2025-11-25 WR

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

⟨ ⁵ᵗʰ▨ ⁴ᵗʰ▨ ³ʳᵈ▨ ²ⁿᵈ▨ ¹ˢᵗ▨ ⁰ᵗʰ▨ ⟩

✅Match

⟨? 4 ⋯ (?−3) ⋯ 0 ⋯⟩ (?≠4)

⟨? ⋯ 3 (?−3) ⋯ 2 ⋯⟩ (?≠3)

3rd → 1|5

#125034_v2.11

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

Proof of 2025-11-25 WR
══════════════════════

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

By ✅「⟨? 4 ⋯ (?−3) ⋯ 0 ⋯⟩ (?≠4)」, plainly we have

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

--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ 1 │ 2 │ 5 │ 0 │ 3 │   │
└───┴───┴───┴───┴───┴───┘

This pattern also requires that [5th] = 5|3. By ✅「⟨? ⋯ 3 (?−3) ⋯ 2 ⋯⟩ (?≠3)」, 3 is not in the left corner. Therefore, [5th] = 5:

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

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

Then, by combining ✅「⟨? 4 ⋯ (?−3) ⋯ 0 ⋯⟩ (?≠4)」 with ✅「⟨? ⋯ 3 (?−3) ⋯ 2 ⋯⟩ (?≠3)」, we have the following required pattern:

(1) ⟨ ⋯ 3 2 ⋯ 0 ⋯⟩.

It follows that 3 is at the 3rd or 2nd positions. By ✅「3rd → 1|5」, 3 is not [3rd]. Therefore, 3 = [2nd]:

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

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

Using (1), we finish by

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

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

Q.E.D.

#125034_v2.11