2026-06-16 WR

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

⟨ ⁵ᵗʰ▨ ⁴ᵗʰ▨ ³ʳᵈ▨ ²ⁿᵈ▨ ¹ˢᵗ▨ ⁰ᵗʰ▨ ⟩

✅Match

⟨⋯ 2 ⋯ 0 ⋯ 3 ⋯⟩

3rd → a, 0th → b, ab=15

{p5, p2, p1} = ? + {0,1,2}

⛔Avoid

⟨⋯ ᵃb ⋯⟩, a+b=7

#125034_v2.13

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

Proof of 2026-06-16 WR
══════════════════════

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

By ✅「3rd → a, 0th → b, ab=15」, we have

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

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

--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ 1 │ 2 │   │ 0 │   │ 4 │
└───┴───┴───┴───┴───┴───┘

Then, to match this and ✅「{p5, p2, p1} = ? + {0,1,2}」 at the same time, we need

(2) {p5, p2, p1} = {0,1,2}.

┌───┬───┬───┬───┬───┬───┐
│5th│4th│3rd│2nd│1st│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╡
│012│   │3 5│012│012│3 5│
└───┴───┴───┴───┴───┴───┘

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

As a result, we have [4th] = 4.

┌───┬───┬───┬───┬───┬───┐
│5th│ 4■│3rd│2nd│1st│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╡
│012│ 4 │3 5│012│012│3 5│
└───┴───┴───┴───┴───┴───┘

Next, consider where to place 2. To avoid ⛔「⟨⋯ ᵃb ⋯⟩, a+b=7」, we need 2 != [5th]. Combining this with ✅「⟨⋯ 2 ⋯ 0 ⋯ 3 ⋯⟩」, we get:

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

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

Finally, using ✅「3rd → a, 0th → b, ab=15」, we finish by

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

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

Q.E.D.

#125034_v2.13