Rearrange the digits in ⟨125034⟩ to meet the rules below.
⟨ ⁵ᵗʰ▨ ⁴ᵗʰ▨ ³ʳᵈ▨ ²ⁿᵈ▨ ¹ˢᵗ▨ ⁰ᵗʰ▨ ⟩
✅Match
⟨ ³ʳᵈa ⁰ᵗʰb ⟩, max⟦a,b⟧ = 4
⟨⋯ Perm(0,2,5) ⋯⟩
5th → a, 0th → b, ab=3+5n
min ⊢5⊣ ≤ 1
⛔Avoid
3rd → a, 1st → b, |a-b|=3
4th|2nd|1st|0th → 2
⟨⋯ 0 ⋯ ? 1 ⋯ (?+1)⟩ (?≠1,0)
#125034_v2.13
┌───┬───┬───┬───┬───┬───┐
│5th│4th│3rd│2nd│1st│0th│▒
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Step 1 │ │ 5 │ │ │ │ │▒
├───┼───┼───┼───┼───┼───┤▒
Step 2 │ │ 5 │ │ │ 4 │ │▒
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Step 3 │ │ 5 │ 2 │ │ 4 │ │▒
├───┼───┼───┼───┼───┼───┤▒
Step 4 │ │ 5 │ 2 │ 0 │ 4 │ │▒
├───┼───┼───┼───┼───┼───┤▒
Step 5 │ 1 │ 5 │ 2 │ 0 │ 4 │ │▒
├───┼───┼───┼───┼───┼───┤▒
Step 6 │ 1 │ 5 │ 2 │ 0 │ 4 │ 3 │▒
└───┴───┴───┴───┴───┴───┘▒
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Proof of 2026-05-12 WR
══════════════════════
Notation: if nth -> a, then we write [nth] = a.
✅「⟨ ³ʳᵈa ⁰ᵗʰb ⟩, max⟦a,b⟧ = 4」 implies
(1) 5 = [5th] | [4th].
If it is [5th], then we cannot match ✅「5th → a, 0th → b, ab=3+5n」. Therefore, 5 = [4th].
┌───┬───┬───┬───┬───┬───┐
│5th│ 4■│3rd│2nd│1st│0th│▒
╞═══╪═══╪═══╪═══╪═══╪═══╡▒
Step 1 │ │ 5 │ │ │ │ │▒
└───┴───┴───┴───┴───┴───┘▒
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--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ 1 │ 2 │ │ 0 │ 3 │ 4 │
└───┴───┴───┴───┴───┴───┘
Next, we consider where to place 4.
(2) We claim that indeed 4 = [1st]:
┌───┬───┬───┬───┬───┬───┐
│5th│4th│3rd│2nd│1st│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╡
│ ▬ │ 5 │ ▬ │ ▬ │ │ ▬ │
└───┴───┴───┴───┴───┴───┘
------------------------------
(3.1) It is proved by contradiction. Firstly, suppose 4 = [5th]:
┌───┬───┬───┬───┬───┬───┐
│ 5▲│4th│3rd│2nd│1st│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╡
│ 4 │ 5 │ │ │ │ │
└───┴───┴───┴───┴───┴───┘
Then ✅「5th → a, 0th → b, ab=3+5n」 implies [0th] = 2:
┌───┬───┬───┬───┬───┬───┐
│5th│4th│3rd│2nd│1st│ 0▲│
╞═══╪═══╪═══╪═══╪═══╪═══╡
│ 4 │ 5 │ │ │ │ 2 │
└───┴───┴───┴───┴───┴───┘
We do not match ✅「⟨⋯ Perm(0,2,5) ⋯⟩」, which is a contradiction.
(3.2) Else if 4 = [3rd]:
┌───┬───┬───┬───┬───┬───┐
│5th│4th│ 3▲│2nd│1st│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╡
│ │ 5 │ 4 │ │ │ │
└───┴───┴───┴───┴───┴───┘
then we cannot match ✅「⟨⋯ Perm(0,2,5) ⋯⟩」.
(3.3) Else if 4 = [2nd]:
┌───┬───┬───┬───┬───┬───┐
│5th│4th│3rd│ 2▲│1st│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╡
│ │ 5 │ │ 4 │ │ │
└───┴───┴───┴───┴───┴───┘
then to match ✅「⟨⋯ Perm(0,2,5) ⋯⟩」, we need {[5th], [3rd]} = {0,2}. But then we cannot match ✅「5th → a, 0th → b, ab=3+5n」.
(3.4) Lastly, if 4 = [0th]:
┌───┬───┬───┬───┬───┬───┐
│5th│4th│3rd│2nd│1st│ 0▲│
╞═══╪═══╪═══╪═══╪═══╪═══╡
│ │ 5 │ │ │ │ 4 │
└───┴───┴───┴───┴───┴───┘
then using ✅「5th → a, 0th → b, ab=3+5n」 and ✅「⟨⋯ Perm(0,2,5) ⋯⟩」, we get
┌───┬───┬───┬───┬───┬───┐
│ 5▲│4th│ 3▲│2nd│1st│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╡
│ 2 │ 5 │ 0 │ │ │ 4 │
└───┴───┴───┴───┴───┴───┘
Two cases follow:
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│5th│4th│3rd│ 2▲│ 1▲│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╡
│ 2 │ 5 │ 0 │ 1 │ 3 │ 4 │
├───┼───┼───┼───┼───┼───┤
│ 2 │ 5 │ 0 │ 3 │ 1 │ 4 │
└───┴───┴───┴───┴───┴───┘
In the former case, we match ⛔「3rd → a, 1st → b, |a-b|=3」. In the latter case, we match ⛔「⟨⋯ 0 ⋯ ? 1 ⋯ (?+1)⟩ (?≠1,0)」. Both are contradictions.
------------------------------
We have verified (2). Accordingly, we get
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│5th│4th│3rd│2nd│ 1■│0th│▒
╞═══╪═══╪═══╪═══╪═══╪═══╡▒
│ │ 5 │ │ │ │ │▒
├───┼───┼───┼───┼───┼───┤▒
Step 2 │ │ 5 │ │ │ 4 │ │▒
└───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ 1 │ 2 │ │ 0 │ 3 │ │
└───┴───┴───┴───┴───┴───┘
Then, to match ✅「5th → a, 0th → b, ab=3+5n」, we need
(4) {[5th], [0th]} = {1,3}.
It implies
(5) {[3rd], [2nd]} = {0,2}.
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│5th│4th│3rd│2nd│1st│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╡
│ │ 5 │0 2│0 2│ 4 │ │
└───┴───┴───┴───┴───┴───┘
In view of ⛔「4th|2nd|1st|0th → 2」, we have
┌───┬───┬───┬───┬───┬───┐
│5th│4th│ 3■│ 2■│1st│0th│▒
╞═══╪═══╪═══╪═══╪═══╪═══╡▒
│ │ 5 │ │ │ 4 │ │▒
├───┼───┼───┼───┼───┼───┤▒
Step 3 │ │ 5 │ 2 │ │ 4 │ │▒
├───┼───┼───┼───┼───┼───┤▒
Step 4 │ │ 5 │ 2 │ 0 │ 4 │ │▒
└───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ 1 │ │ │ │ 3 │ │
└───┴───┴───┴───┴───┴───┘
Finally, using ✅「min ⊢5⊣ ≤ 1」, we finish by
┌───┬───┬───┬───┬───┬───┐
│ 5■│4th│3rd│2nd│1st│ 0■│▒
╞═══╪═══╪═══╪═══╪═══╪═══╡▒
│ │ 5 │ 2 │ 0 │ 4 │ │▒
├───┼───┼───┼───┼───┼───┤▒
Step 5 │ 1 │ 5 │ 2 │ 0 │ 4 │ │▒
├───┼───┼───┼───┼───┼───┤▒
Step 6 │ 1 │ 5 │ 2 │ 0 │ 4 │ 3 │▒
└───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ │ │ │ │ │ │
└───┴───┴───┴───┴───┴───┘
Q.E.D.
#125034_v2.13