Mathematics: Easy and Hard. Why?

Nim is Easy, Chess is Hard - But Why??

In nim, a finite number of piles of finitely many tokens is given. Two players alternate in selecting a pile and removing from it any positive number of tokens, possibly the entire pile. The player first unable to move loses and the opponent wins. Nim is easy. Why? Because it has an easy strategy: write the number of tokens in each pile in binary, and “add” them without carry, an addition also ...

Easy and Hard Conformant Planning

Even under polynomial restrictions on plan length, conformant planning remains a very hard computational problem as plan verification itself can take exponential time. This heavy price cannot be avoided in general although in many cases conformant plans are verifiable efficiently by means of simple forms of disjunctive inference. We report an efficient but incomplete planner capable of solving ...

Distinguishing Easy and Hard Instances

Mathematics and Mathematica Made Easy for Students

We present two add-ons for Mathematica for teaching mathematics to undergraduate and high school students. These two applications, M@th Desktop (MD) and M@th Desktop Tools (MDTools), include several palettes and notebooks covering almost every field. The underlying didactic concept is so-called ”blended learning”, in which these tools are meant to be used as a complement to the professor or tea...

Easy Problems are Sometimes Hard

We present a detailed experimental investigation of the easy-hard-easy phase transition for randomly generated instances of satisfiability problems. Problems in the hard part of the phase transition have been extensively used for benchmarking satisfiability algorithms. This study demonstrates that problem classes and regions of the phase transition previously thought to be easy can sometimes be...