For many years now Baron Günnstefen has gone to a lake every day to hunt ducks. Every day starting on August 1, 2000, he says to his cook: "Today I shot more ducks than two days ago, but fewer than a week ago." For how many days can the baron say this?

(Assume he is never lying.)

I began the problem by drawing 8 boxes in a row. The last box I labeled "a" as the first day, August 1st 2000, then proceeded to label day "b" the third box from the end ("two days ago"), while "c" was used to label the first box ("a week ago"). I continued adding boxes to the end, continuing the alphabetical labeling and wrote down each inequality (last week > present day > two days before) that resulted. Then as I added days to the end, I wrote down the combined inequalities that resulted, all with respect to "a", the first day. On the 7th day, these were my results:

[c,g,h] > [a] > [b,m,i,d,e,n,k,g]

Seeing that this was the first day that a letter appeared on both sides of the inequality (the letter "g"), I reasoned that one cannot reach the 7th day without creating an inconsitency. Therefore, the baron may say the phrase until and including August 6th, 2000.

Posted by Eric on 2003-11-07 14:26:50