A famous demonstration of differences in tasting ability involves phenylthiocarbamide (PTC). For most people PTC tastes bitter, but there are many people for whom it has no taste at all.
This difference is genetically inherited for the most part, although there are cases of identical twins in which one can taste it and the other cannot.
There are also people who cannot taste vinegar. Some taste mannose (a type of sugar) as sweet, others as bitter or salty.
It is interesting to note that some variances in the perception of taste are very much variances in perception. For example, cilantro has aldehydes in its leaves, which are also present in soap. Some people hate cilantro because it tastes like soap, others love it despite it having a soapy flavor, and some love it and have no association with soap. It is unclear how much variation there is in the actual taste, but it is clear that perception (specifically, the perception that "soap is not food!") makes a big difference in cilantro enjoyment. Parmesan cheese also shares this sort of perceptual confusion, as it contains butyric acid, also found in vomit and body odor.
Icicle: Those strips of paper most likely had phenylthiocarbamide or Sodium Benzoate on them.