2026-01-13 WR

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

⟨ ⁵ᵗʰ▨ ⁴ᵗʰ▨ ³ʳᵈ▨ ²ⁿᵈ▨ ¹ˢᵗ▨ ⁰ᵗʰ▨ ⟩

✅Match

max ⊢0⊣ ≥ 4

5th → a, 1st → b, a+b=9

⟨⋯ 5 ⋯ 4 ⋯⟩

⛔Avoid

⟨ − ▧ ▢ ▢ − − ⟩, ▧ ≤ Σ▢

4th → 1|3|4

#125034_v2.11

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

Proof of 2026-01-13 WR
══════════════════════

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

By ✅「5th → a, 1st → b, a+b=9」, we have 

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

Combining this with ✅「⟨⋯ 5 ⋯ 4 ⋯⟩」, we get

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

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

Next, we determine what [4th] is. By ⛔「4th → 1|3|4」, it is 0 or 2. If it is 0, then we cannot avoid ⛔「⟨ − ▧ ▢ ▢ − − ⟩, ▧ ≤ Σ▢」. Therefore, it is 2:

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

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

Then, the only way to avoid ⛔「⟨ − ▧ ▢ ▢ − − ⟩, ▧ ≤ Σ▢」 is to have { [3rd], [2nd] } = {0,1}. It implies [0th] = 3:

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

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

Finally, to match ✅「max ⊢0⊣ ≥ 4」, we finish by

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

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

Q.E.D.

#125034_v2.11