The HangMath game, which I showed the NYC Math Circle teachers, is also an opportunity to talk about the importance of "negative information."
In other words, if someone has asked about "2's in the tens column," the game requires the emcee to fill in ALL the 2's in the tens column. An efficient information user should make use of negative information (i.e, the blanks that still remain blank in the ten's column do NOT contain 2's) as well as positive information (the blanks that have been filled in with 2's in the tens column DO
Thanks to a tip from Professor Moorthy, here's a link to a new twist on a classic puzzle about drawing inferences from negative information: The Case of the Pinocchio Politicians.
You can find many delightful related puzzles that use this kind of reasoning in Martin Gardner's The Unexpected Hanging and Other Paradoxes. Raymond Smullyan's awesome mathematical logic books also have many such puzzles.