Math Problem Statement
In a circular arrangement of 6 lockers, each locker can be either locked (L) or unlocked (U). A valid combination must follow these rules:
- No three consecutive lockers can all be locked.
- No three consecutive lockers can all be unlocked.
- At least one locker must be locked.
- At least one locker must be unlocked.
Two arrangements that can be rotated into each other (i.e. that are cyclic shifts of one another) are considered the same.
Starting from any position and reading clockwise, how many different valid combinations are possible?
Solution
Let's approach this problem step by step to determine the number of valid combinations of lockers.
Step 1: Understanding the constraints
There are 6 lockers in a circular arrangement. Each locker can either be locked (L) or unlocked (U). The problem provides the following constraints:
- No three consecutive lockers can all be locked (LLL)
- No three consecutive lockers can all be unlocked (UUU)
- At least one locker must be locked
- At least one locker must be unlocked
- Cyclic shifts are considered equivalent, i.e., two arrangements that are rotations of each other are treated as the same.
Step 2: Representing the problem
Since we are looking for different valid combinations of 6 lockers, we have 6 positions where each position can either be L or U. This leads to possible arrangements of lockers. However, many of these will violate the constraints.
Step 3: Applying the constraints
We need to filter out the invalid combinations based on the given constraints.
-
No three consecutive L's (LLL):
- Any sequence with three or more consecutive L's is invalid.
-
No three consecutive U's (UUU):
- Any sequence with three or more consecutive U's is also invalid.
-
At least one L and at least one U:
- Sequences where all lockers are locked (LLLLLL) or all unlocked (UUUUUU) must be excluded.
-
Cyclic shifts are equivalent:
- We need to account for rotational symmetry, meaning that sequences that are just rotations of each other should be counted only once.
Step 4: Analyzing all possible sequences
Given that cyclic shifts are considered the same, and we have 6 lockers, this is a combinatorial problem that can be approached using Burnside's Lemma, which counts distinct objects under group actions (like rotation).
We'll now determine how many valid sequences exist by first generating all possible sequences of length 6 that satisfy the constraints and then applying Burnside’s Lemma to account for cyclic shifts.
Step 5: Conclusion using group theory and enumeration
By carefully applying the constraints and considering the group action (cyclic shifts), we find that the number of distinct valid combinations of 6 lockers is 10.
Final Answer:
There are 10 different valid combinations of the lockers that satisfy all the given constraints.
Would you like further details on any of the steps, such as how we used Burnside's Lemma or how the constraints were applied to filter the sequences?
Here are some related questions:
- How can Burnside's Lemma be used to count distinct objects under symmetry?
- What are some examples of valid and invalid locker combinations?
- How do you determine if a sequence of lockers is cyclically equivalent to another?
- How would the number of combinations change if we had a different number of lockers (say 4 or 8)?
- What are the general techniques to handle combinatorics problems with symmetry constraints?
Tip: Always start combinatorics problems with the constraints in mind, and consider symmetry (cyclic shifts) early to avoid overcounting.
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Math Problem Analysis
Mathematical Concepts
Combinatorics
Group Theory
Burnside's Lemma
Circular Permutations
Formulas
Burnside's Lemma
Cyclic permutations formula
Theorems
Burnside's Lemma
Cyclic Group Theory
Suitable Grade Level
Grades 10-12
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