The Math Factor Podcast

2 Comments »

1. czarandy said,

   April 20, 2009 at 7:51 pm

   Show Spoiler ▼

   It seems like you need n+2, if n is the degree of the polynomial.

2. Stephen Morris said,

   April 21, 2009 at 1:41 pm

   Thanks czarandy.  I’ve spent a couple of hours thinking about your answer.   It was fun so thanks!  Also it is good material for a follow up post.

   It isn’t the right answer to the question as posed.  However it looks like (but isn’t necesarrily) the right answer to a very similar question, a third variant.

   I would be very interested in seeing your working if you have time to write it up.

   This is what I think you missed in the problem statement:

   Show Spoiler ▼

   Although the co-efficients are constrained to be fractions there are no constraints on the test numbers.

   The actual answer relies on a neat bit of theory I intend to explain in a follow up post.  It relates to: Show Spoiler ▼

   algebraic and trancendental numbers

   This is my analysis of the third variant, I will write it up properly in a follow up post.

   Show Spoiler ▼

   If we constrain the test numbers to be integers or fractions then there is no solution.  If we knew the degree of the polynomial, n, then we could do it in n+1 test numbers. Otherwise it takes an infinite number of guesses; there is a proof by contradiction as follows:  Suppose the polynomial is p(x) and we can identify it with the test numbers t0, t1, t2, … , tm.  Now put q(x) = (x-t0)(x-t1)…(x-tm).  Clearly q(ti) will be zero for each of the test numbers.  Put r(x) = q(x) + p(x).  r(ti) = p(ti) for each of the test numbers so we cannot distinguish between r(x) and p(x).  Therefore we have a proof by contradiction.

   I’ve just edited this because I’ve realised I’ve missed a case: Show Spoiler ▼

   what if the test numbers can only be algebraic numbers (E.g. 3, 47/236, sqrt(59) but not pi or e)? My proof by contradiction doesn’t work. Is the answer n+2? I need more time to think about it! Would love to see your working!

   Thanks czarandy, that was fun!

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