Another Interview Question

This is an interview question for software engineer position at USAA:

"A train leaves San Antonio for Houston at 60 mph. Another train leaves Houston for San Antonio at 80 mph.  Houston and San Antonio are 300 miles apart.  If a bird leaves San Antonio at 100 mph, and turns around and flies back once it reaches the Houston train, and continues to fly between the two, how far will it have flown when they collide?"

First, we need to draw a line to analyze this:

|<------------------------- 300 ------------------------------|
SA                                                            H
 --------60------->             <--------80 ------------------|                            

When these two trains collide, the distance between them is d = 0, or 60*t = 300 - 80*t, solving this equation we get t  = 300/140  hours = 15/7 = 2.143 hours. Meanwhile, for the bird the equation is: 100*t = 300 - 80*t, or t = 300/180 = 1.667 hours.  This is the time when the bird reaches Houston train and turns around.  How far it has traveled from SA? 100*1.667 = 166.7 miles.  For this duration, SA train has traveled 60*1.667 = 100.02 miles toward Houston.  The distance between the bird (now flying back toward SA) and SA train is = 166.7 - 100.02 = 66.68 miles.  How many minutes before the bird hits the SA train? We use the similar equation:

|<------------------------- 66.68 ------------------------------|
SA                                                            H
 --------60------->             <--------100 -------------------|

Or, 60*T = 66.68 - 100*T T = 66.68/160 = 0.41675 hours. Total travel time for the bird: t + T = 1.667 + 0.41675 = 2.08375 hours (and it occurs before these two trains collide each other).  Total travel distance for the bird: 100 mph * 2.08375 hours = 208.375 miles

Interesting Algorithm question in Facebook Interview

The question is:

"Given a number in range of 1 to n, what is minimum number of guesses needed to find a specific number if you're just given an answer either "higher" or "lower" for each guess you make"

It sounds tricky, but actually the answer is very simple.

Here's the illustration:

the number to be guessed
                                             |  
                                             V
|----------------------------------------------------------------------|
min                              ^                                    max
                                 |
                                 |
                             your guess

Using common method of binary searching, our guess starts from: min + (max-min)/2 or a number in the middle of the range (divide-and-conquer). If our guess is lower than the number, we're given "LOWER" and vice versa.

As the number to be guessed is random, it is possible the number falls right in the middle of the range and matches our first guess.  So the answer of this question is (the key of this question is "minimum number of guesses") is 1.