I recently stumbled upon a mathematical puzzle while trying to work with the limitations of a certain program. Said program effectively has the ability to simulate Bernoulli processes, but only at integer percentages. However, one can also perform bitwise logical operations between the resulting strings, which allows for somewhat finer control of the percentages: up to two extra digits per additional string. (For example, ANDing two independent 25% strings results in a 6.25% string.)
On the other hand, this process isn't perfect, as not all percentages with the right number of digits are covered, even allowing for all $2^{2^n}$ possible truth tables. For instance, for $P(A_k=1)=0.27$, $P(B_k=1)=0.34$, and $P(C_k=1)=0.29$, $P(((A_k\veebar B_k)\lor C_k)=1)=0.592744$, a fairly good (but not ideal) six-digit approximation to the square site percolation coefficient of $0.59274605079210(2)$ (according to this paper cited by Wikipedia).
However, from what I can tell, it is apparently not possible to get to $0.592746$ with just three independent processes; one apparently needs at least four. Has there been any work done on this kind of question, either about the minimum number of processes needed for a particular percentage, or what decimal percentages cannot be produced by the number of processes specified by the two-digits-per-process rule?
If I understand what you're getting at, it can be reformulated in the following purely number-theoretic fashion.
Let $A$, $B$, and $C$ be integers between $-50$ and $50$, and consider the $8$ possible values
$$(50\pm A)(50\pm B)(50\pm C)$$
where the $\pm$ is chosen independently for each factor. Now add up any subset of these $8$ values. Of the $256$ possible sums, the smallest is $0$ (if the subset is empty) and the largest is $1000000$ (if you add up all $8$ values). What you're basically asking is for a characterization of the $6$-digit "target" numbers $N$ that can appear as one of these sums for some choice of $A$, $B$, and $C$. For example,
$$592744=(73\cdot34\cdot71)+(27\cdot66\cdot71)+(73\cdot66\cdot29)+(73\cdot34\cdot29)+(27\cdot66\cdot29)+(27\cdot34\cdot29)$$
where we've taken $A=23$, $B=16$, and $C=21$.
Given this reformulation, the specific question about the target number $N=592746$ can clearly be settled by an exhaustive search through $256\cdot101^3$ possibilities. Whether there is a simpler way to resolve it is not at all clear (to me, at least).
Remarks (added later): As the OP observes in comments, the symmetry between plus and minus allows one to restrict the exhaustive search to $0\le A,B,C\le50$. In fact, it's not necessary to go all the way to $50$, so the search space can be trimmed to size $256\cdot50^3=32{,}000{,}000$. Some target numbers, such as any $N$ that is the product of three two-digit numbers, are obviously achievable; the tricky question is identifying target numbers that are not achievable -- it would be nice to have a proof that $N=592746$ is not achievable (if indeed it isn't) that doesn't amount to saying "an exhaustive search failed to find it."
As far as I know there is no literature on this question. If you generalize it from $3$ integer variables $(A,B,C)$ to $n$, so your target numbers $N$ lies between $0$ and $100^n$, the exhaustive search generalizes to size $2^{2^n}\cdot50^n$, which grows extremely rapidly with $n$, so my offhand guess would be that for some not-very-large $n$, every $2n$-digit target number $N$ is achievable. (Actually, that's already trivially true for $n=1$, but it's probably not true for $n=2$. It's conceivable that "universal" achievability might come and go even for large $n$.)