Discrete Math Quiz: Explaining Solutions

What this quiz covers

This quiz focuses on Explaining Solutions, giving you a quick way to practice the rules, question types, and explanations that matter most for Discrete Math.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. Compare with other cycle detection algorithms, then show equivalent behavior proves correctness
2. Test the algorithm on various graph examples, then generalize from observed patterns to prove correctness
3. Prove loop invariants for the coloring scheme, show that back edges indicate cycles, then verify termination (correct answer)
4. Demonstrate that the algorithm finds at least one cycle in cyclic graphs and reports none in acyclic graphs

Question 2

1. Use Fermat's Little Theorem directly since $11$ is prime, then verify using repeated squaring method
2. Calculate $7^{23} \bmod 11$ using repeated squaring, then verify using Fermat's Little Theorem as independent check (correct answer)
3. Apply the Chinese Remainder Theorem after factoring the modulus into smaller prime components
4. Compute the result using brute force multiplication, then confirm using Euler's totient function properties

Question 3

1. Present the mathematical optimization model first, then explain heuristic adaptations for real-time constraints
2. Demonstrate the algorithm using a simplified example, then address scalability and dynamic constraint handling (correct answer)
3. Focus on cost-benefit analysis of the solution, then briefly mention the underlying algorithmic approach
4. Explain the computer science theory behind graph algorithms, then show how it applies to logistics

Question 4

1. Present counterexample attempts first, then show why they fail, leading to the correct bound
2. Use induction on the number of vertices, handling base cases and inductive steps systematically
3. Establish Euler's formula $v - e + f = 2$, derive face-edge relationships, then apply degree constraints (correct answer)
4. Start with the complete graph $K_5$, demonstrate non-planarity, then generalize to the edge bound

Question 5

1. Worst-case analysis first, then average-case analysis, concluding with best-case scenario examination
2. Empirical timing studies across different input sizes, then theoretical analysis to explain observed patterns
3. Comparison with known algorithms of similar complexity, followed by mathematical verification of bounds
4. Mathematical analysis of the recurrence relation, followed by empirical testing with various input distributions (correct answer)

Question 6

1. Establish planarity properties, apply Four Color Theorem, then show algorithm correctness through inductive reasoning (correct answer)
2. Prove algorithm termination first, then demonstrate that output satisfies coloring constraints for all cases
3. Show the algorithm works for specific examples, then generalize to all planar graphs using pattern recognition
4. Verify that the algorithm produces optimal colorings, then conclude validity from optimality properties

Question 7

1. Trace execution for small inputs, identify the exponential time complexity issue, then implement memoization (correct answer)
2. Profile memory usage first, then analyze time complexity, concluding with alternative algorithm comparison
3. Test with increasingly large inputs until failure, then work backwards to find the breaking point
4. Review algorithm correctness first, then optimize for performance without changing the core logic

Question 8

1. Remove detection system nodes from the graph, then check if remaining graph components are disconnected, ensuring no path exists between critical servers and workstations (correct answer)
2. Calculate shortest paths between all server-workstation pairs while assigning infinite weight to edges connected to detection systems, verifying all paths exceed acceptable threshold
3. Apply depth-first search from each potential entry point, marking detection systems as terminal nodes, and verify that critical servers remain unreachable from workstations
4. Use maximum flow algorithms between server and workstation subgraphs with detection systems as bottleneck nodes, ensuring total flow capacity remains below minimum attack threshold

Question 9

1. The 12 users are articulation points whose removal disconnects the graph, indicating the network has low redundancy with average component size of $\frac{500-12}{47} \approx 10.4$ users
2. The 12 users represent critical bridges in the social graph, and removing them exposes the network's inherent clustering structure with 47 natural communities averaging 10.4 users each
3. The network exhibits scale-free properties where 12 highly connected hubs maintain global connectivity, and their removal fragments the network into 47 communities of approximately equal size
4. These 12 users form a minimum vertex cut that partitions the network, revealing structural weakness where 97.6% of users become isolated in small components averaging 10.4 members (correct answer)

Question 10

1. The error affects only $F(0)$, so $F(5)$ remains correct at 5 since the recursion primarily depends on $F(1)$ for larger values
2. Each Fibonacci number incorporates $F(0)$ exactly once through the recursive structure, so $F(5)$ increases by exactly 1, becoming 6 instead of 5
3. The error propagates through multiple paths: $F(5) = F(4) + F(3)$ where both $F(4)$ and $F(3)$ contain the error, resulting in $F(5) = 8$ instead of 5 (correct answer)
4. The error doubles at each recursive level due to the binary branching structure, so $F(5)$ becomes $5 + 2^4 = 21$ instead of 5

Question 11

1. Map known substitutions W→T, K→H, D→A, L→I, V→S, then apply pattern recognition to deduce remaining mappings for 'VDPH' yields 'SAME'
2. Establish Caesar cipher shift of +3 from the given examples, then apply reverse shift (-3) to each letter: V→S, D→A, P→M, H→E, yielding 'SAME' (correct answer)
3. Use frequency analysis on the given pairs to establish W=T, K=H, D=A, L=I, V=S, then solve 'VDPH' by direct substitution to get 'SAME'
4. Apply the identified pattern where consonants shift by +3 and vowels remain fixed, giving V→S, D→A, P→M, H→E for result 'SAME'

Question 12

1. Multiple MSTs indicate the graph has exactly 3 edges with identical weights that create alternative optimal connections, requiring algorithm modification to handle ties consistently
2. Having 3 MSTs means the original graph contains cycles, and breaking ties differently yields different trees, but all guarantee optimal total distance of 47 miles for infrastructure (correct answer)
3. The existence of multiple MSTs suggests redundancy in the road network, providing 3 equally efficient backup routes if any single road becomes unavailable during deliveries
4. Three different MSTs indicate unstable optimization, requiring additional constraints or weighted priorities to select a unique solution for consistent route planning across delivery teams

Question 13

1. With 6 operational servers, the system needs $\lceil \frac{2}{3} \times 6 \rceil = 4$ servers for consensus, which is achievable, so the system remains functional and fault-tolerant
2. The original system required $\lceil \frac{2}{3} \times 7 \rceil = 5$ servers for consensus, and with 6 servers remaining, this threshold is still achievable, maintaining system safety
3. Byzantine protocols require more than $\frac{2}{3}$ agreement based on the original configuration, so $\frac{2}{3} \times 7 = 4.67$ means 5 servers needed, and 6 remaining servers can still provide this (correct answer)
4. The system dynamically adjusts to $\frac{2}{3} \times 6 = 4$ servers needed for the new configuration, but Byzantine fault tolerance is compromised since the protocol was designed for 7-server scenarios

Question 14

1. The algorithm counts rotational symmetries as distinct solutions: each of the 6 valid solutions generates 4 rotations (0°, 90°, 180°, 270°), giving $6 \times 4 = 24$ total (correct answer)
2. The algorithm double-counts solutions by considering both the original arrangement and its reflection, then applies rotational symmetry: $6 \times 2 \times 2 = 24$ combinations
3. The validation logic incorrectly permutes identical colored blocks, treating arrangements that differ only by swapping same-colored pieces as distinct solutions with factor $\frac{24}{6} = 4$
4. The algorithm fails to account for grid translation invariance, counting each solution at multiple positions within a larger conceptual space, multiplying by factor of 4

Question 15

1. Key 67 hashes to position $67 \bmod 11 = 1$, then probes positions 1,2,3,4 which are all occupied by previous insertions that also hashed to position 1
2. Linear probing creates secondary clustering where probe sequences overlap, causing key 67 to require 4 attempts regardless of its initial hash position due to accumulated collision effects
3. All inserted keys hash to the same initial position creating a cluster, so searching for any non-present key requires probing through the entire cluster length of 4 positions
4. Key 67 hashes to position $67 \bmod 11 = 1$, probes consecutive positions 1,2,3,4 finding them occupied due to clustering from previous collisions, stops at first empty position (correct answer)