combining ✅「1st → 0|1|2|3|6」 with ⛔「1st → 1|2|3」, we see that [1st] = 0|6. Since 6 has been used, we have [1st] = 0.

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

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

Then, to match ✅「⟦2,6⟧ ∋ 0」, we have to place 2 at 0th.

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

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

We consider where to place 5. By ✅「⟨     ⁴ᵗʰa     ¹ˢᵗb ⁰ᵗʰc ⟩, (abc)₁₀ ≤ 461」, it is not at 4th. By ⛔「6th → a, 3rd → b, ab=0+5n」, it is not at 3rd. Accordingly, 5 = [2nd].

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

--- Idle ---
┌───┬───┬───┬───┬───┬───┬───┐
│ 1 │   │   │ 3 │   │   │   │
└───┴───┴───┴───┴───┴───┴───┘

Finally, there is only one way to avoid ⛔「⟨⋯ 1 ⋯ ? 5 ⋯ (?−1)⟩ (?≠5,6)」. We finish by

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

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

Q.E.D.

#125034_v2.2