2024-09-17 Q1(m=6)

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

⟨6th 5th 4th 3rd 2nd 1st 0th⟩

✅Match

Jump(0,1) = 4

Jump(1,6) ≥ 3

min ⊢5⊣ = 3

⟨⋯ 3 ⋯ 5 ⋯⟩

⛔Avoid

min ⊢4⊣ ≥ 2

⟨ ⁶ᵗʰa       ²ⁿᵈb ¹ˢᵗc   ⟩, (abc)₁₀ ≥ 123

#125034_v2.6

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

Proof of 2024-09-17 Q1(m=6)
═══════════════════════════

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

In order to avoid ⛔「⟨ ⁶ᵗʰa       ²ⁿᵈb ¹ˢᵗc   ⟩, (abc)₁₀ ≥ 123」, we need [6th] = 1|0.

(1) We claim that [6th] = 0 actually.

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

Suppose this is not the case. Then [6th] = 1. By ✅「Jump(0,1) = 4」, we have [1st] = 0 as well:

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

--- Idle ---
┌───┬───┬───┬───┬───┬───┬───┐
│   │ 2 │ 6 │ 3 │   │ 4 │ 5 │
└───┴───┴───┴───┴───┴───┴───┘

Then, observe that to avoid ⛔「⟨ ⁶ᵗʰa       ²ⁿᵈb ¹ˢᵗc   ⟩, (abc)₁₀ ≥ 123」, we need [2nd] = 2:

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

It follows that there is only one way to match ✅「Jump(1,6) ≥ 3」:

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

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

and only one way to avoid ⛔「min ⊢4⊣ ≥ 2」:

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

Using ✅「⟨⋯ 3 ⋯ 5 ⋯⟩」, we reach

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

It is a contradiction, however, as we do not match ✅「min ⊢5⊣ = 3」.

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

We have verified our claim in (1). Accordingly, we get

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

--- Idle ---
┌───┬───┬───┬───┬───┬───┬───┐
│ 1 │ 2 │ 6 │ 3 │   │ 4 │ 5 │
└───┴───┴───┴───┴───┴───┴───┘

Plainly, by ✅「Jump(0,1) = 4」 and ✅「Jump(1,6) ≥ 3」, we then get

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

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

Next, note that by combining ✅「min ⊢5⊣ = 3」 with ✅「⟨⋯ 3 ⋯ 5 ⋯⟩」, we have the following required pattern:

(2) ⟨⋯ 3 5 ⋯⟩

There are two possibilities:

      ┌───┬───┬───┬───┬───┬───┬───┐
      │6th│5th│4th│3rd│2nd│1st│0th│
      ╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡
(2.1) │ 0 │ 6 │   │ 3 │ 5 │ 1 │   │
      ├───┼───┼───┼───┼───┼───┼───┤
(2.2) │ 0 │ 6 │ 3 │ 5 │   │ 1 │   │
      └───┴───┴───┴───┴───┴───┴───┘

In view of ✅「min ⊢5⊣ = 3」, case (2.1) does not hold. So, case (2.2) holds, and we get

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

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

To match ✅「min ⊢5⊣ = 3」, we cannot place 2 next to 5. Therefore, we finish by

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

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

Q.E.D.

#125034_v2.6