I ran into the following problem in a contest problem book. I won't reveal the source until later. I was not able to solve this problem and even though the solution is in the back of the book so I know how to solve it now, I'd like to see if anyone can solve this in an elegant and intuitive way and maybe even show how the author might have invented this problem.
There's no prize for solving this, but I will publish all solutions and give link love to all solvers, and I will give special kudos to anyone who can help me to see why the interesting property shown in this problem is true.
While surfing the web I discovered this very cool 10 page Math cheat sheet!
You can download the PDF file here.
Scoring this gem led me to wonder if there were other Math cheat sheets. In particular, I was looking for concise lists of formulas for various subjects. PDF was my first choice for file type.
The best source I found was Paul's Online Math Notes, which has algebra, trig, and calculus cheat sheets.
Last November I wrote about the Trachtenberg system of speed mathematics and showed one of the techniques - multiplying an arbitrarily large number by 12 with great ease. In this article I want to show you why the technique for multiplying by 12 works, share two more of the Trachtenberg multiplication techniques, and give you some direction as to how you can develop your own multiplication techniques.
In that first article about Trachtenberg multiplication I taught you to multiply by 12 by "doubling the digit and adding its neighbor (on the right)." I gave an example of multiplying 346 by 12:
Here's an interesting little problem. I made this one up but I bet it's been thought of by students of number theory and diophantine equations in particular.
Let's say you have two kinds of postage stamps, one with a value of 3 cents, the other with a value of 5 cents. List the postage amounts that you CAN'T make. You can't make 1, 2, 4, or 7 cents.
Let's look at another example. You have a 5 cent stamp and an 8 cent stamp. What values can't you make? You can't make 1, 2, 3, 4, 6, 7, 9, 11, 12, 14, 17, 19, 22, 23, or 27 cents.
For 3 and 5 cent stamps, 7 cents is the largest amount you can't make.
For 5 and 8 cent stamps, 27 cents is the largest amount you can't make.
Can you come up with a simple formula for the largest amount you can't make given two kinds of stamps? What assumption must you make in order for there to be a largest amount you can't make?
Can you explain why your simple formula works?
9 is a most interesting number. I'm sure that's largely because 9 is 1 less than 10 and most of us have 10 fingers (or digits) and we do arithmetic in a base 10 system. I've seen an amazing number of math tricks that take advantage of something called "digital roots", which is closely related to the idea of "casting out nines." I want to introduce you to these two concepts and share some fun Math tricks you can do with this "9 stuff."
The digital root of a number, and this only makes sense for whole numbers, is what you get when you add up all of its digits. So, the digital root of 112 is 1+1+2, or 4. The digital root for 1234 is 1+2+3+4, which is 10. Now, when you're computing digital roots you only want a single digit so in the case of 1234, you add up its digits to get 10 then add 1+0 to get 1. So, 1 is the digital root of 1234.
Last month I blogged about mathemagician Arthur Benjamin and his amazing mental Math feats. Benjamin is a master of doing arithmetic in his head with lots of digits involved. In particular, he's able to square a 5-digit number without writing down partial results. How does he do it? I picked up a copy of Benjamin's Secrets of Mental Math to learn how. Here are the steps Benjamin provides for squaring 46,792.
1. First, Benjamin breaks the number into 46,000 + 792.
2. Then he does a little algebra. If a=46,000 and b=792, then (a+b)^2 = a^2 + 2ab + b^2 = (46,000)^2 + 2(46,000)(792) + 792^2.
3. This simplifies a bit to 1,000,000(46^2) + 2,000(46)(792) + 792^2.
4. Then Benjamin sets out to do the middle product: 2,000(46)(792).
I imagine that many of you are familiar with this remarkable mathematical equation that incorporates what are arguably the five most important mathematical constants into one equation. Yes, phi is missing from this equation. I've been reading this fascinating book, Where Mathematics Comes From. Wikipedia has an article about the book. The book seeks to found a cognitive science of mathematics. While the book is very philosophical and abstract in many places, what fascinated me were the very rich metaphors discussed for a number of common mathematical operations and concepts. These metaphors really helped me to see the conceptual basis for some mathematical processes and operations I took for granted and really helped to ground and deepen my understanding.
Some examples of metaphors:
1. Addition and subtraction. Moving to the right or left along the number line
2. Multiplication of a positive number by a positive number. Moving to the right along the number line by a factor.
3. Multiplication by -1. Rotation to the symmetry point of n.
4. A simple fraction (1/n). "Starting at 1, find a distance d such that by moving distance d toward the origin repeatedly n times, you will reach the origin. 1/n is the point-location at distance d from the origin."
5. Exponential function. The mapping of sums onto products. This explains what it means to raise numbers to non-integer and to negative powers.
There are numerous introductions to Vedic mathematics on the web. I won't be doing a general introduction to Vedic Math now. In this article I want to explore one particular Vedic mathematics technique, using something called bases, to optimize certain multiplication problems.
This technique is extremely powerful and it takes getting used to. It's partially a cookie cutter technique but there's also some thinking involved in selecting proper bases for performing multiplication. Don't get frustrated if you can't understand this technique in one reading. It took me a fair amount of focused attention and practice to understand and appreciate the power of this approach. If there's enough interest I'll produce some videos explaining this Vedic technique.
A popular winter Math problem is the counting of presents given during the 12 days of Christmas.
On the 1st day of Christmas my true love gave to me a partridge in a pear tree.
On the 2nd day of Christmas my true love gave to me 2 turtle doves, and a partridge in a pear tree.
On the 3rd day of Christmas my true love gave to me 3 french hens, 2 turtle doves, and a partridge in a pear tree.
... and so on through day 12.
The problem is to count the number of presents received during the 12 days, and for the truly masochistic, the cost of the twelve day's worth of presents in today's dollars.
I love this type of problem because it's an algebra problem and it involves sums of series, two favorite subjects of mine.
Here's a video on how to quickly square a 2-digit number. The technique is based on this algebra:
If you have a number with digits "ab" then the number is 10a+b.
(10a+b)^2 = 100a^2+20ab+b^2.
If you enjoy this video check out all of the Wild About Math! mathcasts.