Rearrange the digits in ⟨1263045⟩ to meet the rules below.
⟨6th 5th 4th 3rd 2nd 1st 0th⟩
✅Match
Jump(0,1) = 4
Jump(1,6) ≥ 3
min ⊢5⊣ = 3
⟨⋯ 3 ⋯ 5 ⋯⟩
⛔Avoid
min ⊢4⊣ ≥ 2
⟨ ⁶ᵗʰa ²ⁿᵈb ¹ˢᵗc ⟩, (abc)₁₀ ≥ 123
#125034_v2.6
┌───┬───┬───┬───┬───┬───┬───┐
│6th│5th│4th│3rd│2nd│1st│0th│▒
╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡▒
Step 1 │ 0 │ │ │ │ │ │ │▒
├───┼───┼───┼───┼───┼───┼───┤▒
Step 2 │ 0 │ │ │ │ │ 1 │ │▒
├───┼───┼───┼───┼───┼───┼───┤▒
Step 3 │ 0 │ 6 │ │ │ │ 1 │ │▒
├───┼───┼───┼───┼───┼───┼───┤▒
Step 4 │ 0 │ 6 │ 3 │ │ │ 1 │ │▒
├───┼───┼───┼───┼───┼───┼───┤▒
Step 5 │ 0 │ 6 │ 3 │ 5 │ │ 1 │ │▒
├───┼───┼───┼───┼───┼───┼───┤▒
Step 6 │ 0 │ 6 │ 3 │ 5 │ 4 │ 1 │ │▒
├───┼───┼───┼───┼───┼───┼───┤▒
Step 7 │ 0 │ 6 │ 3 │ 5 │ 4 │ 1 │ 2 │▒
└───┴───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
Proof of 2024-09-17 Q1(m=6)
═══════════════════════════
Notation: if nth -> a, then we write [nth] = a.
In order to avoid ⛔「⟨ ⁶ᵗʰa ²ⁿᵈb ¹ˢᵗc ⟩, (abc)₁₀ ≥ 123」, we need [6th] = 1|0.
(1) We claim that [6th] = 0 actually.
------------------------------
Suppose this is not the case. Then [6th] = 1. By ✅「Jump(0,1) = 4」, we have [1st] = 0 as well:
┌───┬───┬───┬───┬───┬───┬───┐
│ 6▲│5th│4th│3rd│2nd│ 1▲│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡
│ 1 │ │ │ │ │ 0 │ │
└───┴───┴───┴───┴───┴───┴───┘
--- Idle ---
┌───┬───┬───┬───┬───┬───┬───┐
│ │ 2 │ 6 │ 3 │ │ 4 │ 5 │
└───┴───┴───┴───┴───┴───┴───┘
Then, observe that to avoid ⛔「⟨ ⁶ᵗʰa ²ⁿᵈb ¹ˢᵗc ⟩, (abc)₁₀ ≥ 123」, we need [2nd] = 2:
┌───┬───┬───┬───┬───┬───┬───┐
│6th│5th│4th│3rd│ 2▲│1st│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡
│ 1 │ │ │ │ 2 │ 0 │ │
└───┴───┴───┴───┴───┴───┴───┘
It follows that there is only one way to match ✅「Jump(1,6) ≥ 3」:
┌───┬───┬───┬───┬───┬───┬───┐
│6th│5th│4th│3rd│2nd│1st│ 0▲│
╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡
│ 1 │ │ │ │ 2 │ 0 │ 6 │
└───┴───┴───┴───┴───┴───┴───┘
--- Idle ---
┌───┬───┬───┬───┬───┬───┬───┐
│ │ │ │ 3 │ │ 4 │ 5 │
└───┴───┴───┴───┴───┴───┴───┘
and only one way to avoid ⛔「min ⊢4⊣ ≥ 2」:
┌───┬───┬───┬───┬───┬───┬───┐
│6th│ 5▲│4th│3rd│2nd│1st│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡
│ 1 │ 4 │ │ │ 2 │ 0 │ 6 │
└───┴───┴───┴───┴───┴───┴───┘
Using ✅「⟨⋯ 3 ⋯ 5 ⋯⟩」, we reach
┌───┬───┬───┬───┬───┬───┬───┐
│6th│5th│ 4▲│ 3▲│2nd│1st│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡
│ 1 │ 4 │ 3 │ 5 │ 2 │ 0 │ 6 │
└───┴───┴───┴───┴───┴───┴───┘
It is a contradiction, however, as we do not match ✅「min ⊢5⊣ = 3」.
------------------------------
We have verified our claim in (1). Accordingly, we get
┌───┬───┬───┬───┬───┬───┬───┐
│ 6■│5th│4th│3rd│2nd│1st│0th│▒
╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡▒
Step 1 │ 0 │ │ │ │ │ │ │▒
└───┴───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
--- Idle ---
┌───┬───┬───┬───┬───┬───┬───┐
│ 1 │ 2 │ 6 │ 3 │ │ 4 │ 5 │
└───┴───┴───┴───┴───┴───┴───┘
Plainly, by ✅「Jump(0,1) = 4」 and ✅「Jump(1,6) ≥ 3」, we then get
┌───┬───┬───┬───┬───┬───┬───┐
│6th│ 5■│4th│3rd│2nd│ 1■│0th│▒
╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡▒
│ 0 │ │ │ │ │ │ │▒
├───┼───┼───┼───┼───┼───┼───┤▒
Step 2 │ 0 │ │ │ │ │ 1 │ │▒
├───┼───┼───┼───┼───┼───┼───┤▒
Step 3 │ 0 │ 6 │ │ │ │ 1 │ │▒
└───┴───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
--- Idle ---
┌───┬───┬───┬───┬───┬───┬───┐
│ │ 2 │ │ 3 │ │ 4 │ 5 │
└───┴───┴───┴───┴───┴───┴───┘
Next, note that by combining ✅「min ⊢5⊣ = 3」 with ✅「⟨⋯ 3 ⋯ 5 ⋯⟩」, we have the following required pattern:
(2) ⟨⋯ 3 5 ⋯⟩
There are two possibilities:
┌───┬───┬───┬───┬───┬───┬───┐
│6th│5th│4th│3rd│2nd│1st│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡
(2.1) │ 0 │ 6 │ │ 3 │ 5 │ 1 │ │
├───┼───┼───┼───┼───┼───┼───┤
(2.2) │ 0 │ 6 │ 3 │ 5 │ │ 1 │ │
└───┴───┴───┴───┴───┴───┴───┘
In view of ✅「min ⊢5⊣ = 3」, case (2.1) does not hold. So, case (2.2) holds, and we get
┌───┬───┬───┬───┬───┬───┬───┐
│6th│5th│ 4■│ 3■│2nd│1st│0th│▒
╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡▒
│ 0 │ 6 │ │ │ │ 1 │ │▒
├───┼───┼───┼───┼───┼───┼───┤▒
Step 4 │ 0 │ 6 │ 3 │ │ │ 1 │ │▒
├───┼───┼───┼───┼───┼───┼───┤▒
Step 5 │ 0 │ 6 │ 3 │ 5 │ │ 1 │ │▒
└───┴───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
--- Idle ---
┌───┬───┬───┬───┬───┬───┬───┐
│ │ 2 │ │ │ │ 4 │ │
└───┴───┴───┴───┴───┴───┴───┘
To match ✅「min ⊢5⊣ = 3」, we cannot place 2 next to 5. Therefore, we finish by
┌───┬───┬───┬───┬───┬───┬───┐
│6th│5th│4th│3rd│ 2■│1st│ 0■│▒
╞═══╪═══╪═══╪═══╪═══╪═══╪═══╡▒
│ 0 │ 6 │ 3 │ 5 │ │ 1 │ │▒
├───┼───┼───┼───┼───┼───┼───┤▒
Step 6 │ 0 │ 6 │ 3 │ 5 │ 4 │ 1 │ │▒
├───┼───┼───┼───┼───┼───┼───┤▒
Step 7 │ 0 │ 6 │ 3 │ 5 │ 4 │ 1 │ 2 │▒
└───┴───┴───┴───┴───┴───┴───┘▒
▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒
--- Idle ---
┌───┬───┬───┬───┬───┬───┬───┐
│ │ │ │ │ │ │ │
└───┴───┴───┴───┴───┴───┴───┘
Q.E.D.
#125034_v2.6