We wish to place 5 stones with distinct weights in increasing order of weight. The stones are indistinguisable (apart from their weights). Nine questions of the form "Is it true that A < B < C?" are allowed (and get a yes/no answer). Is that sufficient?
Answer
no
Solution
There are 120 possible orders for the weights. 20 will fit any given A < B < C. So a "No" answer can exclude at most 20 possible orders. Thus if the answer to the first 5 questions is "No", then there are still 20 possible orders after them. Now if a "No" to the next question leaves k possibilities, then a "Yes" will leave at least 20-k, so one answer must leave at least 10 possibilities. Similarly, one answer to the 7th question must leave at least 5 possibilities, one answer to the 8th question must leave at least 3 possibilities, and one answer to the 9th question must leave at least 2 possibilities.
© John Scholes
jscholes@kalva.demon.co.uk
4 March 2004
Last corrected/updated 4 Mar 04