Dear All,
John Bibby has done the hard part. I had earlier sent the following to Indrajit:
"Dear Indrajit,
Your `door' question is a trick one. The state of door n is changed one time for each divisor of n (counting 1 and n). Since all doors start closed, a door will be open in the final state if and only if it has an odd number of divisors. However, it is well-known that the only numbers which have an odd number of divisors are the perfect squares, 1,4,9,16, etc.. See any book on Number Theory, for example the famous one by Hardy and Wright. [Or see John's email!] Hence the answer to your question about the final number of open doors is the integer part of the square root of 1000, namely, 31.
Best wishes for 2002,
David Williams"
Best wishes for 2002,
David Williams
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