2024-05-28 Q1(m=6)

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

⟨6th 5th 4th 3rd 2nd 1st 0th⟩

✅Match

⟨⋯ 2 ⋯ 4 ⋯ 3 ⋯⟩

Jump(1,4) = 1

5th → a, 3rd → b, |a-b|=4

⛔Avoid

2nd → 2|3|4|5|6

5th|4th|3rd → 2

#125034_v2.5

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

Proof of 2024-05-28 Q1(m=6)
═══════════════════════════

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

Plainly, to avoid ⛔「5th|4th|3rd → 2」, we need

2 = [6th] | [2nd] | [1st] | [0th].

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

Actually, it is 2 = [6th] | [2nd], since ✅「⟨⋯ 2 ⋯ 4 ⋯ 3 ⋯⟩」 implies that there are at least two digits at the right of 2. Therefore, in view of ⛔「2nd → 2|3|4|5|6」, we get 2 = [6th].

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

--- Idle ---
┌───┬───┬───┬───┬───┬───┬───┐
│ 1 │   │ 6 │ 3 │ 0 │ 4 │ 5 │
└───┴───┴───┴───┴───┴───┴───┘

Next, we consider ✅「5th → a, 3rd → b, |a-b|=4」. There are exactly four ways to match it:

(1.1) ([5th], [3rd]) = (5,1);

(1.2) ([5th], [3rd]) = (1,5);

(1.3) ([5th], [3rd]) = (4,0);

(1.4) ([5th], [3rd]) = (0,4).

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

We show that (1.1) holds actually.

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

Observe that in order to match ✅「Jump(1,4) = 1」, neither (1.2) nor (1.3) holds. On the other hand, if (1.4) holds:

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

then we cannot match ✅「Jump(1,4) = 1」 and avoid ⛔「2nd → 2|3|4|5|6」 at the same time. So, (1.4) does not hold as well.

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

We have verified that (1.1) holds. Accordingly, we get

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

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

Then, using ⛔「2nd → 2|3|4|5|6」 and ✅「Jump(1,4) = 1」, we get

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

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

Finally, in view of ✅「⟨⋯ 2 ⋯ 4 ⋯ 3 ⋯⟩」, we finish by

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

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

Q.E.D.

#125034_v2.5