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Grade Vs Difficulty:

The material is roughly for three one to oneandhalf hour Math Circle sessions. Each part is selfcontained and could be done independently. The first two parts are suitable for grade 4 and up. Part III could be done with grade 7 and up.
Part I: Voting Paradoxes and Logic. Beginning with a logic puzzle, students are guided through an argument to prove Shapiro's Theorem: If there is no majority of voters who vote unanimously on all propositions then any proposition could be passed.
Part II: Voting Paradoxes and Symmetry deals with the situation when voters need to choose between three or more alternatives. Students learn about popular voting methods, some wellknown paradoxes. We follow D.Saari's geometric representation of the voting profile for the case of three alternatives and explore symmetries which contribute to the paradoxes.
Part III: Arrow' Theorem outlines the proof of the Arrow's theorem in a few simple, engaging exercises. This proof is is a combination of proofs by Sridhar Ramesh and by Terrence Tao.