"WASHINGTON STATE MATHEMATICS COUNCIL 2006 MIDDLE SCHOOL MATH OLYMPIAD - PDF"
WASHINGTON STATE MATHEMATICS COUNCIL 2006 MIDDLE SCHOOL MATH OLYMPIAD Session I: SEVENTH/EIGHTH GRADE PROBLEM SOLVING 7-8 CODING PROBLEM LVP JSF QCF XJQC RQJS! This is the first line of a paragraph in code you receive by email and of course you want to know what it says. As long as people have been sending information to one another there has been a concern about keeping the information secret. One way is to encode the information. All governments have people devoted to the task of breaking codes so that they can find out the secrets of other people, organizations, and governments. You’re going to take on the task of being a code breaker. For this problem the message being sent is a coded English language message. One of the first bits of information you need to know is what the probabilities are for each letter of the alphabet occurring in a message. Task 1: Describe a method for estimating the probabilities for how often a letter in the alphabet occurs in a piece of English text using the text in the box above. Then use your method to estimate the probabilities for the letters A, E, I, O, N, R, S, and T using the boxed text. These are the most frequently occurring letters in plain English language text. There are a total of 479 letters in the text in the box (we ignore spaces and punctuation). Note: Estimating means your answers may not be exact but you need to address having a good enough estimate. How do you know it is good enough? Task 2: Letter pairs also occur with certain frequencies that can then be converted to probabilities. Describe a method for estimating or computing frequencies of letter pairs in a text. Then use your method for obtain frequencies for letter pairs ER, IN, HE, RE, TH in the text in the box. These are the five most frequently appearing letter pairs in English text. If you choose to estimate explain how you know you have a high degree of accuracy. Task 3: You’re going to try to decode the first line of your coded email message and determine the coded alphabet so that you could decode the rest of the message. The first line reads: LVP JSF QCF XJQC RQJS! Through a computer analysis you get back the information in the tables below about the entire coded message (not just the first line). Using what you have found out in Tasks 1 and 2, plus the information in the tables, try and decode the message. Also fill in as much of Table 3 as you can with the decoded alphabet. Explain your process and your conclusions. WSMC 2006 Middle School Math Olympiad Session 1, Grade 7-8, Page 1 of 2 WASHINGTON STATE MATHEMATICS COUNCIL 2006 MIDDLE SCHOOL MATH OLYMPIAD Session I: SEVENTH/EIGHTH GRADE PROBLEM SOLVING 7-8 Table 1: The number of times each letter appears in the entire coded message expressed as percentage A B C D E F G H I J K L M 0.2 7.4 3.5 1.6 2.8 13 4.4 3 0.9 7.3 0.1 1.9 0.5 N O P Q R S T U V W X Y Z 1.6 1.3 2.7 9.3 6.3 7.7 0.3 2.7 7.4 7.8 2.5 3.5 0.3 Table 2: Frequency in the entire coded message of the letter pairs from the first coded line LV VP JS SF QC CF XJ JQ RQ QJ 1 2 10 17 34 30 2 16 11 3 Table 3: Converting code alphabet to the regular alphabet A B C D E F G H I J K L M N O P Q R S T U V W X Y Z For your convenience the line of code that you want to decode is repeated on this page, together with a place for you to write the message as you decode it. Coded line LVP JSF QCF XJQC RQJS! Decoded Message ___ ___ ___ ____ ____! Your work will be evaluated on: • Your Answer(s): the solutions to each of the three tasks. • Your Understanding of the Problem. Your answer should meet all the requirements of the problem, and demonstrate your understanding of each task. • The Strategies that you used. Your strategies and observations that help you solve each task. • Your Communication. How you presented your solution showing your strategy and answer for each of the three tasks. • Your Reasoning/ Reasonableness checks. How do you know you are on the right track. Make sure to support your answer with calculations and explanations for each of the three tasks. WSMC 2006 Middle School Math Olympiad Session 1, Grade 7-8, Page 2 of 2