Given any finite positive real number A, can you define a set S, in the two-dimensional plane, with area A and a partition of S into infinitely many sequentially numbered subsets S1, S2…. such that any two of these subsets are isometric? Show it or prove it impossible. Isometric in this case will specifically mean two subsets related by a one-to-one mapping that only involves translation and/or rotation in the plane. Partition means that the subsets are pairwise disjoint and that their union equals S. Can you generalize your answer to a higher dimensions?
Know the answer? Send your solution to ar@casact.org.
Desire For Dessert Among Logicians
In this puzzle, a group of N logicians are having dinner at the same table at a restaurant where they can all talk to each other. They have finished the main course, but none of them has any idea which of the others want to have dessert. The waiter stops by their table and asks them, “Do you all want to have dessert?” N – 1 of the logicians each answer in succession, “I don’t know.” The puzzle question was: “How might the Nth logician then answer?”
If any one of the logicians had not wanted dessert, that logician would have answered, “No.” If one of them does not want dessert then it is not true that all of them want dessert. So, the Nth logician will answer, “Yes” if he wants a dessert, since all of the other N-1 logicians must have wanted dessert in order to answer, “I don’t know.” However, if the Nth logician does not want a dessert, he will answer, “No.”
Solutions were also submitted by Shyam Bihari Agarwal, Sean Bailey, John Berglund, Jordan Bonner, Roger Bovard, Samuel James Chilson, Bob Conger, Jon Constable, Stephanie Dobbs, Jacob Flisakowski, Kristen Fox, Kacey L. Gilman, Josh Grode, Othon Hamill, Shira Jacobson, Rich Kollmar, Adina Landesman, Jerry Miccolis, Travis Murnan, Dave Oakden, Greg Ostergren, Sean Porreca, Alexander Rosteck, Michael Schwalen, Kwong Koon Shing, David Spiegler, Rick Sutherland, Scott Swanay, Rob Thomas, David Uhland and David Vogt.
