This was a rather intricate puzzle from Presh Talwalkar. I found his solution a bit hard to follow, so I tried for a clearer presentation.
“A servant has a method to steal wine. He removes 3 cups from a barrel of wine and replaces it with 3 cups of water. The next day he wants more wine, so he does the same thing: he removes 3 cups from the same barrel (now with diluted wine) and replaces it with 3 cups of water. The following day he repeats this one more time, so he has drawn 3 times from the same barrel and has poured back 9 cups of water. At this point the barrel is 50% wine and 50% water. How many cups of wine were originally in the barrel? ”
See the Diluted Wine Puzzle for solutions.
This is another interesting problem from 
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These are three “Coffin” Problems posed by Nakul Dawra on his Youtube site 
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This subject admittedly has only a tenuous relationship to mathematics (via arithmetic), but perhaps it can join more mathematically challenging political topics like voting and gerrymandering. In any case, I was stimulated to consider the idea of reapportioning the US Senate by the % US population of each state by an 8 December 2018 article in the Atlantic by former Congressman John Dingell, who advocated abolishing the Senate. I thought this a bit too Draconian and considered the percent population idea as a better compromise. It turned out I was not alone in having this (obvious) thought: I just came across a more extensive 2 January 2019 Atlantic article by Eric Orts that concurs with my idea about reapportionment of the Senate, discusses the legal ramifications in more detail, and echoes the benefits I mentioned as well as others. See 
This is another problem from the defunct Wall Street Journal Varsity Math Week column.
In light of subsequent events it may be that being a politician requires its own set of skills, but this praise of his profession of engineering before he became president casts the unfortunate Herbert Hoover in a different light. My father brought this surprising excerpt from Hoover’s autobiography to my attention years ago. I have highlighted the part that is especially insightful. Unfortunately, the balance of the chapter praising an engineer’s involvement in government does not fare as well, given the author, though a subsequent engineer US president, whatever his shortcomings, was never faulted for his honesty and moral rectitude. See the 
This 1975 New York Times article by Cyril Stanley Smith left an indelible impression over the years to the point that I wanted to capture it in digital format. I found a 1996 copy online, formatted it more closely to the original article, and found more recent images of the illustrations used in the original article. It is a powerful argument for the benefits of basic research vs. directed research. The pursuit of pure mathematics often is accused of being a product of imagination run rampant with no practical purpose. It is argued that Government expenditures of public moneys for research should be applied more to directed research that has specific practical goals with explicit criteria for success. It has always been difficult to argue otherwise. Smith’s article, however, goes a long way toward a rebuttal, as well as showing the benefits of play and artistic creativity for its own sake. See the 
