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In six-dimensional F-theory/heterotic string theory, half-hypermultiplets arise only when they correspond to particular quaternionic Kähler symmetric spaces, which are mostly associated with the Freudenthal–Tits magic square. Motivated by the intriguing singularity structure previously found in such F-theory models with a gauge group SU (6), SO (12), or E 7, we investigate, as the final magical example, an F-theory on an elliptic fibration over a Hirzebruch surface of the non-split I 6 type, in which the unbroken gauge symmetry is supposed to be Sp (3). We find significant qualitative differences between the previous F-theory models associated with the magic square and the present case. We argue that the relevant half-hypermultiplets arise at the E 6 points, where half-hypermultiplets 20 of SU (6) would have appeared in the split model. We also consider the problem on the non-local matter generation near the D 6 point. After stating what the problem is, we explain why this is so by using the recent result that a split/non-split transition can be regarded as a conifold transition. [ABSTRACT FROM AUTHOR]