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'Weekly Challenge 15: Indexing Problems' printed from http://nrich.maths.org/

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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.