You finish in second place. You would have had to pass the first place racer to have finished in first place.
A completely black dog was strolling down Main street during a
total blackout affecting the entire town. Not a single streetlight had
been on for hours. As the dog crosses the center of the road a Buick
Skylark with 2 broken headlights speeds towards it, but manages to
swerve out of the way just in time. How could the driver see
the dog to swerve in time?
It was during the day
You are a cook in a remote area with no clocks or other way of keeping
time other than a four-minute and a seven-minute hourglass. On the stove
is a pot of boiling water. Jill asks you to cook a nine-minute egg in
exactly 9 minutes, and you know she is a perfectionist and can tell if
you undercook or overcook the egg by even a few seconds. How can you
cook the egg for exactly 9 minutes?
1. Flip both hourglasses over and drop the egg into the water.
2. When the 4-minute timer runs out, flip it over (4 minutes elapsed, 3 remaining on the 7-minute timer).
3. When the 7-minute timer runs out, flip it over. (7 minutes elapsed, 1 remaining in the 4-minute timer)
4. When the 4-minute timer runs out, flip the 7-minute timer over. (8 minutes elapsed. 6 minutes remained in the 7-minute timer, but flipping it over leaves one minute's worth of sand on top. When it runs out exactly nine minutes will have elapsed.
You have just purchased a small company called Company X. Company X has N employees, and everyone is either an engineer or a manager. You know for sure that there are more engineers than managers at the company.
Everyone at Company X knows everyone else's position, and you are able to ask any employee about the position of any other employee. For example, you could approach employee A and ask "Is employee B an engineer or a manager?" You can only direct your question to one employee at a time, and can only ask about one other employee at a time. You're allowed to ask the same employee multiple questions if you want.
Your goal is to find at least one engineer to solve a huge problem that has just hit the company's factory. The problem is so urgent that you only have time to ask N-1 total questions.
The major problem with questioning the employees, however, is that while the engineers will always tell you the truth about other employees' roles, the managers may lie to you if they like. You can assume that the managers will do their best to confuse you.
How can you find at least one engineer by asking at most N-1 questions?
You can find at least one engineer using the following process:
Put all of the employees in a conference room. If there happen to be an even number of employees, pick one at random and send him home for the day so that we start with an odd number of employees. Note that there will still be more engineers than managers after we send this employee home.
Then call them out one at a time in any order. You will be forming them into a line as follows:
Keep doing this until you've called everyone out of the conference room. Notice that at this point, you'll have asked N-1 or less questions (you asked at most one question each time you called an employee out except for the first employee, when you didn't ask a question, so that's at most N-1 questions).
When you're done calling everyone out of the conference room, the person at the front of the line is an engineer. So you've found your engineer!
But the real question: how does this work?
We can prove this works by showing a few things.
First, let's show that if there are any engineers in the line, then they must be in front of any managers.
We'll show this with a proof by contradiction. Assume that there is a manager in front of an engineer somewhere in the line. Then it must have been the case that at some point, that engineer was Employee_Front and that manager was Employee_Next. But then Employee_Front would have said "manager" (since he is an engineer and always tells the truth), and we would have sent them both home. This contradicts their being in the line at all, and thus we know that there can never be a manager in front of an engineer in the line.
So now we know that after the process is done, if there are any engineers in the line, then they will be at the front of the line. That means that all we have to prove now is that there will be at least one engineer in the line at the end of the process, and we'll know that there will be an engineer at the front.
So let's show that there will be at least one engineer in the line. To see why, consider what happens when we ask Employee_Front about Employee_Next, and Employee_Front says "manager". We know for sure that in this case, Employee_Front and Employee_Next are not both engineers, because if this were the case, then Employee_Front would have definitely says "engineer". Put another way, at least one of Employee_Front and Employee_Next is a manager. So by sending them both home, we know we are sending home at least one manager, and thus, we are keeping the balance in the remaining employees that there are more engineers than managers.
Thus, once the process is over, there will be more engineers than managers in the line (this is also sufficient to show that there will be at least one person in the line once the process is over). And so, there must be at least one engineer in the line.
Put altogether, we proved that at the end of the process, there will be at least one engineer in the line and that any engineers in the line must be in front of any managers, and so we know that the person at the front of the line will be an engineer.
My daughter has as many sisters as she has brothers. Each of her
brothers has twice as many sisters as brothers. How many sons and
daughters do I have?
Four daughters and three sons. Each daughter has 3 sisters and 3 brothers, and each brother has 2 brothers and 4 sisters.
To figure it out mathematically, you could use the following two equations where G = the number of girls and B = the number of boys:
G - 1 = B
2(B - 1) = G
Solving for G gives you 4 and plugging that in to G - 1 = B gives you a B of 3.