2023-10-25 WR

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

⟨5th 4th 3rd 2nd 1st 0th⟩

✅Match

5th|4th|3rd|0th → 5

4th → a, 1st → b, |a-b|=1

⟨⋯ Perm(2,4,5) ⋯⟩

⟨ ⁵ᵗʰa     ²ⁿᵈb   ⁰ᵗʰc ⟩, (abc)₁₀ ≤ 341

5th → a, 0th → b, a+b=3+5n

⛔Avoid

⟨⋯ a ⋯ 1 ⋯⟩, a = 0|2|4

#125034_v2.1

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

Proof of 2023-10-25 WR
══════════════════════

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

(1) We first consider where to place 1. By ⛔「⟨⋯ a ⋯ 1 ⋯⟩, a = 0|2|4」 and ✅【⟨⋯ Perm(2,4,5) ⋯⟩】, we see that 1 is at the left of 0,2,4,5. Hence, 1=[4th] or [5th].

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

If 1=[5th], then by ✅【5th → a, 0th → b, a+b=3+5n】, we have [0th]=2:

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

where by ✅【⟨⋯ Perm(2,4,5) ⋯⟩】, the * indicate the positions of 4,5. But then ✅【5th|4th|3rd|0th → 5】 would never be matched, which is a contradiction.

Therefore, it follows from (1) that 1=[4th], and {0,2,4,5}= {[3th],[2nd],[1st],[0th]}. Accordingly, we have [5th]=3 as well.

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

--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│   │ 2 │ 5 │ 0 │   │ 4 │
└───┴───┴───┴───┴───┴───┘

------------------------------

Now, in view of ✅【4th → a, 1st → b, |a-b|=1】, we have [1st]=0 or 2. It cannot be 0, for otherwise there are no 3 consecutive grids available for matching ✅【⟨⋯ Perm(2,4,5) ⋯⟩】:

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

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

Therefore, [1st]=2.

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

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

------------------------------

Next, by ✅【5th|4th|3rd|0th → 5】, we have 5=[3th] or [0th]. No matter which holds, by ✅【⟨⋯ Perm(2,4,5) ⋯⟩】 we have [2nd]=4.

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

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

Finally, in view of ✅【⟨ ⁵ᵗʰa     ²ⁿᵈb   ⁰ᵗʰc ⟩, (abc)₁₀ ≤ 341】, 5 cannot be placed at 0th. As a result, we have

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

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

Q.E.D.

#125034_v2.1