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    <p>The real problem, of course, is the fact that the numbers in the middle of the sum (4 in this case) occur in many combinations (0+4, 1+3, etc.) whereas 0 and 8 have exactly one way to be produced.</p> <p>I don't know how to solve this problem, but I'm going to try to reduce it a bit for you. Some points to consider:</p> <ul> <li>The 0-7 range has 8 possible values, so ultimately the total number of possible situations that you should aim for has to be a multiple of 8. That way you can have an integral number of distributions per value in that codomain.</li> <li>When you take the sum of two density functions, the number of possible situations (not necessarily distinct when you evaluate the sum, just in terms of different permutations of inputs) is equal to the product of the size of each of the input sets.</li> <li>Thus, given two {0,1,2,3,4} sets summed together, you have 5*5=25 possibilities.</li> <li>It will not be possible to get a multiple of eight (see first point) from powers of 5 (see second point, but extrapolate it to any number of sets > 1), so you will need to have a surplus of possible situations in your function and ignore some of them if they occur.</li> <li>The simplest way to do that, as far as I can see at this point, is to use the sum of two {0,1,2,3,4} sets (25 possibilities) and ignore 1 (to leave 24, a multiple of 8).</li> <li>Thus the challenge now has been reduced to this: Find a way to distribute the remaining 24 possibilities among the 8 output values. For this, you'll probably NOT want to use the sum, but rather just the input values.</li> </ul> <p>One way to do that is, imagine a number in base 5 constructed from your input. Ignore 44 (that's your 25th, superfluous value; if you get it, synthesize a new set of inputs) and take the others, modulo 8, and you'll get your 0-7 across 24 different input combinations (3 each), which is an equal distribution.</p>
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