Visiting Europe these past two weeks, I figured out how to convert Celsius to Fahrenheit by trying different formulas in my head. And backwards, too! 
I also remember in the fourth grade when I figured out how to multiply numbers by 9, 99, 999, etc. Good times.
Surely these simple examples pale to those of the more math-minded here on the Dope. What math formulas did you figure out merely by thinking about it (or putting pencil to paper)?
At some point it gets hard to get too specific and retain anonymity.
Saw this once on a DIY show. Practical use of Pythagorus’ theorum.
3 4 5 rule. To check a corner for square. Measure 3 units down one side and make a mark. 4 units down the other and mark. Measure the hypotenuse from mark to mark. Will be 5 if you have a right, 90 degree corner. Very handy when laying out string line for a outdoor project (like a brick patio).
Another way, measure the distance between opposite corners. Then repeat for the other two opposite corners. Will be the same distance if your frame is square.
As an “amateur” I don’t know where to begin, or end. Just so many I’ve figured out. A better proof of the Pythagorean theorem than what was in the textbook, discovered and proved the law of cosines, etc. In my Transformational Geometry class, I figured out a much simpler proof of one of the theorems in the book. Lots of stuff like that.
Too many Math proofs I’ve seen are “gotcha” proofs. “Here, do this magic step for no obvious reason and tada! it all works out!” So I like to come up with my own that offer an insight into the process of how to do the proof itself. Some of these are better than the “gotcha” ones.
As a CS professor, I’ve published lots of papers that have a lot of original work in them. There’s even a Wikipedia page that has my name in it for one of my results which impresses FtGKid2. (And note ultrafilter’s comment.)
But one of my cherished (very) minor ones is the one I came up with when I was 4. Lots of birthdays in my family around the same time. I noticed one sib just turned 8. Twice as old as me. Another turned 10. When would that be a doubling too? Hmm, when I turn 10. Oh: If the difference between siblings is “so much” then the older is twice as old as the younger when the younger is the difference in age. Neat!
:hangs head in shame:
Keep 'em coming! My latest formula* has been, fortunately, proven wrong, and I appreciate the replies.
When taking calculus, I discovered ahead of time that there is a local minimum or maximum* of a function where the derivative is zero. It appeared correct by just looking at a graph of a function, and was verified by simple testing of my hypothesis.
*Or an inflection point but I didn’t figure that one out.
I deduced how to calculate a decompression table for nitrox diving given an air table and percent oxygen in the mix. Other related calculations such as Maximum Operating Depth for a given nitrox mix followed.
Figured out the formula for incidence of a recessive trait (gene present in X% of the population) once, and managed to run into the bottom edge of Bayesian statistics trying to work out the odds on a system of dice rolls I wanted to use for a tabletop game.
Neither was all that difficult. I could have easily looked up either one, had I bothered. I was just bored and had time to kill.
OK… some of you are just mocking me, aren’t y’all?
I discovered that the 3-5-7 triangle has a 120 degree angle, and as a result of this the 3-7-8 and 5-7-8 triangles have a 60 degree angle. There is an infinite series of such triangles with integer sides not having a common factor and with a 120 degree angle, each giving rise to a pair of 60 degree triangles. (See this Wikipedia article.)
The 60 and 120 degree triangles merely flip signs. BTW I figured out that if consecutive numbers add up to a square, those integers and the square root are a Pythogarean Triple.
The formula for angles of a regular polygon’s corners. I figured if it has n sides, then there are n spokes coming out of the center, each being 360/n degrees, and those spokes make several triangles like a pizza, each having 180 total. The other two triangle corners sum to a full polygon corner, so 180-(360/n)=corner angle.
I proved to myself the volume formulas for the cone and sphere (the sphere turned out to be easier). They weren’t very elegant proofs since I don’t have much mathematical background, but they did work.
I do that when I hear German weather reports. If they say it’s 30 degrees Celsius, then I can’t help wanting to know what that is in Fahrenheit. The fact that it’s seven thousand miles away is beside the point.
I never forgot the actual formula, but what I needed was a quick and easy way of doing it in my head, without having to imagine the actual figures on an invisible sheet of paper. It’s a lot easier to multiply something by 10 than by anything else, so I do that to the Celsius number and then subtract it to arrive at nine times that number. Then all I have to do is divide by 5 and add back the 32.
Huh. I just double the celsius #, take off 10% of the previous result, then add 32.
For example, 30C = F
30*2=60
60-6=54
54+32=86
30C=86F
I had my proudest math moment a couple of years ago. I was invited to do a build for a discovery channel series on Da Vinci. They were building a catapult powered by bending wood. I never went past arithmatic but had to figure g foces on the throwing arm, rotaional inertia forces, stored energy and a number of other things and I only had a couple of days to figure it out. My calculations came out right on. Now the catapult was another story but we had to do it the way da vinci designed it.
I figured out why adding subsequent odd numbers gets you the next square, and only belatedly converted it back to algebra (X[sup2[/supo]+2x+1 = (x+1)[sup]2[/sup]. I also did the same thing with figuring out products from other products: I know 25 * 25 = 625, so 26 * 24 = 625 + 25 - 26 = 624.
If course, that’s recently. I don’t remember much about what I did in school or anything.
I figured out an easy way to convert between pounds and ounces, or between price per pound and price per ounce: multiply (or divide) by 2 four times.
P.S. The ultimate, easiest way to convert between F and C is as follows (NOT my derivation):
This works because -40 C = -40 F.
When I lived in Chile, with an unobstructed clifftop view of the Pacific, I figured out the formula to determine the distance to the horizon. I also had to pace off and triangulate the elevation of my house.
I also drew a southern star chart, using the star coordinate tables I found in an old World Almanac from a used bookstore in Valparaiso. This was in the early 80s, unaided by computers or internet.
My Star Chart was a reversal of the standard ones. A standard store-bought star chart is a picture of the sky when held over one’s head looking up. Mine, instead, was like a road map, that when looked down at, showed a plane locating imaginary towers with the stars at their tops. I found it much easier to use.
I did something similar with the percentage of hybrids if we know the percentage of recessive phenotypes is X%