Here is one of the strangest things I've ever heard of. Prof. Paul Kwiat's group at my alma mater, the University of Illinois at Urbana-Champaign, has determined an answer to an algorithm using a quantum computer, without ever running the algorithm (see Nature 439, p. 949-952, 2006). Now, quantum mechanics is bizarre no matter how one looks at it, but this is as counterintuitive as it gets, and is bilt around the phenomenon called quantum entanglement.
The idea is to use an interferometer, which is a device that splits a beam of light down two perpendicular arms. This was the device used in the classic Michelson-Morley experiment that ultimately showed there was no ether, and helped lead to Einstein's relativity theories. If a laser is used as the light source, then one has a system of photons that are all in the same quantum mechanical state. In a quantum mechanical system, computers are being designed where bits, the binary digits that are the 1's and 0's in any computer, may be certain spin states of the particles being used in the system. What's more, quantum mechanics is built around the probabilities of particles being in one state as opposed to another state, and the quantum bits (or qubits) may be placed in superpositions of 1 and 0. In Kwiat's computing system, photons from a laser were entangled.
As Kwiat describes it in Physics Illinois News, "By placing our photon in a quantum superposition of running and not running the search algorithm, we obtained information about the answer even when the photon did not run the search algorithm." This was the first time that a theoretical possibility known as 'counterfactual computing,' or inferring information about an answer even through the computer did not run, has been successfully demonstrated in the lab. The goal is to use this quantum interrogation scheme to reduce noise in larger-scale quantum computers, which is a key technical and engineering problem in this field. Large-scale quantum computing is perhaps the Holy Grail of computing and encryption, and it is experiments like Kwiat's that are leading the way to that type of technology.