Puzzle of Week 1• You have 2 identical straight pieces of rope. • Each takes 1 hour to burn if you light it from 1 side (it doesn’t matter which one).
• Show how we can count 15 minutes. • You cannot use rulers, you cannot fold the ropes, and you cannot estimate accurately where the middle point of each rope is.
• You can ONLY use lighters.
Puzzle of Week 2• The city of Königsberg (now Kaliningrad) has 7
bridges in the following arrangement:
• Can we make a tour through the city that crosses each bridge exactly once?
• HINT: you can model it as a graph problem• Generalization: More importantly and more
interestingly, can you find an algorithm that decides if, in a given graph G, it is possible to walk through the graph so that we cross every edge exactly once? (i.e., solving a postman’s problem)
Puzzle of Week 3
• You have nine coins. Eight are genuine and one is fake. The fake coin is heavier than the other eight. You are allowed to use a balance scale at most two times. Figure out which coin is fake.
• HINT: Think of a Divide & Conquer approach
Puzzle of Week 4Divide & Conquer + Recurrence relations
•Suppose you have n coins, out of which one is fake and the other n-1 coins are genuine. The fake coin is heavier than the rest. You are allowed to use a balance scale to figure out which coin is fake. •Can you think of a recursive algorithm to solve this? (e.g. Divide into 2 parts) •Write a recurrence relation for the number of times you use the scale and solve it•What is the best recursive algorithm you can think of? (For convenience, assume n is a power of 3, n = 3k, for some k)
Puzzle of Week 5
• You are given 12 coins, one of which is fake. You do not know however if it is heavier or lighter than the original ones. You can use a balance scale up to 3 times. Determine the fake coin.
Puzzle of Week 6Definition 1: A tromino is an L-shaped tile
Definition 2: A defective chessboard is a chessboard where one square is broken
Puzzle of Week 61.Consider a 2n x 2n defective
chessboard. Show that it can always be filled up with trominos so that all squares are covered apart from the defective one (think of a recursive algorithm for this)
1.How many trominos do you need? (think of a recursive formula)
Puzzle of Week 7Odd pie fightSuppose a set of n people are in a birthday partyAfter some drinks, each person grabs a piece of a pie, and he plans to throw it at the person who is in the shortest distance from him
Puzzle of Week 7
Odd pie fightSuppose that:n is oddThe distances between the people are all distinctQ1: Show that there is at least one lucky person who will not get hit by a pieQ2: Find an algorithm to identify who is the lucky person. In how much time does your algorithm run?
