Lecture Notes

Problem A

• Objective: Maximize the product of A * B * C by incrementing one number up to 5 times
• Strategy: Always increment the smallest number because
  • Incrementing the smallest of A, B, C yields the largest product increment
• Method: Simple greedy algorithm to find and increment the minimum
• Implementation:
  • Keep track of the smallest element and increment it
  • Verify your approach with various test cases
• Additional Note: Can solve by iterating through all possible increments if unsure

Problem B

• Objective: Combine sticks into the longest one
• Strategy: Combine everything into the longest stick
• Steps:
  • Sort the sticks
  • From a given stick of length a, breaking it into 1s takes a-1 steps
  • Combine with the largest stick afterwards takes a steps
• Optimization: Find the max element in linear time to reduce complexity

Problem C

• Objective: Maximize the expression sum(F) - sum(G)
• Steps:
  • Increase F as early as possible
  • Delay increases in G to the end
  • Use reverse-sorted permutation methods
• Approach:
  • Reverse the order of values for F's prefix
  • Reverse last M values for G

Problem D

• Objective: Navigate through a river avoiding crocodiles
• Strategy: Greedy approach
• Steps:
  • Start from the bank (logs on both sides)
  • Swim across each water tile
  • Jump to the farthest water tile if possible else it's game over
  • Prioritize logs over water to extend jump reach
• Optimization: Can be sped up further if needed

Problem E

• Objective: Count pairs (A, B) such that n*A - B equals a specific value
• Constraints: N up to 100, A and B up to 10,000
• Solution:
  • Iterative calculation with length constraints and prefix checks
  • Loop over A and derive B accordingly
  • Reduce complexity using constraints on length and using String concatenation

Problem F

• Objective: Divide an array into 'bad' segments
• 'Bad' Segment Definition: No subset within a segment has a product exactly equal to X
• Approach: Segment as large as possible until violation occurs
• Steps:
  • Factorize X
  • DP approach for verifying possible products within segment
• Optimization: Faster implementations possible using sets

Problem G

• Objective: Calculate sum of mech of subsets plus one for all subsets
• Approach:
  • Iterate over size and mech of subset
  • Count ways and multiply by X for each segment
  • Ensure the calculations exclude the 'mech' itself
• Implementation:
  • Use combinations to determine subset choices
  • Edge cases: handle empty subsets explicitly