Rearrange the digits in ⟨125034⟩ to meet the rules below.
⟨5th 4th 3rd 2nd 1st 0th⟩
✅Match
⟨⋯ a ⋯ 1 ⋯⟩, a = 2|5
⟨ ⁴ᵗʰa ³ʳᵈb ⁰ᵗʰc ⟩, (abc)₁₀ ≥ 123
⟨ − ▧ ▧ ▢ ▢ − ⟩, Σ▧ ≤ Σ▢
⛔Avoid
⟨ ⁴ᵗʰa ¹ˢᵗb ⟩, (ab)₁₀ ≥ 14
⟨ − ▧ ▧ ▢ − − ⟩, Σ▧ ≤ ▢
⟨⋯ 2 ⋯ 3 ⋯⟩
#125034_v2.10
┌───┬───┬───┬───┬───┬───┐
│5th│4th│3rd│2nd│1st│0th│▒
╞═══╪═══╪═══╪═══╪═══╪═══╡▒
Step 1 │ │ 1 │ │ │ │ │▒
├───┼───┼───┼───┼───┼───┤▒
Step 2 │ │ 1 │ │ │ 2 │ │▒
├───┼───┼───┼───┼───┼───┤▒
Step 3 │ 5 │ 1 │ │ │ 2 │ │▒
├───┼───┼───┼───┼───┼───┤▒
Step 4 │ 5 │ 1 │ │ │ 2 │ 0 │▒
├───┼───┼───┼───┼───┼───┤▒
Step 5 │ 5 │ 1 │ 4 │ │ 2 │ 0 │▒
├───┼───┼───┼───┼───┼───┤▒
Step 6 │ 5 │ 1 │ 4 │ 3 │ 2 │ 0 │▒
└───┴───┴───┴───┴───┴───┘▒
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Proof of 2025-08-26 WR
══════════════════════
To avoid ⛔「⟨ ⁴ᵗʰa ¹ˢᵗb ⟩, (ab)₁₀ ≥ 14」, we need [4th] = 1 or 0. If it is 0, then we cannot match ✅「⟨ ⁴ᵗʰa ³ʳᵈb ⁰ᵗʰc ⟩, (abc)₁₀ ≥ 123」. Therefore, [4th] = 1.
┌───┬───┬───┬───┬───┬───┐
│5th│ 4■│3rd│2nd│1st│0th│▒
╞═══╪═══╪═══╪═══╪═══╪═══╡▒
Step 1 │ │ 1 │ │ │ │ │▒
└───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ │ 2 │ 5 │ 0 │ 3 │ 4 │
└───┴───┴───┴───┴───┴───┘
Then, to avoid ⛔「⟨ ⁴ᵗʰa ¹ˢᵗb ⟩, (ab)₁₀ ≥ 14」, we need
(1) [1st] = 0,2,3.
┌───┬───┬───┬───┬───┬───┐
│5th│4th│3rd│2nd│1st│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╡
│ │ 1 │ │ │ ▬ │ │
└───┴───┴───┴───┴───┴───┘
(1.1) We show that [1st] = 2 actually.
------------------------------
(2.1) Note that if [1st] = 0:
┌───┬───┬───┬───┬───┬───┐
│5th│4th│3rd│2nd│ 1▲│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╡
│ │ 1 │ │ │ 0 │ │
└───┴───┴───┴───┴───┴───┘
then we cannot match ✅「⟨ − ▧ ▧ ▢ ▢ − ⟩, Σ▧ ≤ Σ▢」 and avoid ⛔「⟨ − ▧ ▧ ▢ − − ⟩, Σ▧ ≤ ▢」 at the same time. This shows a contradiction.
(2.2) Else if [1st] = 3:
┌───┬───┬───┬───┬───┬───┐
│5th│4th│3rd│2nd│ 1▲│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╡
│ │ 1 │ │ │ 3 │ │
└───┴───┴───┴───┴───┴───┘
then to avoid ⛔「⟨⋯ 2 ⋯ 3 ⋯⟩」, we need 2 = [0th]:
┌───┬───┬───┬───┬───┬───┐
│5th│4th│3rd│2nd│1st│ 0▲│
╞═══╪═══╪═══╪═══╪═══╪═══╡
│ │ 1 │ │ │ 3 │ 2 │
└───┴───┴───┴───┴───┴───┘
It follows that to match ✅「⟨⋯ a ⋯ 1 ⋯⟩, a = 2|5」, we need [5th] = 5:
┌───┬───┬───┬───┬───┬───┐
│ 5▲│4th│3rd│2nd│1st│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╡
│ 5 │ 1 │ │ │ 3 │ 2 │
└───┴───┴───┴───┴───┴───┘
--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ │ │ │ 0 │ │ 4 │
└───┴───┴───┴───┴───┴───┘
In view of ✅「⟨ ⁴ᵗʰa ³ʳᵈb ⁰ᵗʰc ⟩, (abc)₁₀ ≥ 123」, we reach
┌───┬───┬───┬───┬───┬───┐
│5th│4th│ 3▲│ 2▲│1st│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╡
│ 5 │ 1 │ 4 │ 0 │ 3 │ 2 │
└───┴───┴───┴───┴───┴───┘
However, it fails to match ✅「⟨ − ▧ ▧ ▢ ▢ − ⟩, Σ▧ ≤ Σ▢」, which shows a contradiction.
------------------------------
We have verified (1.1). Accordingly, we get
┌───┬───┬───┬───┬───┬───┐
│5th│4th│3rd│2nd│ 1■│0th│▒
╞═══╪═══╪═══╪═══╪═══╪═══╡▒
│ │ 1 │ │ │ │ │▒
├───┼───┼───┼───┼───┼───┤▒
Step 2 │ │ 1 │ │ │ 2 │ │▒
└───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ │ │ 5 │ 0 │ 3 │ 4 │
└───┴───┴───┴───┴───┴───┘
Then, there is only one way to match ✅「⟨⋯ a ⋯ 1 ⋯⟩, a = 2|5」:
┌───┬───┬───┬───┬───┬───┐
│ 5■│4th│3rd│2nd│1st│0th│▒
╞═══╪═══╪═══╪═══╪═══╪═══╡▒
│ │ 1 │ │ │ 2 │ │▒
├───┼───┼───┼───┼───┼───┤▒
Step 3 │ 5 │ 1 │ │ │ 2 │ │▒
└───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ │ │ │ 0 │ 3 │ 4 │
└───┴───┴───┴───┴───┴───┘
Next, we consider where to place 0. Note that if it is at 2nd:
┌───┬───┬───┬───┬───┬───┐
│5th│4th│3rd│ 2▲│1st│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╡
│ 5 │ 1 │ │ 0 │ 2 │ │
└───┴───┴───┴───┴───┴───┘
--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ │ │ │ │ 3 │ 4 │
└───┴───┴───┴───┴───┴───┘
then we would fail to match ✅「⟨ − ▧ ▧ ▢ ▢ − ⟩, Σ▧ ≤ Σ▢」. On the other hand, by ✅「⟨ ⁴ᵗʰa ³ʳᵈb ⁰ᵗʰc ⟩, (abc)₁₀ ≥ 123」, 0 is not at 3rd. Therefore, 0 = [0th]:
┌───┬───┬───┬───┬───┬───┐
│5th│4th│3rd│2nd│1st│ 0■│▒
╞═══╪═══╪═══╪═══╪═══╪═══╡▒
│ 5 │ 1 │ │ │ 2 │ │▒
├───┼───┼───┼───┼───┼───┤▒
Step 4 │ 5 │ 1 │ │ │ 2 │ 0 │▒
└───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ │ │ │ │ 3 │ 4 │
└───┴───┴───┴───┴───┴───┘
Finally, to avoid ⛔「⟨ − ▧ ▧ ▢ − − ⟩, Σ▧ ≤ ▢」, we finish by
┌───┬───┬───┬───┬───┬───┐
│5th│4th│ 3■│ 2■│1st│0th│▒
╞═══╪═══╪═══╪═══╪═══╪═══╡▒
│ 5 │ 1 │ │ │ 2 │ 0 │▒
├───┼───┼───┼───┼───┼───┤▒
Step 5 │ 5 │ 1 │ 4 │ │ 2 │ 0 │▒
├───┼───┼───┼───┼───┼───┤▒
Step 6 │ 5 │ 1 │ 4 │ 3 │ 2 │ 0 │▒
└───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ │ │ │ │ │ │
└───┴───┴───┴───┴───┴───┘
Q.E.D.
#125034_v2.10