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My students are preparing for their SATs and have problems with a certain type of questions, i.e., questions involving a geometrical figure and $n$ (usually $n = 2,3$) straight lines passing through it. The maximum number of regions that the figure can be divided into is asked.

I generally look at the options given and try a hit and trial method.

Question: Is there any other approach to problems like these which I can recommend?

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See "the lazy caterer's sequence":


          Pie7ths
          (Image from Wikipedia.)
The maximum number of cells in an arrangement of $n$ lines is $$\binom{n}{2} + \binom{n}{1} + \binom{n}{0} \;.$$ For $n=3$, this evaulates to $7$. This sequence is A000124 in OEIS.

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  • $\begingroup$ Thanks! Any other similar formulae for squares/rectangles and triangles. $\endgroup$ – Jeremy Cal Sep 8 '15 at 15:53
  • $\begingroup$ The expression quoted, $n(n+1)/2+1$, is the number of cells in an simple arrangement of $n$ infinite lines, simple in the sense each two lines have a distinct crossing point. Then one lays a figure on top of this arrangement to encompass all the cells. $\endgroup$ – Joseph O'Rourke Sep 8 '15 at 20:01
  • $\begingroup$ I meant that the "lazy caterer's sequence" is meant for finding out the total number of divided regions for any n lines passing through a circle, right. Are there any similar formulae for finding the number of regions a triangle/rectangle is divided into if n lines pass through it? $\endgroup$ – Jeremy Cal Sep 9 '15 at 7:51
  • $\begingroup$ @JeremyCal: What I meant to indicate is that the # quoted is for infinite lines, and it doesn't matter what figure you use to surround the cells, as long as you capture all of the cells. Splitting a cell with the boundary of a figure doesn't help, because you have to exclude at least one cell to split. $\endgroup$ – Joseph O'Rourke Sep 9 '15 at 11:47
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Since Joseph O'Rourke, in his response, has linked to the lazy caterer's sequence, let me also mention its three-dimensional analogue, the cake number.

In this case, we are dividing three-dimensional space with planes. The original question asks:

I generally look at the options given and try a hit and trial method. Is there any other approach to problems like these which I can recommend?

The main reason that I post this is to point to a video from the Mathematical Association of America (MAA, 1966) entitled, George Polya in "Teaching Us A Lesson" or Let Us Teaching Guessing: A Demonstration with George Polya (vimeo link).

(If you open the video, the former title includes a misspelling of Polya's first name, and the latter title lists the year as 1965, though IMDB has it as 1966. But let us not get bogged down in the details!)

Since you ask about approaches to this problem, perhaps there is something to be gleaned from Polya's approach. His advocating for "guessing" may seem similar to your approach of "hit and trial," but Polya distinguishes between wild guessing and careful guessing.

Polya is an important figure in mathematics education (and mentioned with some frequency on MESE; see, e.g., here) and so I recommend this video, one, because it is directly related to the original query, but two, because it provides the opportunity to see (what I — and at least David Bressoud) consider to be a favorite student activity!

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    $\begingroup$ The Polya video is indeed a great one. It's quite dated, of course -- you can see how formal Polya is in his speech. But he was remarkable both for being a first-rate mathematician, and at the same time being genuinely concerned with how students could develop their thought processes. Maybe he was even the first to do this. And I'm not talking about ed school blather, either -- Polya really knew what he was talking about, had real knowledge of his material, and conveyed it with wonderful erudition, in a way that drew people in rather than being intimidating. Watch the video! $\endgroup$ – Carl Offner Sep 17 '15 at 2:31
  • $\begingroup$ @CarlOffner I agree with much of what you write about Polya, but take slight issue with the phrase ed school blather. One of Polya's most prolific advisees (co-advised with Ed Begle) is Jeremy Kilpatrick, who has gone on to do first rate work in education - including working at ed schools, as have his advisees; I can trace at least one sequence of advisor-advisees that goes: Polya - Kilpatrick - J.P. Smith - Edward Silver - Yeping Li (and perhaps beyond), all of whom (after Polya) are still active in the world of mathematics education sans blather. $\endgroup$ – Benjamin Dickman Sep 17 '15 at 4:13

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