I am trying to solve a problem using the general Pigeonhole Principle. The problem statement is as follows:
A coin is flipped three times and the outcomes recorded. So, HTT might be recorded for 1 round of 3 flips. How many times must we flip a coin three times to be guaranteed that there are two identical outcomes?
I am struggling to understand the correct way to solve this problem. When I use the formula: $$ N = k(r - 1) + 1 $$ Where k = 2 for the number of identical outcomes we want, r = 3 for the number of coin flips, I get 5. The answer in my book is written as 9. When I multiply the $$ k(r - 1) $$ part of the equation by 2 and then add 1, I am able to get this answer, but I feel like I am just doing this without any real understanding of what's going on. I thought maybe it's related to the outcome of each individual coin flip (H or T), but I just don't have a firm grasp on this yet. Any hints or guidance with this is greatly appreciated.