2025-05-13 WR

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

⟨5th 4th 3rd 2nd 1st 0th⟩

✅Match

3rd → a, 1st → b, |a-b|=3

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

Jump(2,3) ≥ 2

⟨⋯ 1 ⋯ 4 ⋯ 5 ⋯⟩

#125034_v2.9

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

Proof of 2025-05-13 WR
══════════════════════

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

By ✅「⟨ ⁵ᵗʰ↓ ⁴ᵗʰ↑ ³ʳᵈ↑ ²ⁿᵈ↑ ¹ˢᵗ↓ ⁰ᵗʰ↓ ⟩ after 5−⟨⋯⟩」, we have

(1) {3,4,5} = {[5th], [1st], [0th]}.

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

A fortiori, 4 ∈ {[5th], [1st], [0th]}. Since ✅「⟨⋯ 1 ⋯ 4 ⋯ 5 ⋯⟩」 implies that 4 is not in corners ([5th] or [0th]), we have 4 = [1st]: 

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

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

It then follows from ✅「⟨⋯ 1 ⋯ 4 ⋯ 5 ⋯⟩」 that [0th] = 5:

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

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

Using (1), we get [5th] = 3 as well:

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

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

Then, to match ✅「3rd → a, 1st → b, |a-b|=3」, we get [3rd] = 1:

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

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

Finally, to match ✅「Jump(2,3) ≥ 2」, we finish by

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

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

Q.E.D.

#125034_v2.9