Rearrange the digits in ⟨1263045⟩ to meet the rules below.
⟨6th 5th 4th 3rd 2nd 1st 0th⟩
✅Match
⟨⋯ 1 ⋯ ? 0 ⋯ (?+3)⟩ (?≠0)
6th → a, 3rd → b, ab=0+4n
median {p5, p1, p0} = 1
⟨ ⁶ᵗʰb ⁵ᵗʰc ⁴ᵗʰa ⟩, a > b > c
⛔Avoid
⟨⋯ a ⋯ 1 ⋯⟩, a = 3|4|6
#125034_v2.7
┌───┬───┬───┬───┬───┬───┬───┐
│6th│5th│4th│3rd│2nd│1st│0th│▒
╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡▒
Step 1 │ │ │ │ │ │ 0 │ │▒
├───┼───┼───┼───┼───┼───┼───┤▒
Step 2 │ │ 1 │ │ │ │ 0 │ │▒
├───┼───┼───┼───┼───┼───┼───┤▒
Step 3 │ 2 │ 1 │ │ │ │ 0 │ │▒
├───┼───┼───┼───┼───┼───┼───┤▒
Step 4 │ 2 │ 1 │ │ │ 3 │ 0 │ │▒
├───┼───┼───┼───┼───┼───┼───┤▒
Step 5 │ 2 │ 1 │ │ │ 3 │ 0 │ 6 │▒
├───┼───┼───┼───┼───┼───┼───┤▒
Step 6 │ 2 │ 1 │ │ 4 │ 3 │ 0 │ 6 │▒
├───┼───┼───┼───┼───┼───┼───┤▒
Step 7 │ 2 │ 1 │ 5 │ 4 │ 3 │ 0 │ 6 │▒
└───┴───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
Proof of 2024-10-22 Q1(m=6)
═══════════════════════════
Notation: if Nth -> a, then we write pN = a.
Note that ✅「median {p5, p1, p0} = 1」 implies
(1) 0,1 ∈ {p5, p1, p0}.
A fortiori, 0 ∈ {p5, p1, p0}. Let us consider where to place it.
┌───┬───┬───┬───┬───┬───┬───┐
│6th│5th│4th│3rd│2nd│1st│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡
│ │ ▬ │ │ │ │ ▬ │ ▬ │
└───┴───┴───┴───┴───┴───┴───┘
By ✅「⟨⋯ 1 ⋯ ? 0 ⋯ (?+3)⟩ (?≠0)」, 0 is not at the right corner; and if 0 = p5, then 1 = p6:
┌───┬───┬───┬───┬───┬───┬───┐
│ 6▲│ 5▲│4th│3rd│2nd│1st│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡
│ 1 │ 0 │ │ │ │ │ │
└───┴───┴───┴───┴───┴───┴───┘
But this contradicts (1). Therefore, 0 = p1:
┌───┬───┬───┬───┬───┬───┬───┐
│6th│5th│4th│3rd│2nd│ 1■│0th│▒
╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡▒
Step 1 │ │ │ │ │ │ 0 │ │▒
└───┴───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
--- Idle ---
┌───┬───┬───┬───┬───┬───┬───┐
│ 1 │ 2 │ 6 │ 3 │ │ 4 │ 5 │
└───┴───┴───┴───┴───┴───┴───┘
Combining (1) and ✅「⟨⋯ 1 ⋯ ? 0 ⋯ (?+3)⟩ (?≠0)」, we get 1 = p5 as well:
┌───┬───┬───┬───┬───┬───┬───┐
│6th│ 5■│4th│3rd│2nd│1st│0th│▒
╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡▒
│ │ │ │ │ │ 0 │ │▒
├───┼───┼───┼───┼───┼───┼───┤▒
Step 2 │ │ 1 │ │ │ │ 0 │ │▒
└───┴───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
--- Idle ---
┌───┬───┬───┬───┬───┬───┬───┐
│ │ 2 │ 6 │ 3 │ │ 4 │ 5 │
└───┴───┴───┴───┴───┴───┴───┘
Then, by ⛔「⟨⋯ a ⋯ 1 ⋯⟩, a = 3|4|6」, we have
(2) p6 ∈ {2,5}.
┌───┬───┬───┬───┬───┬───┬───┐
│6th│5th│4th│3rd│2nd│1st│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡
│ ▬ │ 1 │ │ │ │ 0 │ │
└───┴───┴───┴───┴───┴───┴───┘
(2.1) We claim that p6 = 2 actually.
------------------------------
For, if on the contrary p6 = 5:
┌───┬───┬───┬───┬───┬───┬───┐
│ 6▲│5th│4th│3rd│2nd│1st│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡
│ 5 │ 1 │ │ │ │ 0 │ │
└───┴───┴───┴───┴───┴───┴───┘
--- Idle ---
┌───┬───┬───┬───┬───┬───┬───┐
│ │ 2 │ 6 │ 3 │ │ 4 │ │
└───┴───┴───┴───┴───┴───┴───┘
then to match ✅「⟨ ⁶ᵗʰb ⁵ᵗʰc ⁴ᵗʰa ⟩, a > b > c」, we need p4 = 6:
┌───┬───┬───┬───┬───┬───┬───┐
│6th│5th│ 4▲│3rd│2nd│1st│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡
│ 5 │ 1 │ 6 │ │ │ 0 │ │
└───┴───┴───┴───┴───┴───┴───┘
--- Idle ---
┌───┬───┬───┬───┬───┬───┬───┐
│ │ 2 │ │ 3 │ │ 4 │ │
└───┴───┴───┴───┴───┴───┴───┘
But then we cannot match ✅「⟨⋯ 1 ⋯ ? 0 ⋯ (?+3)⟩ (?≠0)」, which is a contradiction.
------------------------------
We have verified our claim in (2.1). Accordingly, we get
┌───┬───┬───┬───┬───┬───┬───┐
│ 6■│5th│4th│3rd│2nd│1st│0th│▒
╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡▒
│ │ 1 │ │ │ │ 0 │ │▒
├───┼───┼───┼───┼───┼───┼───┤▒
Step 3 │ 2 │ 1 │ │ │ │ 0 │ │▒
└───┴───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
--- Idle ---
┌───┬───┬───┬───┬───┬───┬───┐
│ │ │ 6 │ 3 │ │ 4 │ 5 │
└───┴───┴───┴───┴───┴───┴───┘
Next, we consider the value of p0. In view of ✅「⟨⋯ 1 ⋯ ? 0 ⋯ (?+3)⟩ (?≠0)」, we get p0 = 6 and the "?" in that pattern is 3. It follows
┌───┬───┬───┬───┬───┬───┬───┐
│6th│5th│4th│3rd│ 2■│1st│ 0■│▒
╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡▒
│ 2 │ 1 │ │ │ │ 0 │ │▒
├───┼───┼───┼───┼───┼───┼───┤▒
Step 4 │ 2 │ 1 │ │ │ 3 │ 0 │ │▒
├───┼───┼───┼───┼───┼───┼───┤▒
Step 5 │ 2 │ 1 │ │ │ 3 │ 0 │ 6 │▒
└───┴───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
--- Idle ---
┌───┬───┬───┬───┬───┬───┬───┐
│ │ │ │ │ │ 4 │ 5 │
└───┴───┴───┴───┴───┴───┴───┘
Finally, to match ✅「6th → a, 3rd → b, ab=0+4n」, we finish by
┌───┬───┬───┬───┬───┬───┬───┐
│6th│5th│ 4■│ 3■│2nd│1st│0th│▒
╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡▒
│ 2 │ 1 │ │ │ 3 │ 0 │ 6 │▒
├───┼───┼───┼───┼───┼───┼───┤▒
Step 6 │ 2 │ 1 │ │ 4 │ 3 │ 0 │ 6 │▒
├───┼───┼───┼───┼───┼───┼───┤▒
Step 7 │ 2 │ 1 │ 5 │ 4 │ 3 │ 0 │ 6 │▒
└───┴───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
--- Idle ---
┌───┬───┬───┬───┬───┬───┬───┐
│ │ │ │ │ │ │ │
└───┴───┴───┴───┴───┴───┴───┘
Q.E.D.
#125034_v2.7