2025-09-16 WR

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

⟨5th 4th 3rd 2nd 1st 0th⟩

✅Match

⟨? 2 ⋯ (?−3) ⋯ 4 ⋯⟩ (?≠5)

2nd → a, 1st → b, ab=20

⛔Avoid

⟨⋯ Perm(1,5) ⋯⟩

#125034_v2.10

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

Proof of 2025-09-16 WR
══════════════════════

Plainly, ✅「⟨? 2 ⋯ (?−3) ⋯ 4 ⋯⟩ (?≠5)」 gives [4th] = 2:

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

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

On the other hand, by ✅「2nd → a, 1st → b, ab=20」, we have {[2nd], [1st]} = {4,5}:

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

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

Since ✅「⟨? 2 ⋯ (?−3) ⋯ 4 ⋯⟩ (?≠5)」 implies [5th] >= 3, it follows that [5th] = 3:

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

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

Then, to match ✅「⟨? 2 ⋯ (?−3) ⋯ 4 ⋯⟩ (?≠5)」 we need 0 is to the left of 4. So 0 = [3rd], and 1 = [0th] follows:

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

Finally, to avoid ⛔「⟨⋯ Perm(1,5) ⋯⟩」, we finish by

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

Q.E.D.

#125034_v2.10