Insect Galore 豐富的昆蟲

Solution for HKOI2010 Junior Q4 By Gary Sham Insect Galore豐富的昆蟲

Problem Description • Given 4 kinds of insects with different number and year of activity. • Find the earliest year that can observe most insects. • Bees = 2001, 2003, 2005 etc. • Cicadae = 2002, 2005, 2008 etc. • Beetles = 2001, 2002, 2005, 2006 etc. • Butterflies = 2001, 2003, 2006, 2010 etc. • E.g. {2 4 1 0}, answer = 2017 7 • 2017 is the earliest year that can see all insects.

Statistic • #Attempts 46 • #Maximum 100 • #Maximum 5 • #Mean (Attempted) 25.3 • #Minimum 0 • #Std Dev (Attempted) 32.5

Question • How to know which kind of insects will appear that year? • Mathematics • Or simulation and array. • Another question : When to stop??

Observation • There are only 16 states of activity of insects. • Save the result of all states? • Do all states must appear at some time? • The total number of insects must be the sum of them. • If yes......

Solution1 year = 2010, Max = 0 While (true) do sum = 0 if (BEE && Cicadae && Beetles && Butterflies) sum+=BEE+Cicadae+Beetles+Butterflies; if (sum>max) output, break; Time Complexity: ??? Expected score: 50

In fact…… • NO! • The problem is Butterflies. • Butterflies and Cicadae will not appear at the same time. • Prove by Mathematics.

Full Solution • For each state, there are only one answer: YEAR and HOW TO ADD THE NUMBER will not change.  Hardcode • We have to consider whether we choose Butterflies or Cicadae. • Maximum  2021 • Time Complexity: O(1) • Expected Score: 100

Skills • Observation • Mathematics • Hardcode

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Insect Galore 豐富的昆蟲