Algebra Puzzles!

As if the word "algebra" wasn't off-putting enough, my first two problems are normally formed using trains, evoking the stereotypical headache-inducing algebra problem ("One train leaves Tulsa going northeast at 60mph, the other..."). I've replaced the trains in an attempt to make the problems more attractive to the few readers I haven't already alienated with this series of posts.

Man in a Tunnel

A man has traveled 1/4 of the way through a tunnel when he notices a giant killer bunny hopping at a constant rate behind him, some distance outside of the tunnel. Whether he continues forward as fast as he can, or turns around and runs as fast as he can, he will exit the tunnel and miss being devoured by the bunny with zero time to spare at either end. How much faster is the mutant bunny than the man?

Supersonic Bee

Two giant killer bunnies, initially separated by a distance of 25 miles, are both hopping at a rate of 100mph, directly towards one another. A supersonic bee begins on one bunny's nose, and flies back and forth between that bunny's nose and the other's at a rate of 800 miles an hour until the bunnies ultimately collide with each other and the bee. At the point of impact, how far will the bee have traveled?

Heaven and Hell

A giant killer bunny dies and finds itself given three unmarked doors to choose from. One leads directly to heaven, one results in a 1-day stay in hell, and one results in a 2-day stay in hell. If the bunny chooses either of the "hell" doors, it will carry out its hell sentence and return to the same three doors, which at that point will have been randomly shuffled, and the bunny gets to choose again. What is the average time it takes a giant killer bunny to get into heaven?

Man in a Tunnel
SOLUTION

I find this problem interesting because it feels as though you're not given enough information to solve it. How fast can he run? How far away is the bunny?

More data would only be helpful insofar as it would make the problem easier to visualize than by using algebraic variables. So let's say it takes the man 1 minute to reach the end of the tunnel through which he entered. At this rate, it would take him 3 minutes to reach the far end of the tunnel, since it's three times as far. Adding the times for these segments together means it would take a total of 4 minutes for this man to run from one end to the other. How long would it take the bunny to cover the same distance? It would take the bunny 1 minute to reach the near end and 3 minutes to reach the far end, so 2 minutes to cover the entire distance of the tunnel. The bunny is twice as fast as the man.

Supersonic Bee
SOLUTION

Solving this problem hinges on realizing how simple it really is. If it actually required adding together all the successively decreasing distances between the bunnies, solving it would not be fun or interesting, and even most nerdy math types wouldn't bother.

All you need to know is speed = distance / time (or distance = speed x time). The bee's speed is 800mph. How long will he be flying? Traveling toward each other at 100mph, the bunnies can each cover 12.5 miles in 1/8 of an hour. So the bee travels at 800mph for 1/8 of an hour: 100 miles.

Heaven and Hell
SOLUTION

Solving this by algebra is more or less straightforward, but it's still very easy to get lost.

Call the answer - the average time it takes a bunny to get into heaven - X. X is the average of three values: the expected times for each door. Door #1 has an expected time of 0. But how can we calculate the others?

If the bunny walks through door #2, it spends 1 day in hell and starts over again. After that 1 day, it can then expect to wait the average time, X. So the total expected time for door #2 is 1 + X. The same reasoning gets you 2 + X for door #3.

Since X is the average of the three values, you need to add those values and divide by 3. Putting all this together you get the equation:

X = ( [0] + [1+X] + [ 2+X] ) /3

Some simple equation manipulation gets you 3X = 3 + 2X and finally X=3. The average time it takes a giant killer bunny to get into heaven is 3 days.