Blocker sets, orthogonal arrays and their application to combination locks Original Research Article

Let image denote a set of order image and let image be a subset of image. Then image will be called a blocker (of image) if for any element say image of image, there is some element image of image such that image equals image for at least two image. The smallest size of a blocker set image will be denoted by image and the corresponding blocker set will be called a minimal blocker. Honsberger (who credits Schellenberg for the result) essentially proved that image equals image for all image. Using orthogonal arrays, we obtain precise numbers image (and lower bounds in other cases) for a large number of values of both image and image. The case image that is three coordinate places (and small image) corresponds to the usual combination lock. Supposing that we have a defective combination lock with image coordinate places that would open if any two coordinates are correct, the numbers image obtained here give the smallest number of attempts that will have to be made to ensure that the lock can be opened. It is quite obvious that a trivial upper bound for image is image since allowing the first two coordinates to take all the possible values in image will certainly obtain a blocker set. The results in this paper essentially prove that image is no more than about image in many cases and that the upper bound cannot be improved. The paper also obtains precise values of image whenever suitable orthogonal arrays of strength two (that is, mutually orthogonal Latin squares) exist.