Here are my six solutions. flagamuffin found one of them - kudos! 15 = (1 x 5)!! 24 = (2+√4)! 24 = (√ ( 2 x 4!! ) )! (I'm pretty proud of that one!) 36 = 3 x 6! 46 = -√4 + 6!! 48 = ( (4! / 8)! ) !! (another tricky one!) Four of my solutions make use of double factorial. 4!! = 8 and 6!! = 48. I've worked on this on and off for a few days while I've been traveling -- a great airplane puzzle. And I've worked out a lot of strategies and smart ways to think about solutions. Key is not starting with a number and trying to make a solution, but to look at what operations are available and which ones will get near to the result you want. By the way, if you don't like the double factorial (I hear sometimes from people that think it's cheating), it is a real function and I solved a problem some years ago dealing with probability that made use of this function, although I didn't know the function at the time.