Est. 1998 — 27 years of woodworking knowledge
Friday Puzzler -- Find the number
Alex Y
Can you find the smallest positive integer that cannot be described with fewer than 15 English words, or prove the non-existence of such a number?
Examples:
"The number of hydrogen atoms in the universe" is an eight-word description of a very large number, while
"One hundred and thirty-seven million, seven hundred and forty-three thousand, two hundred and eighty-four" is a 17-word description of a much smaller number.
Re: Friday Puzzler -- Find the number
Bill Earl
"or prove the non-existence of such a number? "
But didn't you just do that?
Thanks for the puzzle!
On the other hand...
Alex Y
Can you find the "other hand"? (BTW, I changed the original from 100 words to 15 words, to tempt those who wanted to find a specific number, before I saw your post.)
Answer -- The Berry Number *LINK*
Alex Y
Bill spotted the fact that such a number can't exist, because "The smallest positive integer that cannot be described in fewer than fifteen English words." is a 14-word description of such a number.
On the other hand, there are a finite number of English words, let's say 100,000 = 10^5 for sake of argument. Then there are 10^70 possible ways of combining those words in a phrase shorter than 15 words--a very large number indeed, but still finite. Of course, most of those phrases will not describe a number, but the point is that there is a finite number of positive integers that can be so described. So there must be a smallest number that cannot be so described.
So we appear to have proven both the existence and non-existence of such a number, referred to as the Berry Number.
This paradox is discussed by Bertrand Russell, who credits a junior librarian at Oxford, named G. G. Berry.
Wikipedia article on Berry Paradox