Where does this definition lead?
The rules for this problem are as follows.
(1) Precisely 27 distinct characters (the 26 capital letters of the English alphabet, plus the space) are allowed in the written definitions.
(2) The language used is only English.
(3) The only numbers to be defined are various positive integers.
The problem:
Let us allow any written string using the 27 characters above, provided it defines a unique positive integer. The same integer may be defined by more than one string.
Examples:
TWO
THE POSITIVE SQUARE ROOT OF FOUR
THE ONLY EVEN PRIME NUMBER
All the above strings define the number 2.
THE SQUARE OF FIFTEEN defines the number 225.
THE CUBE ROOT OF SEVEN HUNDRED TWENTY NINE defines the number 9.
Clearly, if you restrict the length of the strings, you limit the number of possible numbers defined. For example, if the strings have to be 100 or fewer characters, there are possible combinations of characters, so there can be no more than possible integers defined by those strings.
Now consider the following string:
THE SMALLEST POSITIVE INTEGER THAT CANNOT BE DEFINED USING ONE HUNDRED OR FEWER CHARACTERS
Can this integer, as defined, be determined?
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1 solution
The string in question has 88 characters.
If the number it defined could be determined, then contrary to the string's assertion, it would be defined using 100 or fewer characters. Obviously such a number exists, and it has to be less than 2 7 1 0 0 , but it cannot be determined what it is.