I would solve it as follows. Think of each light as initially off. Then the cord is pulled once for each divisor of its number. But a number of form p_1^{r_1}p_2^{r_2}...p_j^{r_j} (where the p_i are distinct primes) has (r_1-1)...(r_j-1) factors; this is easily proved. Now it is clear that the number of factors is odd exactly when the number is a perfect square (i.e. all the r_i are even.) But pulling the switch an odd number of times when it is off turns it on. (The problem states they are all initially on, but think of them as being off and turned on for the divisor 1.)
The trouble with enumerating the result - what if it had been 2,000,000,000,000,000,000,000 instead of 20,000?
chas
Update Sorry, of course the number of factors is (r_1+1)...(r_j+1), not (r_1-1)...(r_j-1). I guess I was brain dead when I first posted. The conclusion is the same, though; this is odd exactly when all the r_i are even.

In reply to Re: CarTalk Puzzler by chas
in thread CarTalk Puzzler by freddo411

Title:
Use:  <p> text here (a paragraph) </p>
and:  <code> code here </code>
to format your post, it's "PerlMonks-approved HTML":


  • Posts are HTML formatted. Put <p> </p> tags around your paragraphs. Put <code> </code> tags around your code and data!
  • Titles consisting of a single word are discouraged, and in most cases are disallowed outright.
  • Read Where should I post X? if you're not absolutely sure you're posting in the right place.
  • Please read these before you post! —
  • Posts may use any of the Perl Monks Approved HTML tags:
    a, abbr, b, big, blockquote, br, caption, center, col, colgroup, dd, del, details, div, dl, dt, em, font, h1, h2, h3, h4, h5, h6, hr, i, ins, li, ol, p, pre, readmore, small, span, spoiler, strike, strong, sub, summary, sup, table, tbody, td, tfoot, th, thead, tr, tt, u, ul, wbr
  • You may need to use entities for some characters, as follows. (Exception: Within code tags, you can put the characters literally.)
            For:     Use:
    & &amp;
    < &lt;
    > &gt;
    [ &#91;
    ] &#93;
  • Link using PerlMonks shortcuts! What shortcuts can I use for linking?
  • See Writeup Formatting Tips and other pages linked from there for more info.