A simple result that has something of the flavour of the argument type referred to above is that there is a number nless than 4 billion that can be written as the sum of　four different cubes in ten distinct ways. It is known that 1729=13+123=93+103 is the smallest number that is the sum of　two cubes in two different ways. However, we do not necessarily have to identify the number n in order to know that it must exist. Sometimes it is possible to know for certain that there are solutions to a problem, without actually finding any of　those solutions explicitly.