Rearrange the digits in ⟨125034⟩ to meet the rules below.
⟨ ⁵ᵗʰ▨ ⁴ᵗʰ▨ ³ʳᵈ▨ ²ⁿᵈ▨ ¹ˢᵗ▨ ⁰ᵗʰ▨ ⟩
✅Match
⟨ ⁴ᵗʰb ¹ˢᵗa ⟩, a > b
⟨ ⁴ᵗʰa ²ⁿᵈb ⁰ᵗʰc ⟩, (abc)₁₀ ≥ 154
⟨? ⋯ 4 ⋯ (?−5) ⋯⟩
⛔Avoid
⟨⋯ ᵃb ⋯⟩, a+b=7+2n
⟨ − − ▧ ▧ ▢ − ⟩, Σ▧ ≤ ▢
#125034_v2.15
┌───┬───┬───┬───┬───┬───┐
│5th│4th│3rd│2nd│1st│0th│▒
╞═══╪═══╪═══╪═══╪═══╪═══╡▒
Step 1 │ 5 │ │ │ │ │ │▒
├───┼───┼───┼───┼───┼───┤▒
Step 2 │ 5 │ 2 │ │ │ │ │▒
├───┼───┼───┼───┼───┼───┤▒
Step 3 │ 5 │ 2 │ │ │ 3 │ │▒
├───┼───┼───┼───┼───┼───┤▒
Step 4 │ 5 │ 2 │ │ 4 │ 3 │ │▒
├───┼───┼───┼───┼───┼───┤▒
Step 5 │ 5 │ 2 │ │ 4 │ 3 │ 0 │▒
├───┼───┼───┼───┼───┼───┤▒
Step 6 │ 5 │ 2 │ 1 │ 4 │ 3 │ 0 │▒
└───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
Proof of 2026-11-03 WR
══════════════════════
Notation: if nth -> a, then we write [nth] = a.
By ✅「⟨? ⋯ 4 ⋯ (?−5) ⋯⟩」, plainly we have [5th]=5:
┌───┬───┬───┬───┬───┬───┐
│ 5■│4th│3rd│2nd│1st│0th│▒
╞═══╪═══╪═══╪═══╪═══╪═══╡▒
Step 1 │ 5 │ │ │ │ │ │▒
└───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ 1 │ 2 │ │ 0 │ 3 │ 4 │
└───┴───┴───┴───┴───┴───┘
------------------------------
Consider what [4th] is. Note that
• To match ✅「⟨ ⁴ᵗʰb ¹ˢᵗa ⟩, a > b」, it cannot be 4;
• To match ✅「⟨ ⁴ᵗʰa ²ⁿᵈb ⁰ᵗʰc ⟩, (abc)₁₀ ≥ 154」, it cannot be 0 or 1;
• To avoid ⛔「⟨⋯ ᵃb ⋯⟩, a+b=7+2n」, it cannot be 3.
Consequently, [4th]=2:
┌───┬───┬───┬───┬───┬───┐
│5th│ 4■│3rd│2nd│1st│0th│▒
╞═══╪═══╪═══╪═══╪═══╪═══╡▒
│ 5 │ │ │ │ │ │▒
├───┼───┼───┼───┼───┼───┤▒
Step 2 │ 5 │ 2 │ │ │ │ │▒
└───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ 1 │ │ │ 0 │ 3 │ 4 │
└───┴───┴───┴───┴───┴───┘
------------------------------
Then, using ✅「⟨ ⁴ᵗʰb ¹ˢᵗa ⟩, a > b」, we see that [1st] is 3 or 4. If it is 4, then we would match ⛔「⟨ − − ▧ ▧ ▢ − ⟩, Σ▧ ≤ ▢」, which is not allowed. Therefore, [1st]=3:
┌───┬───┬───┬───┬───┬───┐
│5th│4th│3rd│2nd│ 1■│0th│▒
╞═══╪═══╪═══╪═══╪═══╪═══╡▒
│ 5 │ 2 │ │ │ │ │▒
├───┼───┼───┼───┼───┼───┤▒
Step 3 │ 5 │ 2 │ │ │ 3 │ │▒
└───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ 1 │ │ │ 0 │ │ 4 │
└───┴───┴───┴───┴───┴───┘
------------------------------
Next, we consider where to place 4. By ✅「⟨? ⋯ 4 ⋯ (?−5) ⋯⟩」, 4 is not [0th], and by ⛔「⟨⋯ ᵃb ⋯⟩, a+b=7+2n」, it is not [3rd]. So 4 has to be [2nd]:
┌───┬───┬───┬───┬───┬───┐
│5th│4th│3rd│ 2■│1st│0th│▒
╞═══╪═══╪═══╪═══╪═══╪═══╡▒
│ 5 │ 2 │ │ │ 3 │ │▒
├───┼───┼───┼───┼───┼───┤▒
Step 4 │ 5 │ 2 │ │ 4 │ 3 │ │▒
└───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ 1 │ │ │ 0 │ │ │
└───┴───┴───┴───┴───┴───┘
------------------------------
Finally, using ✅「⟨? ⋯ 4 ⋯ (?−5) ⋯⟩」 once more, we finish by
┌───┬───┬───┬───┬───┬───┐
│5th│4th│ 3■│2nd│1st│ 0■│▒
╞═══╪═══╪═══╪═══╪═══╪═══╡▒
│ 5 │ 2 │ │ 4 │ 3 │ │▒
├───┼───┼───┼───┼───┼───┤▒
Step 5 │ 5 │ 2 │ │ 4 │ 3 │ 0 │▒
├───┼───┼───┼───┼───┼───┤▒
Step 6 │ 5 │ 2 │ 1 │ 4 │ 3 │ 0 │▒
└───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ │ │ │ │ │ │
└───┴───┴───┴───┴───┴───┘
------------------------------
Q.E.D.
#125034_v2.15