Rearrange the digits in ⟨125034⟩ to meet the rules below.
⟨5th 4th 3rd 2nd 1st 0th⟩
✅Match
4th → a, 2nd → b, |a-b|=4
Jump(1,4) = 1
⛔Avoid
min ⊢3⊣ = 1
Jump(2,5) ≥ 1
#125034_v2.8
┌───┬───┬───┬───┬───┬───┐
│5th│4th│3rd│2nd│1st│0th│▒
╞═══╪═══╪═══╪═══╪═══╪═══╡▒
Step 1 │ │ 5 │ │ │ │ │▒
├───┼───┼───┼───┼───┼───┤▒
Step 2 │ │ 5 │ │ 1 │ │ │▒
├───┼───┼───┼───┼───┼───┤▒
Step 3 │ │ 5 │ │ 1 │ │ 4 │▒
├───┼───┼───┼───┼───┼───┤▒
Step 4 │ │ 5 │ │ 1 │ 0 │ 4 │▒
├───┼───┼───┼───┼───┼───┤▒
Step 5 │ 3 │ 5 │ │ 1 │ 0 │ 4 │▒
├───┼───┼───┼───┼───┼───┤▒
Step 6 │ 3 │ 5 │ 2 │ 1 │ 0 │ 4 │▒
└───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
Proof of 2025-01-14 WR
══════════════════════
Notation: if nth -> a, then we write [nth] = a.
By ✅「4th → a, 2nd → b, |a-b|=4」, we see that one of the following holds:
(1) {[4th], [2nd]} = {0,4}; or
(2) {[4th], [2nd]} = {1,5}.
┌───┬───┬───┬───┬───┬───┐
│5th│4th│3rd│2nd│1st│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╡
│ │ ▬ │ │ ▬ │ │ │
└───┴───┴───┴───┴───┴───┘
We show that (2) holds actually.
------------------------------
For, if on the contrary (1) holds, then to match (1) and ✅「Jump(1,4) = 1」 at the same time, we need
┌───┬───┬───┬───┬───┬───┐
│5th│ 4▲│3rd│ 2▲│1st│ 0▲│
╞═══╪═══╪═══╪═══╪═══╪═══╡
│ │ 0 │ │ 4 │ │ │
├───┼───┼───┼───┼───┼───┤
│ │ 0 │ │ 4 │ │ 1 │
└───┴───┴───┴───┴───┴───┘
--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ │ 2 │ 5 │ │ 3 │ │
└───┴───┴───┴───┴───┴───┘
But then we cannot avoid ⛔「Jump(2,5) ≥ 1」, which is a contradiction.
------------------------------
Therefore, (1) does not hold and (2) holds instead. To match (2) and ✅「Jump(1,4) = 1」 at the same time, we get
┌───┬───┬───┬───┬───┬───┐
│5th│ 4■│3rd│ 2■│1st│ 0■│▒
╞═══╪═══╪═══╪═══╪═══╪═══╡▒
Step 1 │ │ 5 │ │ │ │ │▒
├───┼───┼───┼───┼───┼───┤▒
Step 2 │ │ 5 │ │ 1 │ │ │▒
├───┼───┼───┼───┼───┼───┤▒
Step 3 │ │ 5 │ │ 1 │ │ 4 │▒
└───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ │ 2 │ │ 0 │ 3 │ │
└───┴───┴───┴───┴───┴───┘
Then, we consider what [1st] is. If it is 2, then we would match ⛔「Jump(2,5) ≥ 1」; while if it is 3, then we would match ⛔「min ⊢3⊣ = 1」. Therefore, it has to be 0:
┌───┬───┬───┬───┬───┬───┐
│5th│4th│3rd│2nd│ 1■│0th│▒
╞═══╪═══╪═══╪═══╪═══╪═══╡▒
│ │ 5 │ │ 1 │ │ 4 │▒
├───┼───┼───┼───┼───┼───┤▒
Step 4 │ │ 5 │ │ 1 │ 0 │ 4 │▒
└───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ │ 2 │ │ │ 3 │ │
└───┴───┴───┴───┴───┴───┘
Finally, to avoid ⛔「min ⊢3⊣ = 1」, we finish by
┌───┬───┬───┬───┬───┬───┐
│ 5■│4th│ 3■│2nd│1st│0th│▒
╞═══╪═══╪═══╪═══╪═══╪═══╡▒
│ │ 5 │ │ 1 │ 0 │ 4 │▒
├───┼───┼───┼───┼───┼───┤▒
Step 5 │ 3 │ 5 │ │ 1 │ 0 │ 4 │▒
├───┼───┼───┼───┼───┼───┤▒
Step 6 │ 3 │ 5 │ 2 │ 1 │ 0 │ 4 │▒
└───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ │ │ │ │ │ │
└───┴───┴───┴───┴───┴───┘
Q.E.D.
#125034_v2.8