You can solve this week's puzzle with a pie (traditional or pizza) and slicer, a pen and paper, or a little math. The choice is yours.

### Sunday Puzzle #21: Slicing Pie

*One straight slice through one whole pie divides the pie into two pieces. Simple enough. A second straight cut, crossing the first, will produce four pieces. A third cut, directed through the intersection of the first two, will make six pieces –but a third slice, strategically placed, can actually create as many as seven pieces. What is the maximum number of pieces you can produce with six straight cuts? With N straight cuts?*

We'll be back next week with the solution – and a new puzzle! Got a great brainteaser, original or otherwise, that you'd like to see featured? E-mail me with your recommendations. As always, be sure to include "Sunday Puzzle" in the subject line!

*Top image by Tara Jacoby*

### SOLUTION To Sunday Puzzle #20: The Monk And The Mountain

Last week, I told you a story about a monk and a mountain. On the first day, the monk ascended the mountain, beginning at 7 a.m. and reaching the summit at 7 p.m. On the second day, the monk began his descent at 7 a.m. and reached the base of the mountain at 7 p.m. The question I put to you was this: *Is there any point along the path that the monk occupied at precisely the same time on both days? How do you know?*

The answer is yes, there is one and one point only that the monk will occupy at precisely the same time on both days. A number of commenters arrived at this solution, but I believe the first to do so was bewareofgeek, who supplied both an answer and an explanation:

There must be [a point along the path that the monk occupies at precisely the same time on both days].

Assume, instead, that [the monk] is twins. One does the route from the top, the other from the bottom.

At some point, they must meet. There's your point.

We cannot say where along the trail that point is, but we can say that it exists. What's more, because the monk does not ever back-track, he will only cross paths with his "twin" one time, i.e. there is *only *one point on the path that the two can be said to occupy at precisely the same time. A graph, helpfully supplied by commenter Coronal Shadow, illustrates the solution nicely:

Granted, this is an idealized solution. The monk, according to this graph, ascends and descends the mountain at an identical, steady pace, whereas in the puzzle he is said to vary his speed and occasionally stop altogether. No matter. As Coronal Shadow notes, regardless of how the two paths slow or stop, " they *must* cross at some point." When they do, it is necessary that it be at the same time. Again, we can't say for certain where the point will be – all we can say is that it exists.

This is one of those solutions that is not always immediately obvious, even once it's been explained to you. Several of you had it out in the comments (some more politely than others), disagreeing over the "imagine-the-monks-as-twins" solution. While this is the classic, conceptual answer to this riddle. The mathematical answer involves the intermediate value theorem, and checks out (see the solution to question 6, here).

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