2024-11-05 Q1(m=6)

Rearrange the digits in ⟨1263045⟩ to meet the rules below.

⟨6th 5th 4th 3rd 2nd 1st 0th⟩

✅Match

min ⊢3⊣ ≥ 2

⟦2,5⟧ ∋ 3,6

⟨⋯ 0 ⋯ 5 ⋯ 4 ⋯ 2 ⋯⟩

⟦0,3⟧ ∋ 1,6

{p5, p4, p1, p0} = ? + {0,1,2,3}

#125034_v2.7

       ┌───┬───┬───┬───┬───┬───┬───┐
       │6th│5th│4th│3rd│2nd│1st│0th│▒
       ╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡▒
Step 1 │ 0 │   │   │   │   │   │   │▒
       ├───┼───┼───┼───┼───┼───┼───┤▒
Step 2 │ 0 │   │   │   │   │   │ 2 │▒
       ├───┼───┼───┼───┼───┼───┼───┤▒
Step 3 │ 0 │   │   │   │   │ 3 │ 2 │▒
       ├───┼───┼───┼───┼───┼───┼───┤▒
Step 4 │ 0 │ 5 │   │   │   │ 3 │ 2 │▒
       ├───┼───┼───┼───┼───┼───┼───┤▒
Step 5 │ 0 │ 5 │ 4 │   │   │ 3 │ 2 │▒
       ├───┼───┼───┼───┼───┼───┼───┤▒
Step 6 │ 0 │ 5 │ 4 │   │ 6 │ 3 │ 2 │▒
       ├───┼───┼───┼───┼───┼───┼───┤▒
Step 7 │ 0 │ 5 │ 4 │ 1 │ 6 │ 3 │ 2 │▒
       └───┴───┴───┴───┴───┴───┴───┘▒
        ▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒

Proof of 2024-11-05 Q1(m=6)
═══════════════════════════

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

Combining ✅「⟦2,5⟧ ∋ 3,6」 with ✅「⟨⋯ 0 ⋯ 5 ⋯ 4 ⋯ 2 ⋯⟩」, we see that one of the following holds:

⟨⋯ 0 ⋯ 5 ⋯ 3 ⋯ 6 ⋯ 2 ⋯⟩; or 
⟨⋯ 0 ⋯ 5 ⋯ 6 ⋯ 3 ⋯ 2 ⋯⟩.

The latter holds actually, for otherwise we cannot match ✅「⟦0,3⟧ ∋ 1,6」. This pattern then implies that one of the following holds:

(i)  ⟨⋯ 0 ⋯ 6 ⋯ 1 ⋯ 3 ⋯ 2 ⋯⟩; or
(ii) ⟨⋯ 0 ⋯ 1 ⋯ 6 ⋯ 3 ⋯ 2 ⋯⟩.

Together with ✅「⟨⋯ 0 ⋯ 5 ⋯ 4 ⋯ 2 ⋯⟩」, we see that no matter which case happens, we have

(1) 0 is to the left of 1,2,3,4,5,6; and

(2) 2 is to the right of 0,1,3,4,5,6.

Therefore, we get

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

--- Idle ---
┌───┬───┬───┬───┬───┬───┬───┐
│ 1 │   │ 6 │ 3 │   │ 4 │ 5 │
└───┴───┴───┴───┴───┴───┴───┘

Next, we consider ✅「{p5, p4, p1, p0} = ? + {0,1,2,3}」.

┌───┬───┬───┬───┬───┬───┬───┐
│6th│*5 │*4 │3rd│2nd│*1 │*0 │
╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡
│ 0 │   │   │   │   │   │ 2 │
└───┴───┴───┴───┴───┴───┴───┘

We need p5,p4,p1,p0 to be consecutive integers. It follows that S := {p5, p4, p1, p0} is either {1,2,3,4} or {2,3,4,5}. As a consequence,

(3) 3,4 ∈ S; and

(4) 6 ∉ S; equivalently 6 = [3rd] | [2nd].

┌───┬───┬───┬───┬───┬───┬───┐
│6th│*5 │*4 │3rd│2nd│*1 │*0 │
╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡
│ 0 │   │   │ ▬ │ ▬ │   │ 2 │
└───┴───┴───┴───┴───┴───┴───┘

On the other hand, recall that case (i) or (ii) holds. A fortiori, we need to match

(5) ⟨ ⋯ 6 ⋯ 3 ⋯ ⟩.

Combining (3), (4), and (5), we get [1st] = 3:

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

--- Idle ---
┌───┬───┬───┬───┬───┬───┬───┐
│ 1 │   │ 6 │   │   │ 4 │ 5 │
└───┴───┴───┴───┴───┴───┴───┘

Next, we consider the position of 4. By (3), we have 4 ∈ S. Combining this with ✅「⟨⋯ 0 ⋯ 5 ⋯ 4 ⋯ 2 ⋯⟩」, we get

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

--- Idle ---
┌───┬───┬───┬───┬───┬───┬───┐
│ 1 │   │ 6 │   │   │   │   │
└───┴───┴───┴───┴───┴───┴───┘

Finally, in view of ✅「min ⊢3⊣ ≥ 2」, we finish by

       ┌───┬───┬───┬───┬───┬───┬───┐
       │6th│5th│4th│ 3■│ 2■│1st│0th│▒
       ╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡▒
       │ 0 │ 5 │ 4 │   │   │ 3 │ 2 │▒
       ├───┼───┼───┼───┼───┼───┼───┤▒
Step 6 │ 0 │ 5 │ 4 │   │ 6 │ 3 │ 2 │▒
       ├───┼───┼───┼───┼───┼───┼───┤▒
Step 7 │ 0 │ 5 │ 4 │ 1 │ 6 │ 3 │ 2 │▒
       └───┴───┴───┴───┴───┴───┴───┘▒
        ▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒

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

Q.E.D.

#125034_v2.7