2023-12-15 Q1(m=6)

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

⟨6th 5th 4th 3rd 2nd 1st 0th⟩

✅Match

5th → 3

6th → a, 0th → b, ab=24

2nd → 0|5|6

3rd → a, 0th → b, a+b=0+6n

⛔Avoid

5th → a, 1st → b, a+b=8

⟦0,6⟧ ∋ 1,2,5

Jump(0,5) = 1

2nd → a, 1st → b, a+b=6

#125034_v2.2

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

Proof of 2023-12-15 Q1(m=6)
═══════════════════════════

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

Plainly, our first step follows from ✅「5th → 3」. Next, according to ✅「6th → a, 0th → b, ab=24」, we have

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

where {X,Y} = {4,6}.

(1) We proceed to show that (X,Y) = (4,6).

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

If on the contrary (1) does not hold, then (X,Y) = (6,4), and in view of ✅「3rd → a, 0th → b, a+b=0+6n」, we get

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

--- Idle ---
┌───┬───┬───┬───┬───┬───┬───┐
│ 1 │   │   │   │ 0 │   │ 5 │
└───┴───┴───┴───┴───┴───┴───┘

Then, to avoid ⛔「Jump(0,5) = 1」, we need [1st] = 0 or 5. We need to avoid ⛔「5th → a, 1st → b, a+b=8」 too, so [1st] = 0.

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

But then we would match ⛔「⟦0,6⟧ ∋ 1,2,5」, which is a contradiction.

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

We have verified (1). Accordingly, we have:

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

--- Idle ---
┌───┬───┬───┬───┬───┬───┬───┐
│ 1 │ 2 │   │   │ 0 │   │ 5 │
└───┴───┴───┴───┴───┴───┴───┘

It now follows from ✅「3rd → a, 0th → b, a+b=0+6n」 that [3rd] = 0:

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

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

To match ✅「2nd → 0|5|6」, we need [2nd] = 5.

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

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

Finally, to avoid ⛔「2nd → a, 1st → b, a+b=6」, we finish by

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

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

Q.E.D.

#125034_v2.2