2023-11-14 WR

Rearrange the digits in ⟨125034⟩ to meet the rules below.

⟨5th 4th 3rd 2nd 1st 0th⟩

✅Match

4th → a, 3rd → b, a+b=3+5n

Jump(0,2) = 1

5th → a, 1st → b, ab=10

5th → 0|2

⛔Avoid

4th → a, 0th → b, |a-b|=1

#125034_v2.1

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

Proof of 2023-11-14 WR
══════════════════════

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

To begin with, note that ✅【5th → 0|2】 and ✅【Jump(0,2) = 1】 imply {[5th], [3rd]} = {0,2}.

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

To match ✅【5th → a, 1st → b, ab=10】, we cannot have [5th] = 0, so [5th] = 2. Accordingly, our first three steps are:

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

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

Now, to match ✅【4th → a, 3rd → b, a+b=3+5n】, we need [4th] = 3.

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

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

Finally, to avoid ⛔「4th → a, 0th → b, |a-b|=1」, we cannot have 4 = [0th]. Therefore, 4 = [2nd] and we obtain

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

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

Q.E.D.

#125034_v2.1