2026-07-14 WR

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

⟨ ⁵ᵗʰ▨ ⁴ᵗʰ▨ ³ʳᵈ▨ ²ⁿᵈ▨ ¹ˢᵗ▨ ⁰ᵗʰ▨ ⟩

✅Match

⟨⋯ a ⋯ 2 ⋯⟩, a = 1|3

⟨ − − ▧ ▢ ▢ − ⟩, ▧ ≥ Σ▢

5th → a, 0th → b, a+b=1

⛔Avoid

5th → a, 1st → b, |a-b|=2

#125034_v2.13

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

Proof of 2026-07-14 WR
══════════════════════

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

By ✅「5th → a, 0th → b, a+b=1」, we have

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

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

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

As a consequence, [2nd] + [1st] ≥ 2+3 = 5. Combining this with ✅「⟨ − − ▧ ▢ ▢ − ⟩, ▧ ≥ Σ▢」, we have

(2) [3rd] = 5, and

(3) {[2nd], [1st]} = {2,3}.

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

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

So [4th] = 4:

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

We determine the order of 2,3. If 2 is to the left of 3:

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

then there is no way to match ✅「⟨⋯ a ⋯ 2 ⋯⟩, a = 1|3」 and avoid ⛔「5th → a, 1st → b, |a-b|=2」 at the same time. Therefore, actually 2 is to the right of 3:

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

Finally, to avoid ⛔「5th → a, 1st → b, |a-b|=2」, we finish by

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

Q.E.D.

#125034_v2.13