What's Math Got to Do with It?: How Teachers and Parents Can Transform Mathematics Learning and Inspire Success (26 page)

An interesting follow-up question: Is this the least number of steps, or is there a faster way to measure 4 liters? Another follow-up question: Are there any quantities you can’t make with these two jars?

The Rabbit Puzzle

A rabbit falls into a dry well 30 meters deep. Since being at the bottom of a well was not her original plan, she decides to climb out. When she attempts to do so, she finds that after going up 3 meters (and this is the sad part), she slips back 2. Frustrated, she stops where she is for that day and resumes her efforts the following morning—with the same result. How many days does it take her to get out of the well?

Solution

This is a classic example of a “trick” problem. Even if you see the trick, it is easy to make a mistake. One good thing to notice is that the act of climbing and sliding can be greatly simplified. Instead of thinking of it as going up 3 meters and down 2 meters every day, you can just think about it as going up 1 meter. So every day the rabbit goes up 1 meter—which should mean it takes her 30 days to get out, 1 meter each day. But the reason it does not is that on the last day the rabbit actually gets out, she doesn’t slip back down 2 meters. So the rabbit saves itself 2 days, and it only takes 28 days. Can you work out other ways of saying this and seeing why it is actually 28 days?

The Buddhist Monk Puzzle

One morning, exactly at sunrise, a Buddhist monk leaves his temple and begins to climb a tall mountain. The narrow path, no more than a foot or two wide, spiraled around the mountain to a glittering temple at the summit. The monk ascended the path at varying rates of speed, stopping many times along the way to rest and eat the dried fruit he carried with him. He reached the temple shortly before sunset. After several days of fasting he begins his journey back along the same path, starting at sunrise and again walking at variable speeds with many pauses along the way, finally arriving at the lower temple just before sunset. Prove that there is a spot along the path that the monk will occupy on both trips at precisely the same time of day.

Solution

This problem is part of a beautiful class of problems called “fixed-point theorems.” If you like this one, there are many others like it! One of the prettiest ways to see this solution is to imagine the graph of the monk’s journey, with time on the
x
-axis, and position on the
y
-axis. So his journey on the first day might look something like this:

Then, on the same graph we can represent his journey down the mountain:

Note, these two paths may look very different, because he might choose to speed up or slow down at different times. However, since the first path must go from the lower left corner to the upper right corner, and the second path must go from the upper left corner to the lower right corner, they must cross somewhere. And, as you can see, they do cross somewhere. This point marks the time of day and the location in which the monk was in the same place at the same time on both days.

Solution

There are many ways to make some numbers with four 4s, but for some other numbers it is much more difficult. Here’s one set of solutions for the numbers 0 through 20:

0 = 4 – 4 + 4 – 4

1 = 4/4 + 4 – 4

2 = 4/4 + 4/4

3 =
× 4 – 4/4

4 =
+
+ 4 – 4

5 =
+
+ 4/4

6 = 4 +
+ 4 – 4

7 = 4 +
+ 4/4