Rearrange the digits in ⟨1263045⟩ to meet the rules below.
⟨6th 5th 4th 3rd 2nd 1st 0th⟩
✅Match
6th → a, 4th → b, ab=2+6n
⟨⋯ ? ⋯ 0 ⋯ (?+3)⟩ (?≠0)
⟨⋯ 1 ⋯ 4 ⋯ 2 ⋯ 3 ⋯⟩
Sim⟨ ⁶ᵗʰ5 ⁵ᵗʰ4 ⁴ᵗʰ0 ³ʳᵈ2 ²ⁿᵈ1 ¹ˢᵗ6 ⁰ᵗʰ3 ⟩ = 2
⛔Avoid
⟨ ⁶ᵗʰa ⁵ᵗʰb ⟩, max⟦a,b⟧ = 5
⟨⋯ 2 ⋯ ? 6 ⋯ (?+2)⟩ (?≠4)
#125034_v2.6
┌───┬───┬───┬───┬───┬───┬───┐
│6th│5th│4th│3rd│2nd│1st│0th│▒
╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡▒
Step 1 │ │ 4 │ │ │ │ │ │▒
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Step 2 │ 1 │ 4 │ │ │ │ │ │▒
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Step 3 │ 1 │ 4 │ 2 │ │ │ │ │▒
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Step 4 │ 1 │ 4 │ 2 │ │ │ │ 5 │▒
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Step 5 │ 1 │ 4 │ 2 │ │ │ 6 │ 5 │▒
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Step 6 │ 1 │ 4 │ 2 │ 3 │ │ 6 │ 5 │▒
├───┼───┼───┼───┼───┼───┼───┤▒
Step 7 │ 1 │ 4 │ 2 │ 3 │ 0 │ 6 │ 5 │▒
└───┴───┴───┴───┴───┴───┴───┘▒
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Proof of 2024-09-24 Q1(m=6)
═══════════════════════════
Notation: if nth -> a, then we write [nth] = a.
By ✅「⟨⋯ 1 ⋯ 4 ⋯ 2 ⋯ 3 ⋯⟩」, 4 is not at the right corner. Therefore, ✅「⟨⋯ ? ⋯ 0 ⋯ (?+3)⟩ (?≠0)」 implies one of the following holds:
(1)
(a) ⟨⋯ 2 ⋯ 0 ⋯ 5⟩; or
(b) ⟨⋯ 3 ⋯ 0 ⋯ 6⟩.
Combining this with ✅「⟨⋯ 1 ⋯ 4 ⋯ 2 ⋯ 3 ⋯⟩」, we see that no matter which case happens, there are at least four digits at the right of 4, and at least one digit at the left of 4. So, 4 = [5th] or [4th]:
┌───┬───┬───┬───┬───┬───┬───┐
│6th│ 5▲│ 4▲│3rd│2nd│1st│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡
│ │ ▬ │ ▬ │ │ │ │ │
└───┴───┴───┴───┴───┴───┴───┘
(2) We show that 4 = [5th] actually.
------------------------------
Suppose on the contrary 4 = [4th]:
┌───┬───┬───┬───┬───┬───┬───┐
│6th│5th│ 4▲│3rd│2nd│1st│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡
│ │ │ 4 │ │ │ │ │
└───┴───┴───┴───┴───┴───┴───┘
Then, by ✅「6th → a, 4th → b, ab=2+6n」, we have [6th] = 2|5. This is a contradiction however, because
(2.1) if [6th] = 2, then we cannot match ✅「⟨⋯ 1 ⋯ 4 ⋯ 2 ⋯ 3 ⋯⟩」:
┌───┬───┬───┬───┬───┬───┬───┐
│ 6▲│5th│4th│3rd│2nd│1st│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡
│ 2 │ │ 4 │ │ │ │ │
└───┴───┴───┴───┴───┴───┴───┘
(2.2) Else if [6th] = 5, then to match ✅「⟨⋯ 1 ⋯ 4 ⋯ 2 ⋯ 3 ⋯⟩」 we need [5th] = 1:
┌───┬───┬───┬───┬───┬───┬───┐
│ 6▲│ 5▲│4th│3rd│2nd│1st│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡
│ 5 │ 1 │ 4 │ │ │ │ │
└───┴───┴───┴───┴───┴───┴───┘
but then we match ⛔「⟨ ⁶ᵗʰa ⁵ᵗʰb ⟩, max⟦a,b⟧ = 5」.
------------------------------
We have verified our claim in (2). Accordingly, we get
┌───┬───┬───┬───┬───┬───┬───┐
│6th│ 5■│4th│3rd│2nd│1st│0th│▒
╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡▒
Step 1 │ │ 4 │ │ │ │ │ │▒
└───┴───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
--- Idle ---
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│ 1 │ 2 │ 6 │ 3 │ 0 │ │ 5 │
└───┴───┴───┴───┴───┴───┴───┘
Then, using ✅「⟨⋯ 1 ⋯ 4 ⋯ 2 ⋯ 3 ⋯⟩」, we get [6th] = 1. By ✅「6th → a, 4th → b, ab=2+6n」, we get [4th] = 2 as well:
┌───┬───┬───┬───┬───┬───┬───┐
│ 6■│5th│ 4■│3rd│2nd│1st│0th│▒
╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡▒
│ │ 4 │ │ │ │ │ │▒
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Step 2 │ 1 │ 4 │ │ │ │ │ │▒
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Step 3 │ 1 │ 4 │ 2 │ │ │ │ │▒
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▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
--- Idle ---
┌───┬───┬───┬───┬───┬───┬───┐
│ │ │ 6 │ 3 │ 0 │ │ 5 │
└───┴───┴───┴───┴───┴───┴───┘
To continue, we consider ✅「Sim⟨ ⁶ᵗʰ5 ⁵ᵗʰ4 ⁴ᵗʰ0 ³ʳᵈ2 ²ⁿᵈ1 ¹ˢᵗ6 ⁰ᵗʰ3 ⟩ = 2」. To match it, we need exactly one of the following:
(3)
(a) [1st] = 6; or
(b) [0th] = 3.
Observe that (3) cannot happen if case (1)(b) holds. Therefore, case (1)(a) holds, and we get [0th] = 5:
┌───┬───┬───┬───┬───┬───┬───┐
│6th│5th│4th│3rd│2nd│1st│ 0■│▒
╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡▒
│ 1 │ 4 │ 2 │ │ │ │ │▒
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Step 4 │ 1 │ 4 │ 2 │ │ │ │ 5 │▒
└───┴───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
--- Idle ---
┌───┬───┬───┬───┬───┬───┬───┐
│ │ │ 6 │ 3 │ 0 │ │ │
└───┴───┴───┴───┴───┴───┴───┘
Case (3)(b) becomes impossible. So, case (3)(a) holds:
┌───┬───┬───┬───┬───┬───┬───┐
│6th│5th│4th│3rd│2nd│ 1■│0th│▒
╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡▒
│ 1 │ 4 │ 2 │ │ │ │ 5 │▒
├───┼───┼───┼───┼───┼───┼───┤▒
Step 5 │ 1 │ 4 │ 2 │ │ │ 6 │ 5 │▒
└───┴───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
--- Idle ---
┌───┬───┬───┬───┬───┬───┬───┐
│ │ │ │ 3 │ 0 │ │ │
└───┴───┴───┴───┴───┴───┴───┘
Finally, to avoid ⛔「⟨⋯ 2 ⋯ ? 6 ⋯ (?+2)⟩ (?≠4)」, we finish by
┌───┬───┬───┬───┬───┬───┬───┐
│6th│5th│4th│ 3■│ 2■│1st│0th│▒
╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡▒
│ 1 │ 4 │ 2 │ │ │ 6 │ 5 │▒
├───┼───┼───┼───┼───┼───┼───┤▒
Step 6 │ 1 │ 4 │ 2 │ 3 │ │ 6 │ 5 │▒
├───┼───┼───┼───┼───┼───┼───┤▒
Step 7 │ 1 │ 4 │ 2 │ 3 │ 0 │ 6 │ 5 │▒
└───┴───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
--- Idle ---
┌───┬───┬───┬───┬───┬───┬───┐
│ │ │ │ │ │ │ │
└───┴───┴───┴───┴───┴───┴───┘
Q.E.D.
#125034_v2.6