2025-11-04 WR

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

⟨5th 4th 3rd 2nd 1st 0th⟩

✅Match

2nd → a, 1st → b, a+b=8

5th → a, 2nd → b, ab=20

5th|4th|3rd|1st → 2

⛔Avoid

3rd|2nd|1st|0th → 0

#125034_v2.10

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

Proof of 2025-11-04 WR
══════════════════════

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

By ✅「5th → a, 2nd → b, ab=20」, we have

(1) {[5th], [2nd]} = {4,5}.

If [2nd] = 4, then we will fail to match ✅「2nd → a, 1st → b, a+b=8」. Therefore, [2nd] = 5 and [5th] = 4:

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

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

Then, using ✅「2nd → a, 1st → b, a+b=8」, we get [1st] = 3:

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

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

To avoid ⛔「3rd|2nd|1st|0th → 0」, we need 0 = [4th]:

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

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

Finally, to match ✅「5th|4th|3rd|1st → 2」, we finish by

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

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

Q.E.D.

#125034_v2.10