Ezra, again thanks for these qualitatively and quantitatively really amazing insights and mathematical masterpieces. This whole week I’ve been busy and barely able to keep up, so I’m falling way behind in some feedback I’d like to provide, and now before Shabbos there is very little time left. Just very briefly then focusing on one matter, in reply to what you wrote: “Black magic is prohibited in the Torah, so we know it exists. Magic and illusion can be very entertaining and often employs deft sleight of hand. The magic we are about to show you is neither black, nor illusion or sleight of hand, but it is real magic.” Yes the Torah prohibits magic, so yes “magic” exists, but what is it? As I noted in a previous comment, Rambam (Maimonides) held very strongly that there is no “real magic” of any form, only illusions, performed with sleight of hand and trickery of various sorts. Other sages, including Ramban (Nachmonides) held that “real magic” does exist, only the Torah strictly prohibits such practices. The Artscroll Chumash explains (Exodus 7:9): “Pharaoh’s magicians were able to duplicate the wonders performed by Moses, thus supporting the king’s insistence that Moses himself was nothing more than a magician, a profession that was well represented in Egypt. This raises the question of whether Pharaoh’s sorcerers had any real power. Alone among the classic commentators, Rambam (Hil. Avodas Kochavim 11:16), Moreh Nevuchim 3:37) maintains that all magic, even that discussed in Scripture, is sleight of hand and that only foolish and ignorant people believe in it. “The other classic commentators, however, based on copious proofs from the Talmud, dispute his contention. According to then, the sorcery mentioned in the Toah was real, and its practitioners knew how to alter nature and foretell the future by utilizing powers built into Creation. Very briefly, Ramban (Deuteronomy 18:9), Derech Hashem, and others explain that G-d created the universe so that earthly events are regulated by angels and other heavenly forces. G-d also provided that by the use of various sacred or profane incantations, people could harness these heavenly forces and th3ereby override the laws of nature. This is how the Egyptian magicians and others whose feats are related in Scripture and the Talmud were able to perform miracles. It was because of this ability that false prophets were able to mislead people into believing in the power of idols.” For a long time I have been of the opinion that if our sages like the Ramban were living in today’s modern period, when science and technology have made such major advances, and dispelled the darkness of ignorance and superstition, then surely they would recant their previous ideas and come to agree with the Rambam. Recently I found that this theme is expressed clearly in an article entitled “U-Madua Lo Yeresem,” by R’ Asher Benzion Buchman, published in the 2nd volume of “Hakirah, The Flatbush Journal of Jewish Law and Thought.” This crucially important article may be read online at Hakirah’s website “hakirah.org”- http://www.hakirah.org/Vol%202%20Buchman.pdf R’ Buchman explains that Ramban, too, was a rationalist committed to the scientific method, and were Ramban alive today his eyes would tell him that Rambam’s views were correct. The same article (on page 20 of 32 pages) refers to Rambam’s opinion regarding various Holy Names: “Rambam rejects the concept of magical powers in the שמות [Names] as well. The concept of שמות with 42 and 72 letters represents various understandings of השם ייחוד. There are no magical powers in names but there is great spiritual growth in understanding what these various names tell us about the Creator (Moreh Nevuchim 1:62).” The trend in Judaism is that the practice of so-called “white magic” or “practical Kabbalah” is also being strictly discouraged, and this can be elaborated on in another comment BE”H next week. Good Shabbos!