Concrete Mathematics is book written by Donald E. Knuth. It is a nice book for programmers and computer science students. Actually I seldom read mathematic books, for it needs patience. But I really like the writting style of Knuth that is full of interest and humor. Exercises in the book are really big difficult for me, as I have lost all my mathematics execpt that was learned in high school. However, I want to get my hands in those exercises and see how many I can solve at my best try.
Chapter One: Recurrent Problems
Warmups
1. All horses are the same color; we can prove this by induction on the number of horses in a given set. Here's how: " If there's just one horse then it's the same color as itself, so the basis is trivial. For the induction step, assume that there are n horses numbered 1 to n. By the induction hypothesis, horses 1 through n-1 are the same color, and similarly horses 2 through n are the same color. But the middle horses, 2 through n-1, can't change color when they're in different groups; these are horses, not chameleons. So horses 1 and n must be the same color as well, by transitivity. Thus all n horses are the same color; QED." What, if anything, is wrong with this reasoning?
Actually "the one horse" situation is not sufficent to be all the basis that are needed to support the induction, and "the two horses“ situation should be proved to be a basis too. For the induction step involves [2,n-1], which requires n>=3 to satify [2, n-1] is not empty, otherwise the induction step would not hold. So "the two horses" situation is the real key basis for the induction, but unfortunately it can not be proved.