Weekly Challenge 15: Indexing problems

7037 /www/nrich/html/content/id/7037/ 1 2011-02-01T00:00:01 <mdoxml version="1.0"> <ul id="buttonBar"> <li> <a href="http://nrich.maths.org/6331&part=">Warm-up problem</a> </li> <li> <a href="http://nrich.maths.org/6112&part=">Try this next</a> </li> <li> <a href="https://nrich.maths.org/z/node/6799">Discuss</a> </li> <li> <a href="http://nrich.maths.org/2589&part=">Read all about it</a> </li> <li> <a href="http://nrich.maths.org/7056&part=solution">Last week's solution</a> </li> </ul> <div> Consider a very big book, UltraRef, which refers in its index to every book which does not refer to itself in its index. Does UltraRef refer to itself in its index? Be clear as to your explanation. </div> <div class="framework"> Did you know ... ? Paradoxes are a fascinating feature of more advanced considerations of mathematical logic. For a century mathematicians tried to put mathematics on a totally sure logical foundation before realising that this could never be completely achieved.</div> </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> Constructing a really clear argument to this problem can be difficult! Put simply: Suppose that exactly one of the following statements is true: A: Ultra Ref does refer to itself in its index B: Ultra-Ref does not refer to itself in its index Look at each in turn: IF (Ultra-Ref does refer to itself in its index) THEN (Ultra-Ref does not refer to itself in its index) IF (Ultra-Ref does not refer to itself in its index) THEN (Ultra-Ref does refer to itself in its index) In both cases, our supposition is contradicted and Ultra-Ref is therefore a logical impossibility. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> Constructing a really clear argument to this problem can be difficult! Put simply: Suppose that exactly one of the following statements is true: A: Ultra Ref does refer to itself in its index B: Ultra-Ref does not refer to itself in its index) Look at each in turn: IF (Ultra-Ref does refer to itself in its index) THEN (Ultra-Ref does not refer to itself in its index) IF (Ultra-Ref does not refer to itself in its index) THEN (Ultra-Ref does refer to itself in its index) In both cases, our supposition is contradicted and Ultra-Ref is therefore a logical impossibility. </mdoxml> 2 3 0 0 0 0 1 Weekly Challenge 15: Indexing problems A weekly challenge: these are shorter problems aimed at Post-16 students or enthusiastic younger students. Collections Weekly Challenge