Two orthonormal bases A and B are said to be mutually unbiased when, for every pair of vectors |αi〉 ∈ A, |βj〉 ∈ B, |〈αi|βj〉|2 = 1/d, where d is the dimension of the system. We have tried different methods to construct MUB’s in six dimensions. These include numerical searches for orthonormal bases, extension of Wootters and Fields’ method using the concept of field, and trying to partition the tensor product of qubits and qutrits operators to generate MUB’s as their eigenbases as suggested by Lawrence et al. Our studies have yielded no more than three MUB’s, and we are led to the conclusion that either only these three bases exist or the method for constructing such bases is radically different from those used so far. We shall present the difficulties encountered in each of these methods while searching for MUB’s in composite dimensions.