Imagine a three-dimensional toroidal lattice where each point is defined by coordinates (x, y, z), and each coordinate is an integer from 0 to 19. A particle inhabits this lattice and moves every second. In each of the three dimensions, every second it steps either +1 or –1, each with probability 1/2. These movements are entirely independent across dimensions, so at each step, the particle’s position changes by (+/–1, +/–1, +/–1), with all sign combinations equally likely. The lattice wraps around like a torus: moving +1 from 19 takes you to 0, and moving –1 from 0 takes you to 19, all modulo 20.
Suppose the particle begins at the position origin (0, 0, 0).
What is the probability that it never returns to the origin?
What is the average time (expected number of steps) until it first returns to the origin?
Now answer these same two questions supposing the particle starts at some other point distinct from the origin.
How many liars?
At a conference of logicians, five attendees (A, B, C, D and E) are each either a truth teller (always telling the truth) or a liar (always telling lies). They make the following statements:
A says, “Exactly one of us is a liar.”
B says, “Exactly three of us are truth tellers.”
C says, “Exactly three of us are liars.”
D says, “Exactly one of us is a truth teller.”
E says, “All five of us are liars.”
How many liars are there among the five attendees, and who are the liars?
E’s statement implies that E is a liar and there is at least one truth teller. D’s statement is true only if D is the only truth teller. That is consistent with A, B, and C’s statements being false and them all being liars. So only D being a truth teller and the other four being liars is consistent. The statement by A cannot be true as it would make both B and C liars. The statement by B cannot be true as it would make A, C and D liars. The statement by C cannot be true as it would imply A, B and D would be liars, in addition to E being a liar. Therefore, D being the only truth teller is the uniquely consistent solution.
Solutions were also submitted by Daniel Aarhus, Eli Blum, Krishna Chakravartula, Bob Conger, Athanasios Dafulas, Helen Davidson, Stephanie Dobbs,, Y. Ephrathi, Samantha Glover, First Last, Eamonn Long, Edwin Lopez, Juan McNamara, Jonas Meyer, Jerry Miccolis, Ron Miller, Jim Muza, John Noble, Hannah Park, Alan Putney, Misha Rajcoomar, Brad Rosin, Michael Schwalen, Gregory Scruton, Emily Sledge, Clinton Sornberger, David Spiegler, Bob Spitzer, Jeff Subeck, Betty-Jo Walke, Logan Webb and Paul Zotti.
