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    <p>Since there is no deterministic solution for the problem (as stated by @Mark Byers), you can try a probabilistic approach.</p> <p>It's difficult to obtain the original progression, but its rate can be obtained easily by comparing the differences between elements. The difference of original ones will be multiples of rate. </p> <p>Consider you take 2 elements from the list (probability that both of them belongs to the original sequence is <code>1/4</code>), and compute the difference. This difference, with probability of 1/4, will be a multiple of the rate. Decompose it to prime factors and count them (for example, <code>12 = 2^^2 * 3</code> will add 2 to 2's counter and will increment 3's counter).</p> <p>After many such iterations (it looks like a good problem for probabilistic methods, like <a href="http://en.wikipedia.org/wiki/Monte_Carlo_method" rel="nofollow">Monte Carlo</a>), you could analize the counters.</p> <p>If a prime factor belongs to the rate, its counter will be at least <code>num_iteartions/4</code> ( or <code>num_iterations/2</code> if it appears twice).</p> <p>The main problem is that small factors will have large probability on random input (for example, the difference between two random numbers will have 50% probability to be divisible by 2). So you'll have to compensate it: since 3/4 of your differences were random, you'll have to consider that <code>(3/8)*num_iterations</code> of 2's counter must be ignored. Since this also applies to all powers of two, the simpliest way is to pregenerate "white noise mask" by taking the differences only between random numbers.</p> <p>EDIT: let's take this approach further. Consider that you create this "white noise mask" (let's call it <em>spectrum</em>) for random numbers, and consider that it's base-1 spectrum, since their smallest "largest common factor" is 1. By computing it for a differences of the arithmetic sequence, you'll obtain a base-R spectrum, where <code>R</code> is the rate, and it will equivalent to a shifted version of base-1 spectrum. So you have to find the value of R such that </p> <pre><code>your_spectrum ~= spectrum(1)*3/4 + spectrum(R)*1/4 </code></pre> <p>You could also check for largest number <code>R</code> such that at least half of the elements will be equal modulo <code>R</code>.</p>
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