All of these calculations generalize to actual state


lottery games, but counting up the number of ways to match, say, 4-out-of-6 on a 46-ball lottery is more complicated. We showed and briefly explained the combinatorial formulas for doing that, and then calculated the expected value for the Massachusetts Cash Winfall game (in the non-special weeks). I pointed out that my usual winning was $9 on a $7


investment, slightly below the expected value. But that small difference is mitigated by the fact that I never lost at all.


Follow-Up Activity We presented the following rules (the axioms for a pro- jective plane), and asked teachers if they could draw a picture with a finite number of points that satisfied all of the rules:


1. Each pair of distinct points has a unique “line” between them.


2. Each pair of distinct “lines” intersects in a unique point.


3. Each “line” contains at least three points. 4.Tere exist at least three non-collinear points.


Figure 2. Fano plane I pointed out that if we number the points from 1–7,


then each “line” contains three points, as follows: 1-2-3, 1-4-7, 1-5-6, 2-5-7, 2-4-6, 3-4-5, and 4-6-7. By Rule #1, every possible pair of numbers appears


on a line somewhere, and by Rule #2 that pair appears on only one line. So if we use these seven lines as lottery tickets, then


Figure 1. Lines We emphasized that “lines” may not look like


familiar straight lines from Euclidean geometry. Teachers worked on this in groups for a while.


Te facilitators spent a lot of time with each group, clarifying the axioms and the task, and explaining that you can’t satisfy all of the rules with just three points (or fewer): By Rule #3, they would all be on the same


⊆ 10


no matter what the draw is, all three possible pairs are guaranteed to appear in the tickets. Either they are all on the same ticket (jackpot!), or they are on three different tickets (we match 2-out-of-3 three times). We ended the session by regaling the teachers with


an expanded version of the Massachusetts lottery story and answering any lingering questions. Resources and handouts related to this article can be found at www.mathteacherscircle.org/newsletter. ⊆


MTCircular · Summer/Autumn 2016 · American Institute of Mathematics


line, but Rule #4 says they have to be non-collinear. We were careful to draw curvy “lines” at every opportunity. Later, participants presented arguments that we


can’t have just 4 points or 5 points. One group drew a picture with 7 points that they claimed satisfied all of the rules, and we had everyone check it carefully.


Tying It All Together At this point, I introduced the Fano plane, which is the smallest possible example of a projective plane and has 7 points and 7 lines.


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