2023-11-30 Q1(m=6)

Rearrange the digits in ⟨1263045⟩ to meet the rules below.

⟨6th 5th 4th 3rd 2nd 1st 0th⟩

✅Match

⟨? 3 ⋯ (?+6) ⋯ 2 ⋯⟩

2nd → 6

⟨       ³ʳᵈa ²ⁿᵈb ¹ˢᵗc   ⟩, (abc)₁₀ ≥ 532

⟨ ⁶ᵗʰ↑ ⁵ᵗʰ= ⁴ᵗʰ↑ ³ʳᵈ↓ ²ⁿᵈ↓ ¹ˢᵗ↓ ⁰ᵗʰ↑ ⟩ after 6−⟨⋯⟩

#125034_v2.2

2023-11-30 WR

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

⟨5th 4th 3rd 2nd 1st 0th⟩

✅Match

Sim⟨ ⁵ᵗʰ3 ⁴ᵗʰ4 ³ʳᵈ0 ²ⁿᵈ1 ¹ˢᵗ5 ⁰ᵗʰ2 ⟩ ≥ 2

⟨⋯ Perm(0,2) ⋯⟩

⟨⋯ a ⋯ 3 ⋯⟩, a = 0|1|4|5

⟨⋯ 4 ⋯ 0 ⋯⟩

⟨ ⁵ᵗʰa       ¹ˢᵗc ⁰ᵗʰb ⟩, a > b > c

⛔Avoid

5th|3rd|0th → 2

----- Information -----

🔲 「Sim⟨ ⁵ᵗʰ3 ⁴ᵗʰ4 ³ʳᵈ0 ²ⁿᵈ1 ¹ˢᵗ5 ⁰ᵗʰ2 ⟩ ≥ 2」

⟨pᵢ⟩ matches this pattern if and only if #{i: pᵢ = qᵢ} ≥ 2, where ⟨qᵢ⟩ ∶= ⟨340152⟩.

191 permutations match this pattern.

Examples: ⟨150342⟩, ⟨345021⟩, ⟨530124⟩.

🔲 「⟨⋯ Perm(0,2) ⋯⟩」

0,2 are adjacent.

240 permutations match this pattern.

Examples: ⟨302451⟩, ⟨102435⟩, ⟨420531⟩.

🔲 「⟨⋯ a ⋯ 3 ⋯⟩, a = 0|1|4|5」

576 permutations match this pattern.

Examples: ⟨045213⟩, ⟨451302⟩, ⟨023541⟩.

🔲 「⟨⋯ 4 ⋯ 0 ⋯⟩」

360 permutations match this pattern.

Examples: ⟨415230⟩, ⟨143025⟩, ⟨231540⟩.

🔲 「⟨ ⁵ᵗʰa       ¹ˢᵗc ⁰ᵗʰb ⟩, a > b > c」

120 permutations match this pattern.

Examples: ⟨352401⟩, ⟨521403⟩, ⟨530412⟩.

🔳 「5th|3rd|0th → 2」

360 permutations match this pattern.

Examples: ⟨531402⟩, ⟨145032⟩, ⟨510342⟩.

#125034_v2.2

2023-11-28 Q1(m=6)

Rearrange the digits in ⟨1263045⟩ to meet the rules below.

⟨6th 5th 4th 3rd 2nd 1st 0th⟩

✅Match

1st → a, 0th → b, |a-b|=4

⟨   ⁵ᵗʰd ⁴ᵗʰa   ²ⁿᵈb   ⁰ᵗʰc ⟩, a > b > c > d

2nd → 0|1|4

6th → a, 1st → b, a+b=2+5n

⛔Avoid

⟦1,3⟧ ∋ 0,5

#125034_v2.2

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