Conjectures are the warp upon which we weave mathematics. They are not to be confused with guesses. André Weil is reported to have proclaimed in exasperation at a guess that its author had dared elevate to the rank of conjecture, “That is not a conjecture; that is just talking.”

A conjecture may be false, but this must come as a surprise, as an illumination revealing unimagined subtleties. A good conjecture comes with the certainty that it must be true: because it simplifies what had seemed complicated and brings order to what had appeared chaotic, because it carries implications which themselves seem right and may even be verifiable, because it bears the stamp of elegance that the trained observer has learned to recognize as the hallmark of truth.

This story is woven on the strands of fourteen conjectures that I have chosen from the many that arose in the course of investigations of alternating sign matrices.