2191
/www/nrich/html/content/04/03/six1/
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2011-02-01T00:00:01
<mdoxml version="1.0"><br></br>
<p>You may like to read the article on <a href="http://nrich.maths.org/public/viewer.php?obj_id=2198">Morse code</a> before attempting this question.</p>
<p> </p>
<table border="1" cellpadding="0" cellspacing="0" id="tableedit1" width="100%">
<caption>Morse Code</caption>
<tbody>
<tr>
<td>
<div align="center"><strong>A</strong></div>
</td>
<td>
<div align="center"><strong>B</strong></div>
</td>
<td>
<div align="center"><strong>C</strong></div>
</td>
<td>
<div align="center"><strong>D</strong></div>
</td>
<td>
<div align="center"><strong>E</strong></div>
</td>
<td>
<div align="center"><strong>F</strong></div>
</td>
<td>
<div align="center"><strong>G</strong></div>
</td>
<td>
<div align="center"><strong>H</strong></div>
</td>
<td>
<div align="center"><strong>I</strong></div>
</td>
<td>
<div align="center"><strong>J</strong></div>
</td>
<td>
<div align="center"><strong>K</strong></div>
</td>
<td>
<div align="center"><strong>L</strong></div>
</td>
<td>
<div align="center"><strong>M</strong></div>
</td>
</tr>
<tr>
<td>
<div align="center"><strong>.-</strong></div>
</td>
<td>
<div align="center"><strong>-...</strong></div>
</td>
<td>
<div align="center"><strong>-.-.</strong></div>
</td>
<td>
<div align="center"><strong>-..</strong></div>
</td>
<td>
<div align="center"><strong>.</strong></div>
</td>
<td>
<div align="center"><strong>..-.</strong></div>
</td>
<td>
<div align="center"><strong>--.</strong></div>
</td>
<td>
<div align="center"><strong>....</strong></div>
</td>
<td>
<div align="center"><strong>..</strong></div>
</td>
<td>
<div align="center"><strong>.---</strong></div>
</td>
<td>
<div align="center"><strong>-.-</strong></div>
</td>
<td>
<div align="center"><strong>.-..</strong></div>
</td>
<td>
<div align="center"><strong>--</strong></div>
</td>
</tr>
<tr>
<td>
<div align="center"><strong>N</strong></div>
</td>
<td>
<div align="center"><strong>O</strong></div>
</td>
<td>
<div align="center"><strong>P</strong></div>
</td>
<td>
<div align="center"><strong>Q</strong></div>
</td>
<td>
<div align="center"><strong>R</strong></div>
</td>
<td>
<div align="center"><strong>S</strong></div>
</td>
<td>
<div align="center"><strong>T</strong></div>
</td>
<td>
<div align="center"><strong>U</strong></div>
</td>
<td>
<div align="center"><strong>V</strong></div>
</td>
<td>
<div align="center"><strong>W</strong></div>
</td>
<td>
<div align="center"><strong>X</strong></div>
</td>
<td>
<div align="center"><strong>Y</strong></div>
</td>
<td>
<div align="center"><strong>Z</strong></div>
</td>
</tr>
<tr>
<td>
<div align="center"><strong>-.</strong></div>
</td>
<td>
<div align="center"><strong>---</strong></div>
</td>
<td>
<div align="center"><strong>.--.</strong></div>
</td>
<td>
<div align="center"><strong>--.-</strong></div>
</td>
<td>
<div align="center"><strong>.-.</strong></div>
</td>
<td>
<div align="center"><strong>...</strong></div>
</td>
<td>
<div align="center"><strong>-</strong></div>
</td>
<td>
<div align="center"><strong>..-</strong></div>
</td>
<td>
<div align="center"><strong>...-</strong></div>
</td>
<td>
<div align="center"><strong>.--</strong></div>
</td>
<td>
<div align="center"><strong>-..-</strong></div>
</td>
<td>
<div align="center"><strong>-.--</strong></div>
</td>
<td>
<div align="center"><strong>--..</strong></div>
</td>
</tr>
</tbody>
</table>
<p>Translate the following message into Morse code:</p>
<p><strong><em>Codes and hidden meanings</em></strong></p>
<p>If each dit (dot) takes 1 unit of time, a dah (dash) 3 units of time, the pause between letters 3 units and the pause between words 7 units, how long will it take to send this message?</p>
<p>Samuel Morse gave E the symbol with the shortest time value (1dit) because he thought it was the most commonly used letter. I, apparently the next most common letter, uses two dits to represent it. However, the letter analysis was done over 150 years ago and language does change and, of course, it may be entirely different in different languages.</p>
<p>So, might there be a better allocation of symbols today?</p>
<p>To tackle this question try counting the number of times each letter occurs in the article on Morse Code and suggest an alternative coding based on the frequency of each letter (this is called frequency analysis).</p>
<p>Would it take less time to send the message above with your code than with Morse Code? To help you there is a list of the symbols and their time in the hints.</p>
<p>On average would you expect your code to take less time to send a message than the international Morse Code? Could there be a more efficient coding? You might also like to look at the letter frequency graph shown in <a href="../public/viewer.php?obj_id=2197">Claire Ellis' article on Codes</a>.</p>
<p>Of course there is always the problem of whether the style of writing you might use to send a message is the same as that of the article. In fact, if Morse were used today I think it would most likely resemble texting.</p></mdoxml>
<mdoxml version="1.0">
<p>
This problem suited those of you who were very careful...
and some of you made errors along the way ...
</p><p>
Tom and Joseph from D.A.P.S. demonstrate the value of close collaboration:
</p><p>
"Our problem was to find out the units of time in the sentence that had to be decoded -
<strong><em>Codes and hidden meanings</em></strong>.
The way we solved this problem: we compared our notes and found an answer that we both agreed on. Joseph's method was: a space between letters is three units of
time, a space between words are seven units of time, a dah (a line-) is three units of time and a dot is one unit of time. First you add the units of time together to get the first number, then you add the three units of time for each space between letters, and at the end of each word you add seven units of time. Tom's method was to count all the spaces (there were 18 and there were 3 spaces between the words) so the next thing I did was to count the units of time in the letters. My answer was 166 units of time. Me and Joseph compared notes and we got the answer 166."
</p><p>
Jakob Plant, Ryan Campbell and Thomas Jewitt from Dauntsey's Aided Primary School set out their work very clearly:
</p>
<pre>
c o d e s
-.-. --- -.. . ... = 38 Units (including 3 units between letters)
= 7 units (7 units between each word)
a n d
.- -. -.. = 19 units
= 7 units
h i d d e n
.... .. -.. -.. . -. = 36 units
= 7 units
m e a n i n g s
-- . .- -. .. -. --. ... = 52 units
= 166 units
KEY
- =3 units
. =1 units
gap between words = 7 units
gap between letters = 3 units
</pre>
<p style="clear:left;">
One of them added:
"First I worked out the message in
Morse code. I next worked out how
many units there were in each word.
Then I added up the units in the
gaps. Next I added up the totals
from each word and then I added the
units from between the words and I
got 166 units."
</p>
<p >
Well done to you all.
</p>
</mdoxml>
<mdoxml version="1.0">
<p>Working in groups to analyse separate sections of the text may
be a useful whole class activity even if pupils then work in
smaller groups to analyse the data.</p>
<p>By sorting the letters by time values you can get an
approximation to Morse's findings.</p>
<p>A simple frequency count would give a first clue to differences.
Are there any letters whose frequency is significantly different
when the two distributions are matched?</p>
<p>Using the new code and translating is there a "significant"
difference in the times the messages would take?</p>
<p>By weighting letter time values according to their frequency it
should be possible to see whether the average time taken to send a
message using each of the two systems, or between systems generated
by groups of pupils, differ.</p>
</mdoxml>
<mdoxml version="1.0"><br></br>
<p>Copying and pasting the article into a word processor and then
using the "replace" facility will quickly tell you how many of each
letter there is.</p>
<table width="100%" cellspacing="0" cellpadding="0" border="1" id="tableedit1">
<caption>The letters and their time values:</caption>
<tbody>
<tr>
<td>
<div align="center"><strong>A</strong></div>
</td>
<td>
<div align="center"><strong>B</strong></div>
</td>
<td>
<div align="center"><strong>C</strong></div>
</td>
<td>
<div align="center"><strong>D</strong></div>
</td>
<td>
<div align="center"><strong>E</strong></div>
</td>
<td>
<div align="center"><strong>F</strong></div>
</td>
<td>
<div align="center"><strong>G</strong></div>
</td>
<td>
<div align="center"><strong>H</strong></div>
</td>
<td>
<div align="center"><strong>I</strong></div>
</td>
<td>
<div align="center"><strong>J</strong></div>
</td>
<td>
<div align="center"><strong>K</strong></div>
</td>
<td>
<div align="center"><strong>L</strong></div>
</td>
<td>
<div align="center"><strong>M</strong></div>
</td>
</tr>
<tr>
<td>
<div align="center"><strong>.-</strong></div>
</td>
<td>
<div align="center"><strong>-...</strong></div>
</td>
<td>
<div align="center"><strong>-.-.</strong></div>
</td>
<td>
<div align="center"><strong>-..</strong></div>
</td>
<td>
<div align="center"><strong>.</strong></div>
</td>
<td>
<div align="center"><strong>..-.</strong></div>
</td>
<td>
<div align="center"><strong>--.</strong></div>
</td>
<td>
<div align="center"><strong>....</strong></div>
</td>
<td>
<div align="center"><strong>..</strong></div>
</td>
<td>
<div align="center"><strong>.---</strong></div>
</td>
<td>
<div align="center"><strong>-.-</strong></div>
</td>
<td>
<div align="center"><strong>.-..</strong></div>
</td>
<td>
<div align="center"><strong>--</strong></div>
</td>
</tr>
<tr>
<td>
<div align="center">4</div>
</td>
<td>
<div align="center">6</div>
</td>
<td>
<div align="center">8</div>
</td>
<td>
<div align="center">5</div>
</td>
<td>
<div align="center">1</div>
</td>
<td>
<div align="center">6</div>
</td>
<td>
<div align="center">7</div>
</td>
<td>
<div align="center">4</div>
</td>
<td>
<div align="center">2</div>
</td>
<td>
<div align="center">10</div>
</td>
<td>
<div align="center">7</div>
</td>
<td>
<div align="center">6</div>
</td>
<td>
<div align="center">6</div>
</td>
</tr>
<tr>
<td>
<div align="center"><strong>N</strong></div>
</td>
<td>
<div align="center"><strong>O</strong></div>
</td>
<td>
<div align="center"><strong>P</strong></div>
</td>
<td>
<div align="center"><strong>Q</strong></div>
</td>
<td>
<div align="center"><strong>R</strong></div>
</td>
<td>
<div align="center"><strong>S</strong></div>
</td>
<td>
<div align="center"><strong>T</strong></div>
</td>
<td>
<div align="center"><strong>U</strong></div>
</td>
<td>
<div align="center"><strong>V</strong></div>
</td>
<td>
<div align="center"><strong>W</strong></div>
</td>
<td>
<div align="center"><strong>X</strong></div>
</td>
<td>
<div align="center"><strong>Y</strong></div>
</td>
<td>
<div align="center"><strong>Z</strong></div>
</td>
</tr>
<tr>
<td>
<div align="center"><strong>-.</strong></div>
</td>
<td>
<div align="center"><strong>---</strong></div>
</td>
<td>
<div align="center"><strong>.--.</strong></div>
</td>
<td>
<div align="center"><strong>--.-</strong></div>
</td>
<td>
<div align="center"><strong>.-.</strong></div>
</td>
<td>
<div align="center"><strong>...</strong></div>
</td>
<td>
<div align="center"><strong>-</strong></div>
</td>
<td>
<div align="center"><strong>..-</strong></div>
</td>
<td>
<div align="center"><strong>...-</strong></div>
</td>
<td>
<div align="center"><strong>.--</strong></div>
</td>
<td>
<div align="center"><strong>-..-</strong></div>
</td>
<td>
<div align="center"><strong>-.--</strong></div>
</td>
<td>
<div align="center"><strong>--..</strong></div>
</td>
</tr>
<tr>
<td>
<div align="center">4</div>
</td>
<td>
<div align="center">9</div>
</td>
<td>
<div align="center">8</div>
</td>
<td>
<div align="center">10</div>
</td>
<td>
<div align="center">5</div>
</td>
<td>
<div align="center">3</div>
</td>
<td>
<div align="center">3</div>
</td>
<td>
<div align="center">5</div>
</td>
<td>
<div align="center">6</div>
</td>
<td>
<div align="center">7</div>
</td>
<td>
<div align="center">8</div>
</td>
<td>
<div align="center">10</div>
</td>
<td>
<div align="center">8</div>
</td>
</tr>
</tbody>
</table>
<p>Is there much variation in the order of frequency of use of
letters you found in the article and what is implied by the choice
of symbols and time values used by Morse? Is E the most frequent
letter in the article and I the next?</p>
<p>Your coded message might be longer or shorter than when you use
Morse but it is just one possible message. How would you compare
the two systems more generally?</p>
<p></p>
<br></br></mdoxml>
2
3
0
0
1
0
0
Inspector Morse
You may like to read the article on Morse code before attempting
this question. Morse's letter analysis was done over 150 years ago,
so might there be a better allocation of symbols today?
Using, Applying and Reasoning about Mathematics
Selecting and using information
Handling, Processing and Representing Data
Handling data
Handling, Processing and Representing Data
Comparing data
Collecting Data
Data collection
Collecting Data
Types of data