2025-10-07 WR

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

⟨5th 4th 3rd 2nd 1st 0th⟩

✅Match

⟨   ⁴ᵗʰa ³ʳᵈb   ¹ˢᵗc   ⟩, (abc)₁₀ ≥ 234

⟨⋯ 3 ⋯ 2 ⋯⟩

1st → 3

2nd → 2|3|5

⛔Avoid

⟨     ³ʳᵈ↓       ⟩ after 2×⟨→⟩

#125034_v2.10

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

Proof of 2025-10-07 WR
══════════════════════

Plainly, the first step follows from ✅「1st → 3」:

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

--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ 1 │ 2 │ 5 │ 0 │   │ 4 │
└───┴───┴───┴───┴───┴───┘

Then, in view of ✅「⟨⋯ 3 ⋯ 2 ⋯⟩」, we get [0th] = 2:

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

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

Now we have only one way to match ✅「2nd → 2|3|5」, namely [2nd] = 5:

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

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

To match ✅「⟨   ⁴ᵗʰa ³ʳᵈb   ¹ˢᵗc   ⟩, (abc)₁₀ ≥ 234」, we then need [4th] = 4:

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

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

Finally, to avoid ⛔「⟨     ³ʳᵈ↓       ⟩ after 2×⟨→⟩」, we finish by

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

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

Q.E.D.

#125034_v2.10