2026-09-22 WR

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

⟨ ⁵ᵗʰ▨ ⁴ᵗʰ▨ ³ʳᵈ▨ ²ⁿᵈ▨ ¹ˢᵗ▨ ⁰ᵗʰ▨ ⟩

✅Match

⟨? ⋯ 0 ⋯ (?−2) ⋯⟩ (?≠2)

⟨⋯ 4 ⋯ 5 ⋯ 3 ⋯⟩

⛔Avoid

⟨ ▧ ▧ − ▢ − − ⟩, Σ▧ ≥ ▢

⟨ ⁵ᵗʰ↓ ⁴ᵗʰ↑ ³ʳᵈ↑ ²ⁿᵈ↓ ¹ˢᵗ↑ ⁰ᵗʰ↓ ⟩ after 5−⟨⋯⟩

5th|3rd → 1

#125034_v2.15

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

Proof of 2026-09-22 WR
══════════════════════

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

By ✅「⟨? ⋯ 0 ⋯ (?−2) ⋯⟩ (?≠2)」, we have

(1) [5th] = 5 | 4 | 3.

Combining this with ✅「⟨⋯ 4 ⋯ 5 ⋯ 3 ⋯⟩」, we have [5th] = 4 indeed:

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

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

A fortiori, the sum of [5th] and [4th] is at least 4. So, to avoid ⛔「⟨ ▧ ▧ − ▢ − − ⟩, Σ▧ ≥ ▢」, we need 

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

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

Consider which digit [3rd] is. By ✅「⟨⋯ 4 ⋯ 5 ⋯ 3 ⋯⟩」, it is not 3, and by ⛔「5th|3rd → 1」, it is not 1. Therefore, it is 2:

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

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

Finally, to avoid ⛔「⟨ ⁵ᵗʰ↓ ⁴ᵗʰ↑ ³ʳᵈ↑ ²ⁿᵈ↓ ¹ˢᵗ↑ ⁰ᵗʰ↓ ⟩ after 5−⟨⋯⟩」, we finish by

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

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

Q.E.D.

#125034_v2.15