I don't think these are good POTD puzzles, they are straight from a pure maths text. Answers would go like: #1: I think it is Cauchy's famous construction, if they are countable, write them down in order, then construct a new number that is different from the first one in the first decimal place, the second one in the second .... You need a little care to prevent some repeated patterns (9s mainly I htink) you can guarantee to get a number not on the list, QED #2: go around a spiral in Z2 #3: see #2 It is interesting that mathematicians think that there are the same number of rational numbers (that is any number p/q where p and q are integers) as there are integers. There are more real numbers than rational numbers though. You can do funny things with infinities.
Peter "Until the invention of the computer, the machine gun was the device that enabled humans to make the most mistakes in the smallest amount of time."