2026-08-18 WR

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

⟨ ⁵ᵗʰ▨ ⁴ᵗʰ▨ ³ʳᵈ▨ ²ⁿᵈ▨ ¹ˢᵗ▨ ⁰ᵗʰ▨ ⟩

✅Match

⟦3,5⟧ ∋ 0,1

{p3, p2, p1} = ? + {0,1,3}

5th|1st|0th → 5

⛔Avoid

4th → a, 3rd → b, ab=0

5th|4th|3rd → 2

#125034_v2.14

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

Proof of 2026-08-18 WR
══════════════════════

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

We consider where to place 0. By ✅「⟦3,5⟧ ∋ 0,1」, it is not in corners ([5th], [0th]):

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

and by ⛔「4th → a, 3rd → b, ab=0」, it is not [4th] or [3rd]:

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

Therefore,

(1) 0 = [2nd] or [1st].

Combining this with ✅「{p3, p2, p1} = ? + {0,1,3}」, we see that

(2) {p3, p2, p1} = {0,1,3}.

┌───┬───┬───┬───┬───┬───┐
│5th│4th│3rd│2nd│1st│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╡
│   │   │013│013│013│   │
└───┴───┴───┴───┴───┴───┘

Using ⛔「5th|4th|3rd → 2」 now, we get 2 = [0th]:

       ┌───┬───┬───┬───┬───┬───┐
       │5th│4th│3rd│2nd│1st│ 0■│▒
       ╞═══╪═══╪═══╪═══╪═══╪═══╡▒
Step 1 │   │   │013│013│013│ 2 │▒
       └───┴───┴───┴───┴───┴───┘▒
        ▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒

Then, to match ✅「⟦3,5⟧ ∋ 0,1」, we need 3 = [1st]:

       ┌───┬───┬───┬───┬───┬───┐
       │5th│4th│3rd│2nd│ 1■│0th│▒
       ╞═══╪═══╪═══╪═══╪═══╪═══╡▒
       │   │   │013│013│013│ 2 │▒
       ├───┼───┼───┼───┼───┼───┤▒
Step 2 │   │   │0 1│0 1│ 3 │ 2 │▒
       └───┴───┴───┴───┴───┴───┘▒
        ▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒

In view of ✅「5th|1st|0th → 5」, we have 5 = [5th], and 4 = [4th] follows:

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

Finally, to avoid ⛔「4th → a, 3rd → b, ab=0」, we finish by

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

Q.E.D.

#125034_v2.14