Rearrange the digits in ⟨125034⟩ to meet the rules below.
⟨ ⁵ᵗʰ▨ ⁴ᵗʰ▨ ³ʳᵈ▨ ²ⁿᵈ▨ ¹ˢᵗ▨ ⁰ᵗʰ▨ ⟩
✅Match
g(0) = 4
g(1) = 7
g(2) = 12
g(3) = 8
g(4) = 3
g(5) = 5
----- Information -----
Your goal is to find a permutation that satisfies every required pattern ✅ simultaneously.
There are 6 positions labeled 5th down to 0th, arranged left to right in descending order (5th is leftmost, 0th is rightmost). Therefore, ⟨125034⟩ = ⟨ ⁵ᵗʰ1 ⁴ᵗʰ2 ³ʳᵈ5 ²ⁿᵈ0 ¹ˢᵗ3 ⁰ᵗʰ4 ⟩.
It is given that
- [5th] is connected to [2nd]
- [4th] is connected to [3rd] and [2nd]
- [3rd] is connected to [4th], [1st] and [0th]
- [2nd] is connected to [5th], [4th] and [1st]
- [1st] is connected to [3rd], [2nd] and [0th]
- [0th] is connected to [3rd] and [1st]
By "g(X) = Y" we mean that the circles connected to X have numbers which sum to Y.
----- Required patterns -----
✅ 「g(0) = 4」
There are 48 permutations matching this pattern.
Examples: ⟨023415⟩, ⟨241530⟩, ⟨203154⟩, ⟨251430⟩.
✅ 「g(1) = 7」
There are 120 permutations matching this pattern.
Examples: ⟨340215⟩, ⟨253104⟩, ⟨520314⟩, ⟨204153⟩.
✅ 「g(2) = 12」
There are 36 permutations matching this pattern.
Examples: ⟨013524⟩, ⟨013425⟩, ⟨540231⟩, ⟨132054⟩.
✅ 「g(3) = 8」
There are 36 permutations matching this pattern.
Examples: ⟨510324⟩, ⟨023415⟩, ⟨053412⟩, ⟨405231⟩.
✅ 「g(4) = 3」
There are 108 permutations matching this pattern.
Examples: ⟨514320⟩, ⟨352041⟩, ⟨314520⟩, ⟨351024⟩.
✅ 「g(5) = 5」
There are 120 permutations matching this pattern.
Examples: ⟨254103⟩, ⟨402513⟩, ⟨234051⟩, ⟨204531⟩.