How to

When solving these problems, learning by doing is much better than learning by reading. I encourage to you read only as far in the solution as you need, then trying to solve the problem. If you get stuck, try reading a little further. And of course, let me know if you find a better solution!

Friday, March 9, 2012

The expert arrow shooter (or not)

Problem:  A prince comes across a barn with an arrow right in the middle of bull's-eye painted on the side.  He asks who shot these arrows and a young boy comes forward and says he did from 100 paces and can repeat the outcome.  He does and the prince walks away disgusted.  What happened?

Solution:  Let's start by thinking about the obvious answer.  Let's say that the young archer hits the middle of bull's eye targets.  It is highly unlikely the prince would be disgusted, so this probably isn't the solution.

Let's draw a picture, as this often helps of the scenario -- shown in pic 1.

You might add that the young archer insulted the prince or wore another prince's colors, etc., but this isn't a terribly likely scenario, and it isn't the answer.

Well, then, why would the prince be upset?  It is hard to imagine a scenario where the archer walks back 100 paces and hits the bull's-eye and the prince gets disgusted.  The prince would naturally be excited.

So, what could have happened?  Let's assume that the boy 'cheated' somehow.  How could he have cheated.

This actually reduces the question to, "How did the boy cheat?" which is an important reduction from the original problem, and not immediately obvious.

Well, let's think of some ways of cheating.  He could have had someone else shoot -- but that is not shown in the question.  He could have lied initially, but that is also not stated in the question.

Maybe he did something else like have a contraption with a string that steers the arrow to the target, but this too seems a bit far fetched.

Well, let's look at pic 1 (and the question) closer to see if anything comes to mind.  Is anything odd about the picture?  One thing that seems a little odd is the choice of a barn.  The questions states that the arrows are "right in the middle of a bull's-eye painted on the side [of the barn.]"

This is an unusually specific scenario.  It is not 'in the center of a target.' as we might expect.  How can we use this information?  Well, since it is the side of a barn, we know that the barn probably has a big size.  And the target is specifically painted on it, which is also unusual.

Well, this starts to hint at the likely cheat.  The target could have been painted at any time, even after the arrow is shot.

So, the answer become apparent -- the boy hits the side of the barn (an easy shot) and then paints the target around the arrow.  Disgusting!

This is a good example of a problem which is pretty challenging because the answer involves thinking non-logically (painting the arrows after the shot), but it shows that the trick to brainteasers is often: 1) challenging assumption (e.g., target is painted first), 2) listening very closely to all information, even that which seems extraneous, and especially information which is a little odd or overly specific, and 3) pictures can help even in a question where it doesn't appear that it will.

/* Please let me know if there are any errors or alternative solutions.  Thank you, Noah */