WASHINGTON STATE MATHEMATICS COUNCIL 2006 MIDDLE SCHOOL MATH OLYMPIAD - PDF

Pages to are hidden for

"WASHINGTON STATE MATHEMATICS COUNCIL 2006 MIDDLE SCHOOL MATH OLYMPIAD - PDF"

```					                WASHINGTON STATE MATHEMATICS COUNCIL
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.

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?

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.

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
WSMC 2006 Middle School Math Olympiad                              Session 1, Grade 7-8, Page 1 of 2
WASHINGTON STATE MATHEMATICS COUNCIL
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 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