With bizarre interests such as memory techniques and mental math feats, it's not often I run across a kindred soul, even on the internet!
That's why I was thrilled to recently discover Colin Beveridge's Flying Colours Maths site! He started it in 2008, and regular Grey Matters readers will find plenty of interesting items in his blog.
Most of the mathematical feats on this site are told via the stories of the Mathematical Ninja. Looking through these stories, I realize that not only does Colin just tell the stories differently, but he also has a different enough take on mental math that there are feats and principles I've never covered on Grey Matters.
One of the simpler examples of this is Converting Awkward Fractions to Decimals. The principle is simple enough, in that you can scale any fractions up to 17ths (and many beyond that) to get denominator within 5% of 100. From here, scaling the numerator up and making a small adjustment can give you a startlingly accurate decimal.
My favorite Flying Colours feat, however, has to be the Nth Root Feat, best described by Colin himself:
Pick a number* between 1 and 10 – don’t tell me what it is. Pick another number between 1 and 100 – you can tell me that one. Now work out the first number to the power of the second for me on this handy calculator, and I’ll tell you the first number.Even if you've practiced the cube, fifth, and square root feats, you'll realize this is on another level. You'll definitely want to be familiar with logarithms and the previously-mentioned fraction feat before trying this.
The detail and varied approaches in his multi-part series on squaring 3-digit numbers (Part I, Part II, Part III) are wonderful examples of his approach to mental math.
Don't pay attention to only the Mathematical Ninja to the exclusion of all else on the site; there's plenty more to discover! If you've ever been astounded by James Martin's amazing appearance on Countdown, you'll appreciate Colin's down-to-earth analysis of how James made those calculations.
The puzzle about the absent-minded professor and his umbrella is one of the best ways to introduce people to Bayes' theorem, as well. It's easily understandable for most people, and even clarifies some of the trickier aspects.
I don't want to ruin too much, however, so I suggest exploring Flying Colours Maths for yourself! If you find something you find especially interesting, let me know about it down in the comments!