We discuss a generalization of logic puzzles in which truth-tellers and liars are allowed to deviate from their pattern in case of one particular question: "Are you guilty?"

We consider the space [0, n] 3 , imagined as a three dimensional, axis-aligned grid world partitioned into n 3 1 × 1 × 1 unit cubes. Each cube is either considered to be empty, in which case a line of sight can pass through it, or obstructing, in which case no line of sight can pass through it. From a given position, some of these obstructing cubes block one's view of other obstructing cubes, leading to the following extremal problem: What is the largest number of obstructing cubes that can be simultaneously visible from the surface of an observer cube, over all possible choices of which cubes of [0, n] 3 are obstructing? We construct an example of a configuration in which Ω n 8 3 obstructing cubes are visible, and generalize this to an example with Ω n d− 1 d visible obstructing hypercubes for dimension d > 3. Using Fourier analytic techniques, we prove an O n d− 1 d log n upper bound in a reduced visibility setting.

We introduce a new type of coin: the alternator. The alternator can pretend to be either a real or a fake coin (which is lighter than a real one). Each time it is put on a balance scale it switches between pretending to be either a real coin or a fake one.In this paper, we solve the following problem: You are given N coins that look identical, but one of them is the alternator. All real coins weigh the same. You have a balance scale which you can use to find the alternator. What is the smallest number of weighings that guarantees that you will find the alternator?

We discuss a generalization of logic puzzles in which truth-tellers and liars are allowed to deviate from their pattern in case of one particular question: “Are you guilty?”