2026-04-14 WR

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

⟨ ⁵ᵗʰ▨ ⁴ᵗʰ▨ ³ʳᵈ▨ ²ⁿᵈ▨ ¹ˢᵗ▨ ⁰ᵗʰ▨ ⟩

✅Match

⟨⋯ 1 ⋯ ? ⋯ 2 (?+1)⟩ (?≠2,1)

⟨⋯ 5 ⋯ 3 ⋯ 0 ⋯⟩

⛔Avoid

5th → a, 0th → b, |a-b|=3

#125034_v2.12

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

Proof of 2026-04-14 WR
══════════════════════

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

Plainly, by ✅「⟨⋯ 1 ⋯ ? ⋯ 2 (?+1)⟩ (?≠2,1)」, we get

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

--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ 1 │   │ 5 │ 0 │ 3 │ 4 │
└───┴───┴───┴───┴───┴───┘

We consider what [0th] is. By ✅「⟨⋯ 1 ⋯ ? ⋯ 2 (?+1)⟩ (?≠2,1)」, it is not 0,1,2. By ✅「⟨⋯ 5 ⋯ 3 ⋯ 0 ⋯⟩」, it is not 3 or 5. Therefore, it is 4:

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

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

Now ✅「⟨⋯ 1 ⋯ ? ⋯ 2 (?+1)⟩ (?≠2,1)」 means:

(1) ⟨⋯ 1 ⋯ 3 ⋯ 2 4⟩.

Combining this with ✅「⟨⋯ 5 ⋯ 3 ⋯ 0 ⋯⟩」, we have the following required pattern:

(2) ⟨⋯ 1 ⋯ 3 ⋯ 0 ⋯⟩.

From (2) and ✅「⟨⋯ 5 ⋯ 3 ⋯ 0 ⋯⟩」, we see that 3,0 are to the right of 1,5, and that 0 is to the right of 3. There is only one way to do so:

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

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

Finally, to avoid ⛔「5th → a, 0th → b, |a-b|=3」, we finish by

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

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

Q.E.D.

#125034_v2.12