Rearrange the digits in ⟨1263045⟩ to meet the rules below.
⟨6th 5th 4th 3rd 2nd 1st 0th⟩
✅Match
Jump(0,2) = 4
⟨ ⁶ᵗʰ= ⁵ᵗʰ↑ ⁴ᵗʰ↓ ³ʳᵈ↓ ²ⁿᵈ↓ ¹ˢᵗ↑ ⁰ᵗʰ↑ ⟩ after 6−⟨⋯⟩
⟦2,5⟧ ∋ 4,6
4th → 4
#125034_v2.3
Originating from a math proof, 125034 is a game driven by combinatorics, logic, and probability.
2024-02-29 WR
Rearrange the digits in ⟨125034⟩ to meet the rules below.
⟨5th 4th 3rd 2nd 1st 0th⟩
✅Match
⟦1,2⟧ ∋ 3,4
2nd → a, 1st → b, a+b=9
⛔Avoid
Sim⟨ ⁵ᵗʰ2 ⁴ᵗʰ0 ³ʳᵈ5 ²ⁿᵈ3 ¹ˢᵗ4 ⁰ᵗʰ1 ⟩ ≥ 2
⟦1,4⟧ ∋ 5
⟦0,2⟧ ∋ 3
#125034_v2.3
2024-02-27 Q1(m=6)
Rearrange the digits in ⟨1263045⟩ to meet the rules below.
⟨6th 5th 4th 3rd 2nd 1st 0th⟩
✅Match
Sim⟨ ⁶ᵗʰ3 ⁵ᵗʰ6 ⁴ᵗʰ5 ³ʳᵈ1 ²ⁿᵈ2 ¹ˢᵗ4 ⁰ᵗʰ0 ⟩ ≥ 3
⟨⋯ 4 ⋯ ? 0 ⋯ (?+4)⟩ (?≠0)
⟨? 3 ⋯ (?+1) ⋯ 5 ⋯⟩ (?≠3,2)
#125034_v2.3
2024-02-27 WR
Rearrange the digits in ⟨125034⟩ to meet the rules below.
⟨5th 4th 3rd 2nd 1st 0th⟩
✅Match
2nd → a, 0th → b, a+b=1
⟨⋯ 0 ⋯ ? ⋯ 4 (?+1)⟩ (?≠4,3)
⛔Avoid
⟦3,5⟧ ∋ 2
⟨⋯ 3 ⋯ ? 5 ⋯ (?−2)⟩ (?≠5)
⟨⋯ 2 ⋯ 5 ⋯⟩
#125034_v2.3
2024-02-25 Q1(m=6)
Rearrange the digits in ⟨1263045⟩ to meet the rules below.
⟨6th 5th 4th 3rd 2nd 1st 0th⟩
✅Match
6th → a, 0th → b, a+b=7
Jump(4,6) ≤ 2
⟨⋯ ? ⋯ 5 ⋯ (?+2)⟩ (?≠3)
⟨ ⁵ᵗʰa ²ⁿᵈb ¹ˢᵗc ⟩, (abc)₁₀ ≥ 315
⛔Avoid
⟨ ⁶ᵗʰa ⁵ᵗʰb ⟩, (ab)₁₀ ≥ 25
⟨ ⁵ᵗʰb ⁴ᵗʰc ²ⁿᵈa ⟩, a > b > c
#125034_v2.3
2024-02-25 WR
Rearrange the digits in ⟨125034⟩ to meet the rules below.
⟨5th 4th 3rd 2nd 1st 0th⟩
✅Match
⟨⋯ Perm(1,4) ⋯⟩
Jump(0,4) = 1
4th → a, 2nd → b, a+b=3
⟦3,4⟧ ∋ 5
⛔Avoid
5th|3rd|1st → 0
#125034_v2.3
2024-02-23 Q1(m=6)
Rearrange the digits in ⟨1263045⟩ to meet the rules below.
⟨6th 5th 4th 3rd 2nd 1st 0th⟩
✅Match
⟨ ⁶ᵗʰ↑ ⁵ᵗʰ↓ ⁴ᵗʰ↑ ³ʳᵈ↓ ²ⁿᵈ↑ ¹ˢᵗ↓ ⁰ᵗʰ↑ ⟩ after 6−⟨⇌⟩
⟨⋯ 5 ⋯ 6 ⋯⟩
⟨? ⋯ 4 (?−2) ⋯ 0 ⋯⟩ (?≠4,6)
⟨ ⁶ᵗʰa ²ⁿᵈb ⟩, min⟦a,b⟧ = 0
⟨ ²ⁿᵈa ¹ˢᵗb ⟩, min⟦a,b⟧ = 1
#125034_v2.3
2024-02-23 WR
Rearrange the digits in ⟨125034⟩ to meet the rules below.
⟨5th 4th 3rd 2nd 1st 0th⟩
✅Match
⟨⋯ 0 ⋯ 5 ⋯ 3 ⋯⟩
⟨⋯ Perm(1,3,5) ⋯⟩
⛔Avoid
5th|4th|3rd|1st → 2
⟨ ²ⁿᵈa ¹ˢᵗb ⟩, (ab)₁₀ ≥ 23
⟨⋯ Perm(2,3) ⋯⟩
#125034_v2.3
2024-02-22 Q1(m=6)
Rearrange the digits in ⟨1263045⟩ to meet the rules below.
⟨6th 5th 4th 3rd 2nd 1st 0th⟩
✅Match
⟨⋯ Perm(1,2,6) ⋯⟩
6th → a, 3rd → b, a+b=1+6n
⟨ ³ʳᵈa ²ⁿᵈc ¹ˢᵗb ⟩, a > b > c
⛔Avoid
Jump(1,3) ≤ 3
#125034_v2.3
2024-02-22 WR
Rearrange the digits in ⟨125034⟩ to meet the rules below.
⟨5th 4th 3rd 2nd 1st 0th⟩
✅Match
⟨ ³ʳᵈa ¹ˢᵗb ⟩, min⟦a,b⟧ = 1
4th → a, 1st → b, ab=3
⟨ ⁴ᵗʰc ²ⁿᵈa ⁰ᵗʰb ⟩, a > b > c
#125034_v2.3
2024-02-20 Q1(m=6)
Rearrange the digits in ⟨1263045⟩ to meet the rules below.
⟨6th 5th 4th 3rd 2nd 1st 0th⟩
✅Match
⟨ ⁶ᵗʰa ⁵ᵗʰb ²ⁿᵈc ⟩, (abc)₁₀ ≤ 203
Sim⟨ ⁶ᵗʰ2 ⁵ᵗʰ4 ⁴ᵗʰ6 ³ʳᵈ5 ²ⁿᵈ0 ¹ˢᵗ1 ⁰ᵗʰ3 ⟩ ≥ 2
Jump(3,5) = 2
⟨ ⁶ᵗʰa ⁵ᵗʰb ²ⁿᵈc ⟩, (abc)₁₀ ≥ 132
⛔Avoid
⟨ ³ʳᵈa ¹ˢᵗb ⟩, (ab)₁₀ ≥ 24
#125034_v2.3
2024-02-20 WR
Rearrange the digits in ⟨125034⟩ to meet the rules below.
⟨5th 4th 3rd 2nd 1st 0th⟩
✅Match
4th → a, 1st → b, |a-b|=1
⟨⋯ 1 ⋯ ? ⋯ 2 (?+2)⟩ (?≠2,0)
⛔Avoid
2nd → 0|1|4
⟨ ⁵ᵗʰc ¹ˢᵗb ⁰ᵗʰa ⟩, a > b > c
2nd → 0|1|5
#125034_v2.3
2024-02-18 Q1(m=6)
Rearrange the digits in ⟨1263045⟩ to meet the rules below.
⟨6th 5th 4th 3rd 2nd 1st 0th⟩
✅Match
4th → a, 3rd → b, a+b=0+5n
⟨⋯ ? ⋯ 4 ⋯ (?+6)⟩
⟨⋯ Perm(0,1,5) ⋯⟩
4th → a, 2nd → b, ab=0+4n
⟨ ⁵ᵗʰa ³ʳᵈb ⁰ᵗʰc ⟩, (abc)₁₀ ≤ 132
⟨? ⋯ 5 ⋯ (?+3) ⋯⟩ (?≠2)
#125034_v2.3
2024-02-18 WR
Rearrange the digits in ⟨125034⟩ to meet the rules below.
⟨5th 4th 3rd 2nd 1st 0th⟩
✅Match
⟨ ³ʳᵈa ¹ˢᵗb ⟩, min⟦a,b⟧ = 1
⟨ ⁴ᵗʰa ²ⁿᵈb ¹ˢᵗc ⟩, (abc)₁₀ ≤ 130
⟦0,2⟧ ∋ 1,5
⟨ ³ʳᵈa ¹ˢᵗb ⟩, a > b
⛔Avoid
⟦2,3⟧ ∋ 1,5
4th|2nd|1st → 3
#125034_v2.3
2024-02-16 Q1(m=6)
Rearrange the digits in ⟨1263045⟩ to meet the rules below.
⟨6th 5th 4th 3rd 2nd 1st 0th⟩
✅Match
⟨ ⁶ᵗʰ↑ ⁵ᵗʰ↓ ⁴ᵗʰ↓ ³ʳᵈ↑ ²ⁿᵈ= ¹ˢᵗ↑ ⁰ᵗʰ↓ ⟩ after 6−⟨⋯⟩
Jump(5,6) = 4
⟦1,5⟧ ∋ 2,4
⛔Avoid
⟨ ⁶ᵗʰb ¹ˢᵗa ⟩, a > b
#125034_v2.3
2024-02-16 WR
Rearrange the digits in ⟨125034⟩ to meet the rules below.
⟨5th 4th 3rd 2nd 1st 0th⟩
✅Match
⟨ ³ʳᵈ↑ ²ⁿᵈ↑ ⟩ after 3×⟨→⟩
⟨ ⁴ᵗʰa ²ⁿᵈb ⟩, max⟦a,b⟧ = 4
⟨ ⁴ᵗʰa ³ʳᵈb ⟩, min⟦a,b⟧ = 1
⟨⋯ ? ⋯ 0 ⋯ (?−3)⟩ (?≠3)
⛔Avoid
⟨⋯ Perm(3,5) ⋯⟩
Jump(2,3) = 0
#125034_v2.3
2024-02-15 Q1(m=6)
Rearrange the digits in ⟨1263045⟩ to meet the rules below.
⟨6th 5th 4th 3rd 2nd 1st 0th⟩
✅Match
⟨ ⁶ᵗʰ↑ ⁵ᵗʰ↓ ⁴ᵗʰ↑ ³ʳᵈ= ²ⁿᵈ↓ ¹ˢᵗ↓ ⁰ᵗʰ↑ ⟩ after 6−⟨⋯⟩
6th → a, 5th → b, |a-b|=2
Sim⟨ ⁶ᵗʰ4 ⁵ᵗʰ5 ⁴ᵗʰ6 ³ʳᵈ2 ²ⁿᵈ3 ¹ˢᵗ1 ⁰ᵗʰ0 ⟩ = 1
2nd → 6
#125034_v2.3
2024-02-15 WR
Rearrange the digits in ⟨125034⟩ to meet the rules below.
⟨5th 4th 3rd 2nd 1st 0th⟩
✅Match
⟨⋯ 3 ⋯ 0 ⋯ 5 ⋯⟩
5th → a, 2nd → b, |a-b|=1
⟨ ⁵ᵗʰ= ⁴ᵗʰ↓ ³ʳᵈ↑ ²ⁿᵈ↑ ¹ˢᵗ↓ ⁰ᵗʰ= ⟩ after 5−⟨⇌⟩
⛔Avoid
Jump(1,5) = 2
⟨ ⁵ᵗʰ↑ ⁴ᵗʰ↑ ³ʳᵈ↓ ²ⁿᵈ↑ ¹ˢᵗ↓ ⁰ᵗʰ↓ ⟩ after ⟨⇌⟩
#125034_v2.3
2024-02-13 Q1(m=6)
Rearrange the digits in ⟨1263045⟩ to meet the rules below.
⟨6th 5th 4th 3rd 2nd 1st 0th⟩
✅Match
⟨ ⁵ᵗʰd ³ʳᵈa ¹ˢᵗc ⁰ᵗʰb ⟩, a > b > c > d
⟨? ⋯ 1 ⋯ (?−2) ⋯⟩ (?≠3)
⟦0,4⟧ ∋ 3
⟨⋯ 1 ⋯ 2 ⋯ 3 ⋯⟩
⛔Avoid
⟨? ⋯ 1 (?−6) ⋯ 0 ⋯⟩
0 ∾ 3 ∾ 6
⟨⋯ Perm(0,6) ⋯⟩
----- Information -----
🔲 「⟨ ⁵ᵗʰd ³ʳᵈa ¹ˢᵗc ⁰ᵗʰb ⟩, a > b > c > d」
210 permutations match this pattern.
Examples: ⟨5016234⟩, ⟨3026415⟩, ⟨6024513⟩.
🔲 「⟨? ⋯ 1 ⋯ (?−2) ⋯⟩ (?≠3)」
?, 1, (?−2) are three different numbers.
1440 permutations match this pattern.
Examples: ⟨4035162⟩, ⟨4503126⟩, ⟨5130642⟩.
🔲 「⟦0,4⟧ ∋ 3」
The closed interval given by 0 and 4 contains 3.
1680 permutations match this pattern.
Examples: ⟨6013254⟩, ⟨6013542⟩, ⟨4361205⟩.
🔲 「⟨⋯ 1 ⋯ 2 ⋯ 3 ⋯⟩」
840 permutations match this pattern.
Examples: ⟨4126305⟩, ⟨1462035⟩, ⟨4015623⟩.
🔳 「⟨? ⋯ 1 (?−6) ⋯ 0 ⋯⟩」
?, 1, (?−6) are three different numbers.
120 permutations match this pattern.
Examples: ⟨6254103⟩, ⟨6253410⟩, ⟨6510324⟩.
🔳 「0 ∾ 3 ∾ 6」
0,3,6 are in the same cycle.
A permutation can be decomposed into cycles. For example, ⟨125034⟩ has two cycles: (135) and (420), as (1st→3, 3rd→5, 5th→1) and (4th→2, 2nd→0, 0th→4).
1680 permutations match this pattern.
Examples: ⟨3125604⟩, ⟨2051463⟩, ⟨5432061⟩.
🔳 「⟨⋯ Perm(0,6) ⋯⟩」
0,6 are adjacent.
1440 permutations match this pattern.
Examples: ⟨2351064⟩, ⟨5241063⟩, ⟨4135260⟩.
#125034_v2.3
2024-02-13 WR
Rearrange the digits in ⟨125034⟩ to meet the rules below.
⟨5th 4th 3rd 2nd 1st 0th⟩
✅Match
⟨? ⋯ 0 ⋯ (?+2) ⋯⟩ (?≠0)
⟨ ⁴ᵗʰa ³ʳᵈb ¹ˢᵗc ⟩, (abc)₁₀ ≤ 142
⛔Avoid
⟨⋯ a ⋯ 1 ⋯⟩, a = 2|3|4
⟨⋯ 4 ⋯ 2 ⋯⟩
⟨? 0 ⋯ (?+3) ⋯ 3 ⋯⟩ (?≠0)
5th|4th|2nd|1st → 5
⟨ ²ⁿᵈ↑ ⟩ after ⟨→⟩
#125034_v2.3
2024-02-10 Q1(m=6)
Rearrange the digits in ⟨1263045⟩ to meet the rules below.
⟨6th 5th 4th 3rd 2nd 1st 0th⟩
✅Match
⟦1,3⟧ ∋ 0,2,4,5
⟨ ⁵ᵗʰa ⁴ᵗʰb ⟩, min⟦a,b⟧ = 2
⟨⋯ ? ⋯ 2 ⋯ (?−4)⟩ (?≠6)
⛔Avoid
⟨⋯ 4 ⋯ a ⋯⟩, a = 0|2|3|5
⟨⋯ 6 ⋯ a ⋯⟩, a = 2|3|4
#125034_v2.3
2024-02-10 WR
Rearrange the digits in ⟨125034⟩ to meet the rules below.
⟨5th 4th 3rd 2nd 1st 0th⟩
✅Match
Jump(3,4) = 0
⟨ ¹ˢᵗa ⁰ᵗʰb ⟩, max⟦a,b⟧ = 4
Jump(1,3) = 3
5th → a, 0th → b, ab=0+5n
⛔Avoid
3rd → 0
3rd → a, 0th → b, |a-b|=1
#125034_v2.3
2024-02-09 Q1(m=6)
Rearrange the digits in ⟨1263045⟩ to meet the rules below.
⟨6th 5th 4th 3rd 2nd 1st 0th⟩
✅Match
⟦1,2⟧ ∋ 3,5
Jump(0,1) ≥ 1
⟨? ⋯ 3 ⋯ (?−1) ⋯⟩ (?≠3,4)
6th|5th|4th|3rd|0th → 1
⛔Avoid
⟦1,6⟧ ∋ 0
Jump(1,3) ≤ 1
⟨ ⁶ᵗʰb ³ʳᵈa ⟩, a > b
⟨⋯ ? ⋯ 4 ⋯ (?−1)⟩ (?≠4,5)
#125034_v2.3
2024-02-09 WR
Rearrange the digits in ⟨125034⟩ to meet the rules below.
⟨5th 4th 3rd 2nd 1st 0th⟩
✅Match
⟨? ⋯ 5 (?+3) ⋯ 4 ⋯⟩ (?≠2)
⛔Avoid
5th → 1|4
5th|1st → 2
⟨ ⁵ᵗʰc ¹ˢᵗa ⁰ᵗʰb ⟩, a > b > c
⟨ ³ʳᵈa ¹ˢᵗb ⟩, min⟦a,b⟧ = 1
#125034_v2.3
2024-02-08 Q1(m=6)
Rearrange the digits in ⟨1263045⟩ to meet the rules below.
⟨6th 5th 4th 3rd 2nd 1st 0th⟩
✅Match
3rd → a, 2nd → b, |a-b|=3
⟨⋯ Perm(1,3,4) ⋯⟩
⟦4,6⟧ ∋ 0
⟨ ⁴ᵗʰc ²ⁿᵈd ¹ˢᵗa ⁰ᵗʰb ⟩, a > b > c > d
⛔Avoid
Jump(0,1) = 1
#125034_v2.3
2024-02-08 WR
Rearrange the digits in ⟨125034⟩ to meet the rules below.
⟨5th 4th 3rd 2nd 1st 0th⟩
✅Match
⟨ ²ⁿᵈa ⁰ᵗʰb ⟩, min⟦a,b⟧ = 2
4th → a, 1st → b, |a-b|=2
⟨⋯ 0 ⋯ ? 4 ⋯ (?−3)⟩ (?≠4)
#125034_v2.3
2024-02-06 Q1(m=6)
Rearrange the digits in ⟨1263045⟩ to meet the rules below.
⟨6th 5th 4th 3rd 2nd 1st 0th⟩
✅Match
⟨ ⁶ᵗʰa ²ⁿᵈb ⟩, min⟦a,b⟧ = 1
⟨? 6 ⋯ (?+4) ⋯ 5 ⋯⟩ (?≠2)
Jump(2,4) = 0
⟦1,2⟧ ∋ 3,4,6
⛔Avoid
3rd → 3|4
#125034_v2.3
2024-02-06 WR
Rearrange the digits in ⟨125034⟩ to meet the rules below.
⟨5th 4th 3rd 2nd 1st 0th⟩
✅Match
⟨ ⁰ᵗʰ↑ ⟩ after 3×⟨←⟩
⟨ ⁵ᵗʰc ²ⁿᵈb ⁰ᵗʰa ⟩, a > b > c
⟨ ⁴ᵗʰ↑ ⟩ after 2×⟨←⟩
⛔Avoid
⟨⋯ 0 ⋯ ? ⋯ 5 (?+1)⟩ (?≠4)
⟨ ⁵ᵗʰc ¹ˢᵗb ⁰ᵗʰa ⟩, a > b > c
5th → a, 0th → b, ab=8
⟨⋯ Perm(0,1,5) ⋯⟩
#125034_v2.3
2024-02-04 Q1(m=6)
Rearrange the digits in ⟨1263045⟩ to meet the rules below.
⟨6th 5th 4th 3rd 2nd 1st 0th⟩
✅Match
⟦1,4⟧ ∋ 0
⟨ ³ʳᵈa ⁰ᵗʰb ⟩, min⟦a,b⟧ = 2
⟨⋯ 2 ⋯ 3 ⋯⟩
5th → a, 2nd → b, ab=18
⛔Avoid
⟨⋯ 0 ⋯ ? ⋯ 5 (?+4)⟩ (?≠1)
#125034_v2.3
2024-02-04 WR
Rearrange the digits in ⟨125034⟩ to meet the rules below.
⟨5th 4th 3rd 2nd 1st 0th⟩
✅Match
⟨⋯ Perm(0,2) ⋯⟩
5th → a, 2nd → b, |a-b|=3
Jump(3,4) = 3
⟨⋯ Perm(0,1,3) ⋯⟩
#125034_v2.3
2024-02-02 Q1(m=6)
Rearrange the digits in ⟨1263045⟩ to meet the rules below.
⟨6th 5th 4th 3rd 2nd 1st 0th⟩
✅Match
⟨⋯ 1 ⋯ ? 4 ⋯ (?+4)⟩ (?≠0)
0 ∾ 1 ∾ 2 ∾ 3 ∾ 5
⟨⋯ ? ⋯ 4 ⋯ (?+5)⟩
1st → a, 0th → b, a+b=2+5n
⟨ ⁶ᵗʰa ⁴ᵗʰb ⟩, a > b
⛔Avoid
⟦3,5⟧ ∋ 2,6
#125034_v2.3
2024-02-02 WR
Rearrange the digits in ⟨125034⟩ to meet the rules below.
⟨5th 4th 3rd 2nd 1st 0th⟩
✅Match
5th → 1
Jump(2,4) = 1
5th → a, 0th → b, |a-b|=3
1st → 3|4|5
⛔Avoid
5th|1st → 5
⟨⋯ 0 ⋯ 5 ⋯ 3 ⋯⟩
#125034_v2.3
2024-02-01 Q1(m=6)
Rearrange the digits in ⟨1263045⟩ to meet the rules below.
⟨6th 5th 4th 3rd 2nd 1st 0th⟩
✅Match
⟨⋯ a ⋯ 1 ⋯⟩, a = 2|3|6
5th → a, 0th → b, |a-b|=1
⟨⋯ 1 ⋯ 6 ⋯ 5 ⋯ 0 ⋯⟩
⟨ ⁶ᵗʰa ⁴ᵗʰb ¹ˢᵗc ⟩, (abc)₁₀ ≤ 310
⟨⋯ 0 ⋯ a ⋯⟩, a = 1|2|4|5|6
#125034_v2.3
2024-02-01 WR
Rearrange the digits in ⟨125034⟩ to meet the rules below.
⟨5th 4th 3rd 2nd 1st 0th⟩
✅Match
5th|2nd|1st → 0
⟨? ⋯ 5 ⋯ (?+3) ⋯⟩ (?≠2)
⛔Avoid
⟨Perm(0,2,5) Perm(1,3,4)⟩
⟨⋯ 1 ⋯ ? ⋯ 4 (?−3)⟩ (?≠4)
⟨⋯ 5 ⋯ 4 ⋯ 3 ⋯⟩
⟨ ¹ˢᵗ↑ ⟩ after ⟨←⟩
⟨⋯ ? ⋯ 5 ⋯ (?+3)⟩ (?≠2)
⟨? ⋯ 4 ⋯ (?+3) ⋯⟩ (?≠1)
⟨⋯ 5 ⋯ 1 ⋯ 4 ⋯⟩
5th|2nd → 2
#125034_v2.3
Subscribe to:
Posts (Atom)