![]() To start, we represent each element of as a list of lists of elements of, corresponding to the disjoint cycles in the permutation of. So how can we do this? My approach was to use inclusion-exclusion, counting those colorings fixed by none of the symmetries in. In other words, given the (dihedral) group of symmetries acting on colorings of the set of possible key card hole positions, we are counting only regular orbits of this action– i.e., those orbits whose colorings are “fully asymmetric,” having a trivial stabilizer. The idea is that, referring again to the figure above, key cards may only have patterns of holes like the example on the far right, without any rotation or reflection symmetries. ![]() Now how many distinct key cards and corresponding room locks are possible? This way, the sensing hardware in the lock only needs to “look for” a single pattern of holes. Which leads to the problem that motivated this post: to reduce cost, let’s modify the second requirement above– but still retaining the first requirement– so that a manufacturer-provided key card will only open its assigned manufacturer-provided lock when inserted into the slot in a single correct orientation labeled on the key card. (For example, the key card on the left in the figure above “looks the same” to the sensor in any orientation the key card in the middle, however, may present any of four distinct patterns of scanned holes and the key card on the right “looks different” in each of its eight possible rotated or flipped orientations.) Among us keycard software#However, these locks are expensive: the second requirement above means that each lock must contain not only the sensing hardware to scan the pattern of holes in a key card, but also the software to compare that detected pattern against the eight possibly distinct rotations and reflections of the pattern that unlocks the door. When is even, the cycle index (recently worked out in a similar problem here) isĮvaluating at yields a total of 8,590,557,312 distinct key cards– and corresponding hotel room door locks– that the manufacturer can provide. The problem as stated above is a relatively straightforward application of Pólya counting, using the cycle index of the dihedral group of symmetries of the key card acting on (2-colorings of) the grid of possible holes in the card. How many distinct safely-locked rooms can the manufacturer support? A manufacturer-provided key card will open its assigned manufacturer-provided lock when inserted into the slot in any orientation.A manufacturer-provided key card will only open its assigned manufacturer-provided lock and no other and.The lock manufacturer agrees to provide locks and corresponding key cards for each room, with the following requirements: ![]() Examples of hotel key cards each with a 6×6 grid of 36 possible holes. Each key card has a pattern of up to 36 holes aligned with a 6×6 grid of sensors in the lock that may “scan” the key card in any orientation. When you meet with a lock manufacturer, he shows you some examples of his innovative square key card design, with the “feature” that a key card may be safely inserted into the slot in a door lock in any of its eight possible orientations: any of the four edges of the square key card may be inserted first, with either side of the key card facing up. (I have vague childhood memories of family vacations and my parents letting me use just such an exotic gadget to unlock our hotel room door.) ![]() ![]() Each key card will have a unique pattern of holes in it, so that when a card is inserted into the corresponding room’s door lock, a system of LEDs and detectors inside the lock will only recognize that unique pattern of holes as an indication to unlock the door. Suppose that you are the owner of a new hotel chain, and that you want to implement a mechanical key card locking system on all of the hotel room doors. ![]()
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