Rearrange the digits in ⟨125034⟩ to meet the rules below.
⟨ ⁵ᵗʰ▨ ⁴ᵗʰ▨ ³ʳᵈ▨ ²ⁿᵈ▨ ¹ˢᵗ▨ ⁰ᵗʰ▨ ⟩
✅Match
⟨ ▧ ▧ ▧ − ▢ − ⟩, Σ▧ ≤ ▢
min ⊢3⊣ = 3
Sim⟨ ⁵ᵗʰ0 ⁴ᵗʰ2 ³ʳᵈ3 ²ⁿᵈ4 ¹ˢᵗ5 ⁰ᵗʰ1 ⟩ ≤ 1
⛔Avoid
⟨⋯ 2 ⋯ a ⋯⟩, a = 1|3|5
#125034_v2.14
┌───┬───┬───┬───┬───┬───┐
│5th│4th│3rd│2nd│1st│0th│▒
╞═══╪═══╪═══╪═══╪═══╪═══╡▒
Step 1 │ │ │ │ │ │ 2 │▒
├───┼───┼───┼───┼───┼───┤▒
Step 2 │ │ │ │ │ 5 │ 2 │▒
├───┼───┼───┼───┼───┼───┤▒
Step 3 │ │ │ │ 3 │ 5 │ 2 │▒
├───┼───┼───┼───┼───┼───┤▒
Step 4 │ │ │ 4 │ 3 │ 5 │ 2 │▒
├───┼───┼───┼───┼───┼───┤▒
Step 5 │ │ 0 │ 4 │ 3 │ 5 │ 2 │▒
├───┼───┼───┼───┼───┼───┤▒
Step 6 │ 1 │ 0 │ 4 │ 3 │ 5 │ 2 │▒
└───┴───┴───┴───┴───┴───┘▒
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Proof of 2026-07-28 WR
══════════════════════
To avoid ⛔「⟨⋯ 2 ⋯ a ⋯⟩, a = 1|3|5」, we need
(1) 2 = [2nd] | [1st] | [0th].
┌───┬───┬───┬───┬───┬───┐
│5th│4th│3rd│2nd│1st│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╡
│ │ │ │ ▬ │ ▬ │ ▬ │
└───┴───┴───┴───┴───┴───┘
If 2 = [2nd], then ⛔「⟨⋯ 2 ⋯ a ⋯⟩, a = 1|3|5」 implies 1,3,5 are [5th], [4th], [3rd]:
┌───┬───┬───┬───┬───┬───┐
│5th│4th│3rd│2nd│1st│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╡
│135│135│135│ 2 │ │ │
└───┴───┴───┴───┴───┴───┘
--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ │ │ │ 0 │ │ 4 │
└───┴───┴───┴───┴───┴───┘
We will fail to match ✅「⟨ ▧ ▧ ▧ − ▢ − ⟩, Σ▧ ≤ ▢」, which is a contradiction. Similarly, if 2 = [1st]:
┌───┬───┬───┬───┬───┬───┐
│5th│4th│3rd│2nd│1st│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╡
│ │ │ │ │ 2 │ │
└───┴───┴───┴───┴───┴───┘
--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ 1 │ │ 5 │ 0 │ 3 │ 4 │
└───┴───┴───┴───┴───┴───┘
then we also cannot match ✅「⟨ ▧ ▧ ▧ − ▢ − ⟩, Σ▧ ≤ ▢」.
As 2 is not [2nd] or [1st], by (1) we have 2 = [0th]:
┌───┬───┬───┬───┬───┬───┐
│5th│4th│3rd│2nd│1st│ 0■│▒
╞═══╪═══╪═══╪═══╪═══╪═══╡▒
Step 1 │ │ │ │ │ │ 2 │▒
└───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ 1 │ │ 5 │ 0 │ 3 │ 4 │
└───┴───┴───┴───┴───┴───┘
Next, we consider where to place 5. To match ✅「⟨ ▧ ▧ ▧ − ▢ − ⟩, Σ▧ ≤ ▢」, it cannot be [5th], [4th], or [3rd], so
(2) 5 = [2nd] or [1st].
┌───┬───┬───┬───┬───┬───┐
│5th│4th│3rd│2nd│1st│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╡
│ / │ / │ / │ │ │ 2 │
└───┴───┴───┴───┴───┴───┘
If 5 = [2nd]:
┌───┬───┬───┬───┬───┬───┐
│5th│4th│3rd│2nd│1st│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╡
│ │ │ │ 5 │ │ 2 │
└───┴───┴───┴───┴───┴───┘
--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ 1 │ │ │ 0 │ 3 │ 4 │
└───┴───┴───┴───┴───┴───┘
then there is only one way to match ✅「⟨ ▧ ▧ ▧ − ▢ − ⟩, Σ▧ ≤ ▢」:
┌───┬───┬───┬───┬───┬───┐
│5th│4th│3rd│2nd│1st│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╡
│013│013│013│ 5 │ 4 │ 2 │
└───┴───┴───┴───┴───┴───┘
But we will not match ✅「min ⊢3⊣ = 3」. This shows a contradiction.
Therefore, it follows from (2) that 5 = [1st]:
┌───┬───┬───┬───┬───┬───┐
│5th│4th│3rd│2nd│ 1■│0th│▒
╞═══╪═══╪═══╪═══╪═══╪═══╡▒
│ │ │ │ │ │ 2 │▒
├───┼───┼───┼───┼───┼───┤▒
Step 2 │ │ │ │ │ 5 │ 2 │▒
└───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ 1 │ │ │ 0 │ 3 │ 4 │
└───┴───┴───┴───┴───┴───┘
Then, to match ✅「min ⊢3⊣ = 3」, we need one of the following:
┌───┬───┬───┬───┬───┬───┐
│5th│4th│3rd│2nd│1st│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╡
(3) │ 3 │ 4 │ │ │ 5 │ 2 │
├───┼───┼───┼───┼───┼───┤
(4) │ │ │ 4 │ 3 │ 5 │ 2 │
└───┴───┴───┴───┴───┴───┘
If case (3) holds, then we cannot match ✅「⟨ ▧ ▧ ▧ − ▢ − ⟩, Σ▧ ≤ ▢」. Therefore, case (4) holds, and we get
┌───┬───┬───┬───┬───┬───┐
│5th│4th│ 3■│ 2■│1st│0th│▒
╞═══╪═══╪═══╪═══╪═══╪═══╡▒
│ │ │ │ │ 5 │ 2 │▒
├───┼───┼───┼───┼───┼───┤▒
Step 3 │ │ │ │ 3 │ 5 │ 2 │▒
├───┼───┼───┼───┼───┼───┤▒
Step 4 │ │ │ 4 │ 3 │ 5 │ 2 │▒
└───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ 1 │ │ │ 0 │ │ │
└───┴───┴───┴───┴───┴───┘
Finally, to match ✅「Sim⟨ ⁵ᵗʰ0 ⁴ᵗʰ2 ³ʳᵈ3 ²ⁿᵈ4 ¹ˢᵗ5 ⁰ᵗʰ1 ⟩ ≤ 1」, we finish by
┌───┬───┬───┬───┬───┬───┐
│ 5■│ 4■│3rd│2nd│1st│0th│▒
╞═══╪═══╪═══╪═══╪═══╪═══╡▒
│ │ │ 4 │ 3 │ 5 │ 2 │▒
├───┼───┼───┼───┼───┼───┤▒
Step 5 │ │ 0 │ 4 │ 3 │ 5 │ 2 │▒
├───┼───┼───┼───┼───┼───┤▒
Step 6 │ 1 │ 0 │ 4 │ 3 │ 5 │ 2 │▒
└───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ │ │ │ │ │ │
└───┴───┴───┴───┴───┴───┘
Q.E.D.
#125034_v2.14