Rearrange the digits in ⟨1263045⟩ to meet the rules below.
⟨6th 5th 4th 3rd 2nd 1st 0th⟩
✅Match
⟨⋯ 5 ⋯ ? ⋯ 6 (?−4)⟩ (?≠6)
⟨⋯ ? ⋯ 4 ⋯ (?−1)⟩ (?≠4,5)
⟨⋯ Perm(0,2,4) ⋯⟩
Jump(1,2) = 4
⛔Avoid
0 ∾ 2 ∾ 3 ∾ 6
⟨⋯ 0 ⋯ 4 ⋯ 5 ⋯ 1 ⋯⟩
#125034_v2.7
┌───┬───┬───┬───┬───┬───┬───┐
│6th│5th│4th│3rd│2nd│1st│0th│▒
╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡▒
Step 1 │ │ │ │ │ │ 6 │ │▒
├───┼───┼───┼───┼───┼───┼───┤▒
Step 2 │ │ │ │ │ │ 6 │ 1 │▒
├───┼───┼───┼───┼───┼───┼───┤▒
Step 3 │ │ 2 │ │ │ │ 6 │ 1 │▒
├───┼───┼───┼───┼───┼───┼───┤▒
Step 4 │ │ 2 │ 4 │ │ │ 6 │ 1 │▒
├───┼───┼───┼───┼───┼───┼───┤▒
Step 5 │ │ 2 │ 4 │ 0 │ │ 6 │ 1 │▒
├───┼───┼───┼───┼───┼───┼───┤▒
Step 6 │ │ 2 │ 4 │ 0 │ 5 │ 6 │ 1 │▒
├───┼───┼───┼───┼───┼───┼───┤▒
Step 7 │ 3 │ 2 │ 4 │ 0 │ 5 │ 6 │ 1 │▒
└───┴───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
Proof of 2024-12-10 Q1(m=6)
═══════════════════════════
Notation: if nth -> a, then we write [nth] = a.
Plainly, our step 1 follows from ✅「⟨⋯ 5 ⋯ ? ⋯ 6 (?−4)⟩ (?≠6)」:
┌───┬───┬───┬───┬───┬───┬───┐
│6th│5th│4th│3rd│2nd│ 1■│0th│▒
╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡▒
Step 1 │ │ │ │ │ │ 6 │ │▒
└───┴───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
--- Idle ---
┌───┬───┬───┬───┬───┬───┬───┐
│ 1 │ 2 │ │ 3 │ 0 │ 4 │ 5 │
└───┴───┴───┴───┴───┴───┴───┘
Then, to match ✅「Jump(1,2) = 4」, we have to use the 5th and 0th positions:
┌───┬───┬───┬───┬───┬───┬───┐
│6th│ 5▲│4th│3rd│2nd│1st│ 0▲│
╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡
│ │ ▬ │ │ │ │ 6 │ ▬ │
└───┴───┴───┴───┴───┴───┴───┘
By ✅「⟨⋯ 5 ⋯ ? ⋯ 6 (?−4)⟩ (?≠6)」, we have [0th]<=1. Therefore, we get
┌───┬───┬───┬───┬───┬───┬───┐
│6th│ 5■│4th│3rd│2nd│1st│ 0■│▒
╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡▒
│ │ │ │ │ │ 6 │ │▒
├───┼───┼───┼───┼───┼───┼───┤▒
Step 2 │ │ │ │ │ │ 6 │ 1 │▒
├───┼───┼───┼───┼───┼───┼───┤▒
Step 3 │ │ 2 │ │ │ │ 6 │ 1 │▒
└───┴───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
--- Idle ---
┌───┬───┬───┬───┬───┬───┬───┐
│ │ │ │ 3 │ 0 │ 4 │ 5 │
└───┴───┴───┴───┴───┴───┴───┘
Then, note that ✅「⟨⋯ ? ⋯ 4 ⋯ (?−1)⟩ (?≠4,5)」 becomes
(1) ⟨⋯ 2 ⋯ 4 ⋯ 1⟩.
Combining this with ✅「⟨⋯ Perm(0,2,4) ⋯⟩」, there are two possibilities:
┌───┬───┬───┬───┬───┬───┬───┐
│6th│5th│ 4▲│ 3▲│2nd│1st│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡
(2) │ │ 2 │ 4 │ │ │ 6 │ 1 │
├───┼───┼───┼───┼───┼───┼───┤
(3) │ │ 2 │ │ 4 │ │ 6 │ 1 │
└───┴───┴───┴───┴───┴───┴───┘
(4) We show that (2) holds actually.
------------------------------
If (3) holds on the contrary, then by ✅「⟨⋯ Perm(0,2,4) ⋯⟩」, we get
┌───┬───┬───┬───┬───┬───┬───┐
│6th│5th│ 4▲│3rd│2nd│1st│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡
│ │ 2 │ 0 │ 4 │ │ 6 │ 1 │
└───┴───┴───┴───┴───┴───┴───┘
--- Idle ---
┌───┬───┬───┬───┬───┬───┬───┐
│ │ │ │ 3 │ │ │ 5 │
└───┴───┴───┴───┴───┴───┴───┘
To avoid ⛔「⟨⋯ 0 ⋯ 4 ⋯ 5 ⋯ 1 ⋯⟩」, we reach
┌───┬───┬───┬───┬───┬───┬───┐
│ 6▲│5th│4th│3rd│ 2▲│1st│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡
│ 5 │ 2 │ 0 │ 4 │ 3 │ 6 │ 1 │
└───┴───┴───┴───┴───┴───┴───┘
It is a contradiction, however, as it matches ⛔「0 ∾ 2 ∾ 3 ∾ 6」.
------------------------------
We have verified (4). Accordingly, we get
┌───┬───┬───┬───┬───┬───┬───┐
│6th│5th│ 4■│3rd│2nd│1st│0th│▒
╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡▒
Step 4 │ │ 2 │ 4 │ │ │ 6 │ 1 │▒
└───┴───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
--- Idle ---
┌───┬───┬───┬───┬───┬───┬───┐
│ │ │ │ 3 │ 0 │ │ 5 │
└───┴───┴───┴───┴───┴───┴───┘
Then, we consider where to place 0. To match ✅「⟨⋯ Perm(0,2,4) ⋯⟩」 and avoid ⛔「⟨⋯ 0 ⋯ 4 ⋯ 5 ⋯ 1 ⋯⟩」 at the same time, we have to place it at 3rd:
┌───┬───┬───┬───┬───┬───┬───┐
│6th│5th│4th│ 3■│2nd│1st│0th│▒
╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡▒
│ │ 2 │ 4 │ │ │ 6 │ 1 │▒
├───┼───┼───┼───┼───┼───┼───┤▒
Step 5 │ │ 2 │ 4 │ 0 │ │ 6 │ 1 │▒
└───┴───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
--- Idle ---
┌───┬───┬───┬───┬───┬───┬───┐
│ │ │ │ 3 │ │ │ 5 │
└───┴───┴───┴───┴───┴───┴───┘
Finally, to avoid ⛔「0 ∾ 2 ∾ 3 ∾ 6」, we place 5 at [2nd] to form the cycle 2 ∾ 5. We finish by
┌───┬───┬───┬───┬───┬───┬───┐
│ 6■│5th│4th│3rd│ 2■│1st│0th│▒
╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡▒
│ │ 2 │ 4 │ 0 │ │ 6 │ 1 │▒
├───┼───┼───┼───┼───┼───┼───┤▒
Step 6 │ │ 2 │ 4 │ 0 │ 5 │ 6 │ 1 │▒
├───┼───┼───┼───┼───┼───┼───┤▒
Step 7 │ 3 │ 2 │ 4 │ 0 │ 5 │ 6 │ 1 │▒
└───┴───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
--- Idle ---
┌───┬───┬───┬───┬───┬───┬───┐
│ │ │ │ │ │ │ │
└───┴───┴───┴───┴───┴───┴───┘
Q.E.D.
#125034_v2.7