2026-08-11 WR

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

⟨ ⁵ᵗʰ▨ ⁴ᵗʰ▨ ³ʳᵈ▨ ²ⁿᵈ▨ ¹ˢᵗ▨ ⁰ᵗʰ▨ ⟩

✅Match

median {p2, p1, p0} = 4

Jump(0,5) = 3

⛔Avoid

⟨⋯ ᵃb ⋯⟩, a+b=6

⟨       ²ⁿᵈa ¹ˢᵗb   ⟩, (ab)₁₀ ≥ 20

#125034_v2.14

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

Proof of 2026-08-11 WR
══════════════════════

✅「median {p2, p1, p0} = 4」 implies

(1) 4,5 ∈ {p2, p1, p0}.

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

To avoid ⛔「⟨       ²ⁿᵈa ¹ˢᵗb   ⟩, (ab)₁₀ ≥ 20」, we need 4,5 != p2. Therefore,

(2) {4,5} = {p1, p0}.

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

By ⛔「⟨⋯ ᵃb ⋯⟩, a+b=6」, we have 5!=p1, so 5=p0 and 4=p1:

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

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

Then, using ✅「Jump(0,5) = 3」, we have

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

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

Now there is only one way to avoid ⛔「⟨       ²ⁿᵈa ¹ˢᵗb   ⟩, (ab)₁₀ ≥ 20」:

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

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

Finally, using ⛔「⟨⋯ ᵃb ⋯⟩, a+b=6」 once more, we finish by

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

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

Q.E.D.

#125034_v2.14