Here's what WebMD has to say about it: www.webmd.com/diet/features/the-grapefruit
It's a fad crash diet that your grandmother probably tried. It has absolutely nothing to do with the Mayo Clinics; that name is a flat-out lie/scam that got attached to this diet in chain letters in the 1970s. If you just want to lose a few pounds to fit into a dress, it's not going to kill you (but it's not necessarily easier than any other crash diet.) Grapefruit is useful on a crash diet because it's slow and messy to eat and it's filling for the amount of calories it has. But it's a boring, unpleasant way to live for 12 days just to lose a few pounds that will come back.
If you need to lose a significant amount of weight and improve your health on a permanent basis, you might try the REAL Mayo Clinic Diet. www.mayoclinic.com/health/mayo-clinic-diet
Of course, the real Mayo Clinic Diet is really just a balanced health diet. It's no different from what SparkPeople recommends, the difference being that you don't have to buy a book to get the information from SparkPeople. If you really want a specific diet mapped out on paper, you could almost certainly get the Mayo Clinic Diet book from your public library.