Rearrange the digits in ⟨125034⟩ to meet the rules below.
⟨5th 4th 3rd 2nd 1st 0th⟩
✅Match
⟨ ▧ − − − ▢ ▢ ⟩, ▧ ≥ Σ▢
⟦0,2⟧ ∋ 4
⟨⋯ 0 ⋯ ? ⋯ 1 (?+3)⟩ (?≠1)
⛔Avoid
⟨ ⁵ᵗʰa ⁴ᵗʰc ¹ˢᵗb ⟩, a > b > c
#125034_v2.9
┌───┬───┬───┬───┬───┬───┐
│5th│4th│3rd│2nd│1st│0th│▒
╞═══╪═══╪═══╪═══╪═══╪═══╡▒
Step 1 │ │ │ │ │ 1 │ │▒
├───┼───┼───┼───┼───┼───┤▒
Step 2 │ 5 │ │ │ │ 1 │ │▒
├───┼───┼───┼───┼───┼───┤▒
Step 3 │ 5 │ │ │ │ 1 │ 3 │▒
├───┼───┼───┼───┼───┼───┤▒
Step 4 │ 5 │ │ 4 │ │ 1 │ 3 │▒
├───┼───┼───┼───┼───┼───┤▒
Step 5 │ 5 │ │ 4 │ 0 │ 1 │ 3 │▒
├───┼───┼───┼───┼───┼───┤▒
Step 6 │ 5 │ 2 │ 4 │ 0 │ 1 │ 3 │▒
└───┴───┴───┴───┴───┴───┘▒
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Proof of 2025-06-10 WR
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Notation: if nth -> a, then we write [nth] = a.
Plainly, in view of ✅「⟨⋯ 0 ⋯ ? ⋯ 1 (?+3)⟩ (?≠1)」, we first have
┌───┬───┬───┬───┬───┬───┐
│5th│4th│3rd│2nd│ 1■│0th│▒
╞═══╪═══╪═══╪═══╪═══╪═══╡▒
Step 1 │ │ │ │ │ 1 │ │▒
└───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ │ 2 │ 5 │ 0 │ 3 │ 4 │
└───┴───┴───┴───┴───┴───┘
The preceding pattern also suggests that [1st] + [0th] >= 1+3 = 4. Therefore, to match ✅「⟨ ▧ − − − ▢ ▢ ⟩, ▧ ≥ Σ▢」, we need [5th] = 4 or 5.
Since 4 is not in the left corner by ✅「⟦0,2⟧ ∋ 4」, we have [5th] = 5:
┌───┬───┬───┬───┬───┬───┐
│ 5■│4th│3rd│2nd│1st│0th│▒
╞═══╪═══╪═══╪═══╪═══╪═══╡▒
│ │ │ │ │ 1 │ │▒
├───┼───┼───┼───┼───┼───┤▒
Step 2 │ 5 │ │ │ │ 1 │ │▒
└───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ │ 2 │ │ 0 │ 3 │ 4 │
└───┴───┴───┴───┴───┴───┘
Then, we consider what [0th] is. As ✅「⟨⋯ 0 ⋯ ? ⋯ 1 (?+3)⟩ (?≠1)」 implies that [0th]>=3 and [0th]!=4, it has to be 3:
┌───┬───┬───┬───┬───┬───┐
│5th│4th│3rd│2nd│1st│ 0■│▒
╞═══╪═══╪═══╪═══╪═══╪═══╡▒
│ 5 │ │ │ │ 1 │ │▒
├───┼───┼───┼───┼───┼───┤▒
Step 3 │ 5 │ │ │ │ 1 │ 3 │▒
└───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ │ 2 │ │ 0 │ │ 4 │
└───┴───┴───┴───┴───┴───┘
Next, to match ✅「⟦0,2⟧ ∋ 4」, we need that 4 is placed between 0 and 2. There is only one way to do so:
┌───┬───┬───┬───┬───┬───┐
│5th│4th│ 3■│2nd│1st│0th│▒
╞═══╪═══╪═══╪═══╪═══╪═══╡▒
│ 5 │ │ │ │ 1 │ 3 │▒
├───┼───┼───┼───┼───┼───┤▒
Step 4 │ 5 │ │ 4 │ │ 1 │ 3 │▒
└───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ │ 2 │ │ 0 │ │ │
└───┴───┴───┴───┴───┴───┘
Finally, to avoid ⛔「⟨ ⁵ᵗʰa ⁴ᵗʰc ¹ˢᵗb ⟩, a > b > c」, we finish by
┌───┬───┬───┬───┬───┬───┐
│5th│ 4■│3rd│ 2■│1st│0th│▒
╞═══╪═══╪═══╪═══╪═══╪═══╡▒
│ 5 │ │ 4 │ │ 1 │ 3 │▒
├───┼───┼───┼───┼───┼───┤▒
Step 5 │ 5 │ │ 4 │ 0 │ 1 │ 3 │▒
├───┼───┼───┼───┼───┼───┤▒
Step 6 │ 5 │ 2 │ 4 │ 0 │ 1 │ 3 │▒
└───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ │ │ │ │ │ │
└───┴───┴───┴───┴───┴───┘
Q.E.D.
#125034_v2.9