Summary: Optical, or diffuse tomography, refers to the use of low-energy probes to obtain images of highly scattering media. The inverse problem for one of the earliest and crudest models of optical tomography amounts to reconstructing the one-step transition probability matrix for a Markov chain (with three kinds of states) from boundary measurements. This model is too simple and too general to faithfully reflect the physics of diffuse tomography but could be of interest in other set-ups. It gives a difficult class of nonlinear inverse problems for certain networks with a complex pattern of connections which are motivated by the diffuse tomography picture. A remarkable feature of this simple model is that, at least for systems arising from very coarse tomographic discretizations, it gives an exactly solvable system of nonlinear equations; i.e. a certain number of unknowns are expressible in terms of the data and a number of free parameters. The advantages of this rather uncommon accident are clear: for instance, it is possible to go beyond iterative methods of solution, which are very common for nonlinear problems. This opens the door for a careful study of the ill-conditioned nature of the problem, a subject that is not touched upon here. The lure of an explicit inversion formula, especially in a nonlinear problem is too much of a temptation to pass on, and this paper takes a step in that direction. What we get here is really a careful procedure which might eventually be put together as an explicit inversion formula. Existing recursive algorithms in the two-dimensional case hinge on a very complete study of a \(2\times 2\) system, for which it has been previously shown that using photon count and time-of-flight information all the unknown parameters are given analytically up to an explicitly described eight-dimensional gauge. Here we consider, in detail, the three-dimensional situation and present the solution of a very general \(2\times 2\times 2\) system. There are a total of 288 unknowns and if only photon count is used there are 576 pieces of data. A total of 48 of the unknowns can be prescribed arbitrarily and then the remaining 240 unknowns can be solved uniquely in terms of these free parameters and the data. Most of this is done by writing down explicit formulae as in the two-dimensional case. When (the first moment of) time of flight is also used, we show that all parameters are determined up to a 24-dimensional gauge. We show that, unless physical assumptions are made on the model, this freedom in picking 24 parameters cannot be eliminated. The results given here should be the basis of a recursive algorithm for larger systems. This would require using some good ideas from multiresolution analysis, domain decomposition and/or multigrid methods.