7206
/www/nrich/html/content/id/7206/
1
2011-12-20T10:17:08
<mdoxml version="1.0"><p><em>This problem follows on from <a href="/7482">Secret Transmissions</a>, so if you haven't had a go at it yet, you should try it first.</em></p>
<p> </p>
<p>In the problem <a href="/7482">Secret Transmissions</a>, you were invited to explore a system for detecting and correcting errors in transmissions.</p>
<p> </p>
<p>Imagine you now had to send five 'bits' (0 or 1) of information, instead of just four. Can you devise a system of error detection and correction that will allow your message to be corrected if there is at most one error in transmission?</p>
<p>How many check digits would you need?</p>
<p>What if you had more than five 'bits'? Can your method be generalised?</p>
<p>If you were sending an n-bit message, how many check digits would you need?</p>
<p> </p>
<p><strong>Extension:</strong></p>
<p>Suppose there were two errors in transmission. Can you find an error detection system that would alert you to this, and enable you to correct the message?</p>
<p><strong>Very challenging extension</strong></p>
<p>Suppose each digit of the message might be 'flipped' (a 0 switched to a 1 or vice versa) with probability p=0.1. Explore the likelihood of messages appearing to be transmitted correctly but actually arriving with errors that can't be detected. Can you devise a system where the correct message could be retrieved 99.99% of the time?</p>
<p> </p></mdoxml>
<mdoxml version="1.0"><h3>Why do this problem?</h3>
<p>This problem continues the theme of error detection and correction from the field of Information Theory explored in the problem <a href="/7482">Secret Transmissions</a>.</p>
<p> </p>
<h3>Possible approach</h3>
<p>Begin by giving students some time to try the problem <a href="/7482">Secret Transmissions</a>. Once they have had a go at making sense of and understanding the error detection and correction method, set them the challenge:</p>
<p>"What if I wanted to send more than four digits? Can you come up with a way of extending the error detection and correction method?"</p>
<p>Invite students to work together in small groups to try out their ideas, and once they have come up with a possible solution, encourage them to compose simple binary strings and 'transmit' them with one bit switched for someone else in the group to detect and correct.</p>
<p>Finally, allow some time for discussion of the methods that emerged.</p>
<p> </p>
<h3>Key questions</h3>
<p>What do you notice about the position of the check digits in the message?<br></br>
Where might you put the next check digit in a longer message?<br></br>
How can you determine which message digits 'belong' to each check digit?<br></br>
</p>
<h3>Possible extension</h3>
<p>The extension tasks suggested in the problem should offer a challenge to any student who wants to explore further.<br></br>
</p>
<h3>Possible support</h3>
<p>See the Teachers' Notes to <a href="/7482/note">Secret Transmissions</a> for some suggestions of how to scaffold the original task.</p></mdoxml>
<mdoxml version="1.0"><p>In <a href="/7482">Secret Transmissions</a>, the check digits were in position 1, 2 and 4 of the 7 digits, and the message digits were in position 3, 5, 6 and 7.</p>
<p>Digits 1, 3, 5 and 7 contained an even number of 1s.</p>
<p>Digits 2, 3, 6 and 7 contained an even number of 1s.</p>
<p>Digits 4, 5, 6 and 7 contained an even number of 1s.</p>
<p><br></br>
What do you notice about the position of the check digits in the message?<br></br>
Where might you put the next check digit in a longer message?<br></br>
How can you determine which message digits 'belong' to each check digit?</p></mdoxml>
2
4
0
0
0
0
1
More Secret Transmissions
In 'Secret Transmissions', Agent X could send four-digit codes error free. Can you devise an error-correcting system for codes with more than four digits?
Applications
Codes and cryptography
Numbers and the Number System
Number bases