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witzar

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10. drinking in the bar

I agree with Rainman. The proper version comes from "What is the Name of this Book" by Raymond Smullyan and is called 'The Drinking Principle'. Smullyan also states a dual paradox: There is at least one person such that if anybody drinks, then he does, and more dramatic version of TDP: There is a woman on earth such that if she becomes sterile, the whole human race will die out.

Well done!
14. n-colored straight lines on painted plane

(1b) solved. Well done.
15. Summing a summed sequence

i believe that this is the sum of a different sequence The sequence is the same, but there is an error in reasoning. True. 3 x False. True.
16. Summing a summed sequence

i like this approach. What is your reasoning?

(1a) solved
19. n-colored straight lines on painted plane

(1a) Paint the plane using 3 colors in such way that no 3-colored straight line exists. (1b) Prove that if plane is painted with 4 colors, then a 3-colored straight line exists. (2a) Paint the plane using as many colors as you can in such way that no 4-colored straight line exists. (2b) Let m be the number of colors you used in (2a). Prove that if plane is painted with (m+1) colors, then a 4-colored straight line exists. (3) What about 5-colored straight lines? (4) What about n-colored straight lines? Note: I don't have solutions to all of the problems above. Partial solutions are welcome.

22. 3 x 4 Connect-The-Dots Game

Good idea but does that guarantee a win?

24. 3 x 4 Connect-The-Dots Game

If I understand correctly, the following pattern is not considered "crossing": __Ω_Ω_Ω_Ω_Ω__ The loops are closed, but it is "touching" not "crossing".
25. who pays for the drinks?

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