2023-11-28 WR

Rearrange the digits in ⟨125034⟩ to meet the rules below.

⟨5th 4th 3rd 2nd 1st 0th⟩

✅Match

⟦3,5⟧ ∋ 0,2

4th → a, 1st → b, a+b=5

⟨⋯ Perm(2,5) ⋯⟩

⛔Avoid

⟨   ⁴ᵗʰa   ²ⁿᵈb ¹ˢᵗc   ⟩, (abc)₁₀ ≥ 152

⟦0,4⟧ ∋ 1,2

----- Information -----

🔲 「⟦3,5⟧ ∋ 0,2」

The closed interval given by 3 and 5 contains 0, 2.

120 permutations match this pattern.

Examples: ⟨340215⟩, ⟨302514⟩, ⟨132405⟩.

🔲 「4th → a, 1st → b, a+b=5」

144 permutations match this pattern.

Examples: ⟨420135⟩, ⟨251403⟩, ⟨134520⟩.

🔲 「⟨⋯ Perm(2,5) ⋯⟩」

2,5 are adjacent.

240 permutations match this pattern.

Examples: ⟨014325⟩, ⟨025143⟩, ⟨345210⟩.

🔳 「⟨   ⁴ᵗʰa   ²ⁿᵈb ¹ˢᵗc   ⟩, (abc)₁₀ ≥ 152」

(abc)₁₀ ∶= 100a + 10b + c.

498 permutations match this pattern.

Examples: ⟨345012⟩, ⟨531024⟩, ⟨520314⟩.

🔳 「⟦0,4⟧ ∋ 1,2」

The closed interval given by 0 and 4 contains 1, 2.

120 permutations match this pattern.

Examples: ⟨052134⟩, ⟨425103⟩, ⟨453120⟩.

#125034_v2.2

       ┌───┬───┬───┬───┬───┬───┐
       │5th│4th│3rd│2nd│1st│0th│▒
       ╞═══╪═══╪═══╪═══╪═══╪═══╡▒
Step 1 │   │ 0 │   │   │   │   │▒
       ├───┼───┼───┼───┼───┼───┤▒
Step 2 │   │ 0 │   │   │ 5 │   │▒
       ├───┼───┼───┼───┼───┼───┤▒
Step 3 │ 3 │ 0 │   │   │ 5 │   │▒
       ├───┼───┼───┼───┼───┼───┤▒
Step 4 │ 3 │ 0 │   │ 2 │ 5 │   │▒
       ├───┼───┼───┼───┼───┼───┤▒
Step 5 │ 3 │ 0 │ 4 │ 2 │ 5 │   │▒
       ├───┼───┼───┼───┼───┼───┤▒
Step 6 │ 3 │ 0 │ 4 │ 2 │ 5 │ 1 │▒
       └───┴───┴───┴───┴───┴───┘▒
        ▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒

Proof of 2023-11-28 WR
══════════════════════

Notation: if nth -> a, then we write [nth] = a.

By ⛔「⟨   ⁴ᵗʰa   ²ⁿᵈb ¹ˢᵗc   ⟩, (abc)₁₀ ≥ 152」, we have [4th] = 1 or 0. Together with ✅「4th → a, 1st → b, a+b=5」, there are only two possibilities:

(i) [4th] = 1, [1st] = 4;
(ii) [4th] = 0, [1st] = 5.

(1) We show that (i) is not possible. Suppose on the contrary (i) holds:

┌───┬───┬───┬───┬───┬───┐
│5th│ 4▲│3rd│2nd│ 1▲│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╡
│   │ 1 │   │   │ 4 │   │
└───┴───┴───┴───┴───┴───┘

By ✅「⟨⋯ Perm(2,5) ⋯⟩」, we see that

┌───┬───┬───┬───┬───┬───┐
│5th│4th│*3 │*2 │1st│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╡
│   │ 1 │ x │ y │ 4 │   │
└───┴───┴───┴───┴───┴───┘

where {x,y} = {2,5}. To avoid ⛔「⟨   ⁴ᵗʰa   ²ⁿᵈb ¹ˢᵗc   ⟩, (abc)₁₀ ≥ 152」, we need (x,y) = (5,2):

┌───┬───┬───┬───┬───┬───┐
│5th│4th│ 3▲│ 2▲│1st│0th│
╞═══╪═══╪═══╪═══╪═══╪═══╡
│   │ 1 │ 5 │ 2 │ 4 │   │
└───┴───┴───┴───┴───┴───┘

But then ✅「⟦3,5⟧ ∋ 0,2」 could not be matched, which is a contradiction.

------------------------------

We have verified our claim in (1). Accordingly, we obtain:

       ┌───┬───┬───┬───┬───┬───┐
       │5th│ 4■│3rd│2nd│ 1■│0th│▒
       ╞═══╪═══╪═══╪═══╪═══╪═══╡▒
Step 1 │   │ 0 │   │   │   │   │▒
       ├───┼───┼───┼───┼───┼───┤▒
Step 2 │   │ 0 │   │   │ 5 │   │▒
       └───┴───┴───┴───┴───┴───┘▒
        ▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒

--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ 1 │ 2 │   │   │ 3 │ 4 │
└───┴───┴───┴───┴───┴───┘

Now, to match ✅「⟦3,5⟧ ∋ 0,2」, we need 0 to be between 5 and 3. Thus

       ┌───┬───┬───┬───┬───┬───┐
       │ 5■│4th│3rd│2nd│1st│0th│▒
       ╞═══╪═══╪═══╪═══╪═══╪═══╡▒
       │   │ 0 │   │   │ 5 │   │▒
       ├───┼───┼───┼───┼───┼───┤▒
Step 3 │ 3 │ 0 │   │   │ 5 │   │▒
       └───┴───┴───┴───┴───┴───┘▒
        ▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒

--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ 1 │ 2 │   │   │   │ 4 │
└───┴───┴───┴───┴───┴───┘

Next, we consider where to place 2. By ✅「⟨⋯ Perm(2,5) ⋯⟩」, 2,5 are adjacent, and by ✅「⟦3,5⟧ ∋ 0,2」, 2 is between 5 and 3. Therefore, we have

       ┌───┬───┬───┬───┬───┬───┐
       │5th│4th│3rd│ 2■│1st│0th│▒
       ╞═══╪═══╪═══╪═══╪═══╪═══╡▒
       │ 3 │ 0 │   │   │ 5 │   │▒
       ├───┼───┼───┼───┼───┼───┤▒
Step 4 │ 3 │ 0 │   │ 2 │ 5 │   │▒
       └───┴───┴───┴───┴───┴───┘▒
        ▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒

--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│ 1 │   │   │   │   │ 4 │
└───┴───┴───┴───┴───┴───┘

Finally, note that there is only one way to avoid ⛔「⟦0,4⟧ ∋ 1,2」. We reach

       ┌───┬───┬───┬───┬───┬───┐
       │5th│4th│ 3■│2nd│1st│ 0■│▒
       ╞═══╪═══╪═══╪═══╪═══╪═══╡▒
       │ 3 │ 0 │   │ 2 │ 5 │   │▒
       ├───┼───┼───┼───┼───┼───┤▒
Step 5 │ 3 │ 0 │ 4 │ 2 │ 5 │   │▒
       ├───┼───┼───┼───┼───┼───┤▒
Step 6 │ 3 │ 0 │ 4 │ 2 │ 5 │ 1 │▒
       └───┴───┴───┴───┴───┴───┘▒
        ▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒

--- Idle ---
┌───┬───┬───┬───┬───┬───┐
│   │   │   │   │   │   │
└───┴───┴───┴───┴───┴───┘

Q.E.D.

#125034_v2.2