2025-12-16 WR

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

⟨ ⁵ᵗʰ▨ ⁴ᵗʰ▨ ³ʳᵈ▨ ²ⁿᵈ▨ ¹ˢᵗ▨ ⁰ᵗʰ▨ ⟩

✅Match

⟨   ⁴ᵗʰa     ¹ˢᵗb   ⟩, min⟦a,b⟧ = 1

0 ∾ 3 ∾ 4 ∾ 5

⟨ − − ▧ − ▢ ▢ ⟩, ▧ ≥ Σ▢

⛔Avoid

⟨     ³ʳᵈa ²ⁿᵈb   ⁰ᵗʰc ⟩, (abc)₁₀ ≥ 425

#125034_v2.11

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

Proof of 2025-12-16 WR
══════════════════════

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

By ✅「⟨   ⁴ᵗʰa     ¹ˢᵗb   ⟩, min⟦a,b⟧ = 1」, 0 is [5th] or [0th]. If it is [0th] then we cannot match ✅「0 ∾ 3 ∾ 4 ∾ 5」. Therefore, 0 = [5th]:

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

--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ 1 │ 2 │ 5 │   │ 3 │ 4 │
└───┴───┴───┴───┴───┴───┘

We consider what [3rd] is. By ✅「⟨ − − ▧ − ▢ ▢ ⟩, ▧ ≥ Σ▢」, it is not 1 or 2. To match ✅「0 ∾ 3 ∾ 4 ∾ 5」, it is not 3, and to avoid ⛔「⟨     ³ʳᵈa ²ⁿᵈb   ⁰ᵗʰc ⟩, (abc)₁₀ ≥ 425」, it is not 5. Therefore, [3rd] = 4:

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

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

Then, to match ✅「⟨ − − ▧ − ▢ ▢ ⟩, ▧ ≥ Σ▢」, we need {[1st], [0th]} = {1,2} or {1,3}. A fortiori, 1 = [1st] or [0th]. Using ✅「⟨   ⁴ᵗʰa     ¹ˢᵗb   ⟩, min⟦a,b⟧ = 1」, we have 1 = [1st]:

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

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

Finally, to avoid ⛔「⟨     ³ʳᵈa ²ⁿᵈb   ⁰ᵗʰc ⟩, (abc)₁₀ ≥ 425」, we finish by

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

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

Q.E.D.

#125034_v2.11