Born in Far Rockaway, New York, he is the author of many books on recreational mathematics, recreational logic, etc. Most notably, one is titled What is the name of this book?. It would hardly be considered a spoiler to reveal that the name of the book is, What is the name of this book?.
Many of his logic problems are extensions of classic puzzles. Those involving knights (who always tell the truth) and knaves (who always lie) are based on the story of the two doors & two guards, one who lies and one who doesn't. One door leads to heaven and one to hell, and the puzzle is to find out which door leads to heaven by asking one of the guards a question. One way to do this is to ask "Which door would the other guard say leads to hell?".
In more complex puzzles, he introduces characters who may lie or tell the truth (referred to as "normals"), and furthermore instead of answering "yes" or "no", use words which mean "yes" or "no" interchangeably depending on which type of person they are. In his Transylvania puzzles, half of the inhabitants are insane, and believe only false things, whereas the other half are sane and believe only true things. In addition, humans always tell the truth, and vampires always lie. For example, an insane vampire will believe a false thing (2 + 2 is not 4) but will then lie about it, and say that it is. A sane vampire knows 2 + 2 is 4, but will lie and say it isn't. And mutatis mutandis for humans. Thus everything said by a sane human or an insane vampire is true, while everything said by an insane human or a sane vampire is false.
His book Forever Undecided popularises Gödel's incompleteness theorems by phrasing them in terms of reasoners and their beliefs, rather than formal systems and what can be proved in them. For example, if a native of a knight/knave island says to a sufficiently self-aware reasoner "You will never believe that I am a knight", the reasoner cannot believe either that the native is a knight or that he is a knave without becoming inconsistent (i.e. holding two contradictory beliefs). The equivalent theorem is that for any formal system S, there exists a mathematical statement which can be interpreted as "This statement is not provable in formal system S". If the system S is consistent, neither the statement nor its opposite will be provable in it.