OT Last week's CarTalk puzzler

[Charles Hartman]

I absolutely promise not to post on this topic any more. But having  
said something stupid I have to take it back, which requires posting,  
& trying to avoid saying something stupid . . . I think this is  
called karma.

What I neatly demonstrated had nothing much to do with the problem,  
because I was counting prime factors.

For a proof that the Ray Theorem is true, I refer anyone interested  
to a friend & colleague who's a mathematician and whom I should have  
asked before I posted anything, and who wrote as shown below.

Charles, shutting up


================= snip =============

Anyway, hereÕs an argument. Consider the positive integer n > 1.  
There are two cases to
consider.

1. Suppose Ãn is not an integer. (Then, of course, Ãn is not even  
rational but that is
another story.) Now n has at least one factor that is smaller than  
Ãn, namely, 1. And,
clearly, for each distinct factor of n smaller than Ãn, there is a  
distinct factor of n larger
than Ãn such that their product is n. Thus, if n is not a perfect  
square, n must have an
even number of distinct factors.

2. Suppose Ãn is an integer, i.e., n is a perfect square. Then  
arguing as above, for each
distinct factor of n smaller than Ãn, there is a distinct factor of n  
larger than Ãn such
that their product is n. Therefore, other than the factor Ãn itself,  
n has an even number
of distinct factors. But, Ãn, in this case, is also a factor on n.  
Therefore, if n is a perfect
square, n must have an odd number of distinct factors.

This proves that a positive integer n has an odd number of distinct  
factors if and only if
n is odd.

Now, since 1412 = 19881 and 1422 = 20164, there must be exactly 141  
lights that are left