Rearrange the digits in ⟨125034⟩ to meet the rules below.
⟨ ⁵ᵗʰ▨ ⁴ᵗʰ▨ ³ʳᵈ▨ ²ⁿᵈ▨ ¹ˢᵗ▨ ⁰ᵗʰ▨ ⟩
✅Match
⟨⋯ Perm(0,1,5) ⋯⟩
min ⊢4⊣ ≤ 2
⛔Avoid
Jump(0,2) ≤ 2
⟦0,3⟧ ∋ 1,5
⟨⋯ 0 ⋯ a ⋯⟩, a = 3|4|5
#125034_v2.11
┌───┬───┬───┬───┬───┬───┐
│5th│4th│3rd│2nd│1st│0th│▒
╞═══╪═══╪═══╪═══╪═══╪═══╡▒
Step 1 │ │ │ │ │ 0 │ │▒
├───┼───┼───┼───┼───┼───┤▒
Step 2 │ 2 │ │ │ │ 0 │ │▒
├───┼───┼───┼───┼───┼───┤▒
Step 3 │ 2 │ │ │ │ 0 │ 1 │▒
├───┼───┼───┼───┼───┼───┤▒
Step 4 │ 2 │ │ │ 5 │ 0 │ 1 │▒
├───┼───┼───┼───┼───┼───┤▒
Step 5 │ 2 │ 4 │ │ 5 │ 0 │ 1 │▒
├───┼───┼───┼───┼───┼───┤▒
Step 6 │ 2 │ 4 │ 3 │ 5 │ 0 │ 1 │▒
└───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
Proof of 2026-01-20 WR
══════════════════════
Notation: if nth -> a, then we write [nth] = a.
We consider where to place 0. To avoid ⛔「⟨⋯ 0 ⋯ a ⋯⟩, a = 3|4|5」, we need
(1) 3,4,5 are to the left of 0.
It implies
(2) 0 = [2nd] | [1st] | [0th].
(3) We show that 0 = [1st] actually.
------------------------------
(3.1) If 0 = [2nd], then by (1), we have
┌───┬───┬───┬───┬───┬───┐
│5th│4th│3rd│2nd│1st│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╡
│345│345│345│ 0 │1 2│1 2│
└───┴───┴───┴───┴───┴───┘
and we will match ⛔「Jump(0,2) ≤ 2」, which is not allowed.
(3.2) Else if 0 = [0th], then to match ✅「⟨⋯ Perm(0,1,5) ⋯⟩」, we need
┌───┬───┬───┬───┬───┬───┐
│5th│4th│3rd│2nd│1st│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╡
│ │ │ │1 5│1 5│ 0 │
└───┴───┴───┴───┴───┴───┘
and we will match ⛔「⟦0,3⟧ ∋ 1,5」, which is also not allowed.
------------------------------
We have verified (3) and get
┌───┬───┬───┬───┬───┬───┐
│5th│4th│3rd│2nd│ 1■│0th│▒
╞═══╪═══╪═══╪═══╪═══╪═══╡▒
Step 1 │ │ │ │ │ 0 │ │▒
└───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ 1 │ 2 │ 5 │ │ 3 │ 4 │
└───┴───┴───┴───┴───┴───┘
Then, in view of ⛔「Jump(0,2) ≤ 2」, we have
┌───┬───┬───┬───┬───┬───┐
│ 5■│4th│3rd│2nd│1st│0th│▒
╞═══╪═══╪═══╪═══╪═══╪═══╡▒
│ │ │ │ │ 0 │ │▒
├───┼───┼───┼───┼───┼───┤▒
Step 2 │ 2 │ │ │ │ 0 │ │▒
└───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ 1 │ │ 5 │ │ 3 │ 4 │
└───┴───┴───┴───┴───┴───┘
In view of (1), we have 1 = [0th] as well:
┌───┬───┬───┬───┬───┬───┐
│5th│4th│3rd│2nd│1st│ 0■│▒
╞═══╪═══╪═══╪═══╪═══╪═══╡▒
│ 2 │ │ │ │ 0 │ │▒
├───┼───┼───┼───┼───┼───┤▒
Step 3 │ 2 │ │ │ │ 0 │ 1 │▒
└───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ │ │ 5 │ │ 3 │ 4 │
└───┴───┴───┴───┴───┴───┘
There is only one way to match ✅「⟨⋯ Perm(0,1,5) ⋯⟩」:
┌───┬───┬───┬───┬───┬───┐
│5th│4th│3rd│ 2■│1st│0th│▒
╞═══╪═══╪═══╪═══╪═══╪═══╡▒
│ 2 │ │ │ │ 0 │ 1 │▒
├───┼───┼───┼───┼───┼───┤▒
Step 4 │ 2 │ │ │ 5 │ 0 │ 1 │▒
└───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ │ │ │ │ 3 │ 4 │
└───┴───┴───┴───┴───┴───┘
Finally, to match ✅「min ⊢4⊣ ≤ 2」, we finish by
┌───┬───┬───┬───┬───┬───┐
│5th│ 4■│ 3■│2nd│1st│0th│▒
╞═══╪═══╪═══╪═══╪═══╪═══╡▒
│ 2 │ │ │ 5 │ 0 │ 1 │▒
├───┼───┼───┼───┼───┼───┤▒
Step 5 │ 2 │ 4 │ │ 5 │ 0 │ 1 │▒
├───┼───┼───┼───┼───┼───┤▒
Step 6 │ 2 │ 4 │ 3 │ 5 │ 0 │ 1 │▒
└───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ │ │ │ │ │ │
└───┴───┴───┴───┴───┴───┘
Q.E.D.
#125034_v2.11