MHB Can Ricky's Guests Turn the Tables? - June 5

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Ricky's dinner guests mistakenly sat in each other's assigned seats, creating a seating arrangement challenge. The discussion revolves around whether it's possible to rearrange the guests so that at least two are in their correct positions. Participants are encouraged to share their solutions, which will be posted on June 5. There is also a suggestion to address previous challenges before moving on to new ones. The thread will be reopened for further discussion after the solutions are shared.
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For her dinner, Ricky invited many persons to be seated around a round table. She prepared a lovely table plan but all of her guests arrived and took a seat so that everyone was sitting in front of somebody else’s place. Is it possible to turn the tableso at least two guests are sitting in the right place?

- All solutions will be posted June 5.
 
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I think you should go ahead and post the solutions to the previous challenges before posting more. Once you do that, then this thread will be re-opened. :D
 

Thread 'Trigonometry problem of interest'

Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...
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