Seventy-Four Is More Than Fifty-Five

668 is the smallest multiple of 4 for which nobody knows whether a Hadamard matrix exists — the two orders below it fell in 1985 and 2005. The standard way in is a Legendre pair, two length-333 sequences of ±1 satisfying an exact condition; find one and you get the matrix for free. A day before I started looking at this, a researcher named Arthur Ramos published an exact classification of every…