2024-01-16 Q1(m=6)

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

⟨6th 5th 4th 3rd 2nd 1st 0th⟩

✅Match

Jump(2,4) = 3

⟨Perm(0,3,4,6) Perm(1,2,5)⟩

⟨⋯ 0 ⋯ ? ⋯ 1 (?+3)⟩ (?≠1)

⛔Avoid

4th → a, 2nd → b, a+b=2+4n

5th → a, 2nd → b, |a-b|=2

5th → 3

#125034_v2.2

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

Proof of 2024-01-16 Q1(m=6)
═══════════════════════════

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

Plainly, our first step follows from ✅「⟨⋯ 0 ⋯ ? ⋯ 1 (?+3)⟩ (?≠1)」.

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

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

Next, by ✅「⟨Perm(0,3,4,6) Perm(1,2,5)⟩」, we have [0th] = 2|5. Since [0th] >= 3 by ✅「⟨⋯ 0 ⋯ ? ⋯ 1 (?+3)⟩ (?≠1)」, we see that [0th] = 5.

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

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

Then, ✅「⟨Perm(0,3,4,6) Perm(1,2,5)⟩」 implies [2nd] = 2, and it follows from ✅「Jump(2,4) = 3」 that [6th] = 4:

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

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

We determine the value of [5th]. To avoid ⛔「5th → 3」, it is not 3, and to avoid ⛔「5th → a, 2nd → b, |a-b|=2」, it is not 0. So, [5rh] = 6.

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

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

Finally, to avoid ⛔「4th → a, 2nd → b, a+b=2+4n」, we finish by

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

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

Q.E.D.

#125034_v2.2