**Update:** Here is a combined version of all three parts: Three puzzles on mathematics computations and games. Thanks for the remarks and corrections. More corrections and comments welcome.

Dear all, here is the draft of the second third of my paper for ICM 2018. Corrections and comments are very welcome! This part is around voting games and election rules, Boolean functions and their Fourier representation, noise stability and sensitivity especially of percolation, and a little circuit complexity and PCP. Corrections and comments are most welcome.

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Just a tangential association with respect to 2.2.2. I have been exploring questions related to pivotal variables (“Differences that Make a Difference” or “Difference In ⟹ Difference Out”) by means of logical analogues to partial and total differentials.

For example, letting the partial differential operator sends a function with fiber to a function whose fiber consists of all the places where a change in the value of makes a change in the value of

Ref: Differential Logic

Dear Gil,

I enjoyed looking at this essay. It might be nice to add references “locally” for each of the three essays.

Regards,

Joe

Dear Joe,

This is a good point and I am really thinking about the best reference policy. Right now each part refers to 60 – 70 references and the space limitation makes this impossible. But the way the references appears in the text (names + year of publication) makes it quite easy to search so I plan to include only 5-10 references mainly for crucial recent papers for each part. Practically I will get to the bibliography when I will combine the parts next week. But later I may add the relevant references to each part in the blog posts.

Dear Gil,

Did you see this vote counting story!

https://www.washingtonpost.com/local/virginia-politics/democrat-wins-va-house-seat-in-recount-by-single-vote-creating-50-50-tie-in-legislature/2017/12/19/3ff227ae-e43e-11e7-ab50-621fe0588340_story.html

Amazing.

Regards,

Joe

Flip … Flop …

https://www.washingtonpost.com/local/virginia-politics/court-tosses-out-one-vote-victory-in-recount-that-had-briefly-ended-a-republican-majority-in-virginia/2017/12/20/ed979a70-e5b9-11e7-a65d-1ac0fd7f097e_story.html?tid=pm_local_pop&utm_term=.3caef24934c9

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