Why can't one compute a paired t test knowing only the mean and SD (or SEM) and n for each group?

Paired t tests account for pairing. Without knowing which "before" value pairs with which "after" value, the paired t test is impossible. The first step in computing a paired t test is to compute the difference for after-before for each participant or experiment. This is impossible without knowing the raw data.

The graphs below show two sets of paired data with two paired t tests with very different P values. The first (before) data are identical in the two graphs. The second (after) data are also identical, but the order is different. The mean, SD and SEM of the "after" data are identical in the two. 

Prism file