Rearrange the digits in ⟨125034⟩ to meet the rules below.
⟨5th 4th 3rd 2nd 1st 0th⟩
✅Match
min {p4, p1, p0} = 2
5th → a, 1st → b, ab=0
⟨⋯ ? ⋯ 4 ⋯ (?−3)⟩ (?≠4)
3rd → a, 1st → b, a+b=2+5n
⛔Avoid
⟨ ⁴ᵗʰa ³ʳᵈc ¹ˢᵗb ⟩, a > b > c
#125034_v2.10
┌───┬───┬───┬───┬───┬───┐
│5th│4th│3rd│2nd│1st│0th│▒
╞═══╪═══╪═══╪═══╪═══╪═══╡▒
Step 1 │ 0 │ │ │ │ │ │▒
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Step 2 │ 0 │ │ │ │ │ 2 │▒
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Step 3 │ 0 │ │ │ 1 │ │ 2 │▒
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Step 4 │ 0 │ 5 │ │ 1 │ │ 2 │▒
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Step 5 │ 0 │ 5 │ 4 │ 1 │ │ 2 │▒
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Step 6 │ 0 │ 5 │ 4 │ 1 │ 3 │ 2 │▒
└───┴───┴───┴───┴───┴───┘▒
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Proof of 2025-11-11 WR
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Notation: if nth -> a, then we write [nth] = a.
By ✅「5th → a, 1st → b, ab=0」, we have
(1) 0 = [5th] or [1st].
If 0 = [1st] then we cannot match ✅「min {p4, p1, p0} = 2」. Therefore, 0 = [5th]:
┌───┬───┬───┬───┬───┬───┐
│ 5■│4th│3rd│2nd│1st│0th│▒
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Step 1 │ 0 │ │ │ │ │ │▒
└───┴───┴───┴───┴───┴───┘▒
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--- Idle ---
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│ 1 │ 2 │ 5 │ │ 3 │ 4 │
└───┴───┴───┴───┴───┴───┘
Given that 0 has been used, ✅「⟨⋯ ? ⋯ 4 ⋯ (?−3)⟩ (?≠4)」 implies that [0th] = 2:
┌───┬───┬───┬───┬───┬───┐
│5th│4th│3rd│2nd│1st│ 0■│▒
╞═══╪═══╪═══╪═══╪═══╪═══╡▒
│ 0 │ │ │ │ │ │▒
├───┼───┼───┼───┼───┼───┤▒
Step 2 │ 0 │ │ │ │ │ 2 │▒
└───┴───┴───┴───┴───┴───┘▒
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--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ 1 │ │ 5 │ │ 3 │ 4 │
└───┴───┴───┴───┴───┴───┘
Then, to match ✅「3rd → a, 1st → b, a+b=2+5n」, we need
(2) { [3rd] , [1st] } = {3,4}.
┌───┬───┬───┬───┬───┬───┐
│5th│4th│3rd│2nd│1st│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╡
│ 0 │ │3 4│ │3 4│ 2 │
└───┴───┴───┴───┴───┴───┘
--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ 1 │ │ 5 │ │ │ │
└───┴───┴───┴───┴───┴───┘
A fortiori, 1 = [4th] or [2nd]. If it is [4th], then we cannot match ✅「min {p4, p1, p0} = 2」. Therefore, 1 = [2nd], and 5 = [4th] follows:
┌───┬───┬───┬───┬───┬───┐
│5th│ 4■│3rd│ 2■│1st│0th│▒
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│ 0 │ │ │ │ │ 2 │▒
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Step 3 │ 0 │ │ │ 1 │ │ 2 │▒
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Step 4 │ 0 │ 5 │ │ 1 │ │ 2 │▒
└───┴───┴───┴───┴───┴───┘▒
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--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ │ │ │ │ 3 │ 4 │
└───┴───┴───┴───┴───┴───┘
Finally, to avoid ⛔「⟨ ⁴ᵗʰa ³ʳᵈc ¹ˢᵗb ⟩, a > b > c」, we finish by
┌───┬───┬───┬───┬───┬───┐
│5th│4th│ 3■│2nd│ 1■│0th│▒
╞═══╪═══╪═══╪═══╪═══╪═══╡▒
│ 0 │ 5 │ │ 1 │ │ 2 │▒
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Step 5 │ 0 │ 5 │ 4 │ 1 │ │ 2 │▒
├───┼───┼───┼───┼───┼───┤▒
Step 6 │ 0 │ 5 │ 4 │ 1 │ 3 │ 2 │▒
└───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ │ │ │ │ │ │
└───┴───┴───┴───┴───┴───┘
Q.E.D.
#125034_v2.10