Rearrange the digits in ⟨1263045⟩ to meet the rules below.
⟨6th 5th 4th 3rd 2nd 1st 0th⟩
✅Match
⟨⋯ 6 ⋯ 0 ⋯⟩
⟨⋯ Perm(0,1,4) ⋯⟩
Jump(4,5) = 1
5th → a, 1st → b, ab=0
⛔Avoid
Jump(2,3) ≥ 2
min ⊢3⊣ ≥ 2
⟨⋯ 2 ⋯ 6 ⋯⟩
#125034_v2.6
┌───┬───┬───┬───┬───┬───┬───┐
│6th│5th│4th│3rd│2nd│1st│0th│▒
╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡▒
Step 1 │ │ │ │ │ │ 0 │ │▒
├───┼───┼───┼───┼───┼───┼───┤▒
Step 2 │ │ │ │ 1 │ │ 0 │ │▒
├───┼───┼───┼───┼───┼───┼───┤▒
Step 3 │ │ │ │ 1 │ 4 │ 0 │ │▒
├───┼───┼───┼───┼───┼───┼───┤▒
Step 4 │ │ │ 3 │ 1 │ 4 │ 0 │ │▒
├───┼───┼───┼───┼───┼───┼───┤▒
Step 5 │ │ │ 3 │ 1 │ 4 │ 0 │ 5 │▒
├───┼───┼───┼───┼───┼───┼───┤▒
Step 6 │ 6 │ │ 3 │ 1 │ 4 │ 0 │ 5 │▒
├───┼───┼───┼───┼───┼───┼───┤▒
Step 7 │ 6 │ 2 │ 3 │ 1 │ 4 │ 0 │ 5 │▒
└───┴───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
Proof of 2024-10-08 Q1(m=6)
═══════════════════════════
Notation: if nth -> a, then we write [nth] = a.
We begin with an observation. To avoid ⛔「min ⊢3⊣ ≥ 2」, we need
(1) 3 is adjacent to 1 or 0.
We start the proof. By ✅「5th → a, 1st → b, ab=0」, we have 0 = [5th] | [1st].
(2) We claim that 0 = [1st].
------------------------------
For, suppose on the contrary 0 = [5th]. By ✅「⟨⋯ 6 ⋯ 0 ⋯⟩」, we have
┌───┬───┬───┬───┬───┬───┬───┐
│ 6▲│ 5▲│4th│3rd│2nd│1st│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡
│ 6 │ 0 │ │ │ │ │ │
└───┴───┴───┴───┴───┴───┴───┘
Then, in view of ✅「⟨⋯ Perm(0,1,4) ⋯⟩」 and ✅「Jump(4,5) = 1」, two cases follow:
┌───┬───┬───┬───┬───┬───┬───┐
│6th│5th│ 4▲│ 3▲│ 2▲│ 1▲│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡
(3) │ 6 │ 0 │ 1 │ 4 │ │ 5 │ │
├───┼───┼───┼───┼───┼───┼───┤
(4) │ 6 │ 0 │ 4 │ 1 │ 5 │ │ │
└───┴───┴───┴───┴───┴───┴───┘
Both cases give contradiction, however, as we have to match (1).
------------------------------
We have verified our claim in (2). Accordingly, we get
┌───┬───┬───┬───┬───┬───┬───┐
│6th│5th│4th│3rd│2nd│ 1■│0th│▒
╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡▒
Step 1 │ │ │ │ │ │ 0 │ │▒
└───┴───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
--- Idle ---
┌───┬───┬───┬───┬───┬───┬───┐
│ 1 │ 2 │ 6 │ 3 │ │ 4 │ 5 │
└───┴───┴───┴───┴───┴───┴───┘
Next, by ✅「⟨⋯ Perm(0,1,4) ⋯⟩」, there are two possibilities:
┌───┬───┬───┬───┬───┬───┬───┐
│6th│5th│4th│3rd│2nd│1st│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡
(5) │ │ │ │ │ ▬ │ 0 │ ▬ │
├───┼───┼───┼───┼───┼───┼───┤
(6) │ │ │ │ ▬ │ ▬ │ 0 │ │
└───┴───┴───┴───┴───┴───┴───┘
where "▬" are occupied by 1,4.
If case (5) holds, then (1) implies
┌───┬───┬───┬───┬───┬───┬───┐
│6th│5th│4th│3rd│ 2▲│1st│ 0▲│
╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡
│ │ │ │ │ 1 │ 0 │ 4 │
└───┴───┴───┴───┴───┴───┴───┘
But then we cannot match ✅「Jump(4,5) = 1」. Therefore, case (5) does not hold and case (6) holds instead. Then, in view of (1), there are three cases:
┌───┬───┬───┬───┬───┬───┬───┐
│6th│5th│ 4▲│ 3▲│ 2▲│1st│ 0▲│
╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡
(7) │ │ │ │ 4 │ 1 │ 0 │ 3 │
├───┼───┼───┼───┼───┼───┼───┤
(8) │ │ │ │ 1 │ 4 │ 0 │ 3 │
├───┼───┼───┼───┼───┼───┼───┤
(9) │ │ │ 3 │ 1 │ 4 │ 0 │ │
└───┴───┴───┴───┴───┴───┴───┘
Note that if case (7) or (8) holds, then we cannot avoid ⛔「Jump(2,3) ≥ 2」. Therefore, case (9) holds actually. We get
┌───┬───┬───┬───┬───┬───┬───┐
│6th│5th│ 4■│ 3■│ 2■│1st│0th│▒
╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡▒
│ │ │ │ │ │ 0 │ │▒
├───┼───┼───┼───┼───┼───┼───┤▒
Step 2 │ │ │ │ 1 │ │ 0 │ │▒
├───┼───┼───┼───┼───┼───┼───┤▒
Step 3 │ │ │ │ 1 │ 4 │ 0 │ │▒
├───┼───┼───┼───┼───┼───┼───┤▒
Step 4 │ │ │ 3 │ 1 │ 4 │ 0 │ │▒
└───┴───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
--- Idle ---
┌───┬───┬───┬───┬───┬───┬───┐
│ │ 2 │ 6 │ │ │ │ 5 │
└───┴───┴───┴───┴───┴───┴───┘
Then, by ✅「Jump(4,5) = 1」, we have
┌───┬───┬───┬───┬───┬───┬───┐
│6th│5th│4th│3rd│2nd│1st│ 0■│▒
╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡▒
│ │ │ 3 │ 1 │ 4 │ 0 │ │▒
├───┼───┼───┼───┼───┼───┼───┤▒
Step 5 │ │ │ 3 │ 1 │ 4 │ 0 │ 5 │▒
└───┴───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
--- Idle ---
┌───┬───┬───┬───┬───┬───┬───┐
│ │ 2 │ 6 │ │ │ │ │
└───┴───┴───┴───┴───┴───┴───┘
Finally, in view of ⛔「⟨⋯ 2 ⋯ 6 ⋯⟩」, we finish by
┌───┬───┬───┬───┬───┬───┬───┐
│ 6■│ 5■│4th│3rd│2nd│1st│0th│▒
╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡▒
│ │ │ 3 │ 1 │ 4 │ 0 │ 5 │▒
├───┼───┼───┼───┼───┼───┼───┤▒
Step 6 │ 6 │ │ 3 │ 1 │ 4 │ 0 │ 5 │▒
├───┼───┼───┼───┼───┼───┼───┤▒
Step 7 │ 6 │ 2 │ 3 │ 1 │ 4 │ 0 │ 5 │▒
└───┴───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
--- Idle ---
┌───┬───┬───┬───┬───┬───┬───┐
│ │ │ │ │ │ │ │
└───┴───┴───┴───┴───┴───┴───┘
Q.E.D.
#125034_v2.6