Have you ever been stumped by a math puzzle or problem?
Mathematician James Tanton understands that feeling, and he's designed an entire course to help you attack those seemingly impossible challenges!
I've mentioned James Tanton before (see previous mentions here), especially in the context of his Curriculum Inspirations video puzzles.
On the main Curriculum Inspirations page, he's included a useful list of 10 strategies for attacking such problems. They're taught as essays, such as this one for Strategy #1: Engage in Successful Flailing.
Recently, however, James Tanton has begun creating more lively video explanations of each strategy. Much of the advice can also apply to many real-world problems, as well. Check them out below:
Strategy #1: Engage in Successful Flailing
Strategy #2: Do Something
Strategy #3: Engage in Wishful Thinking
Strategy #4: Draw a Picture
Strategy #5: Solve a Smaller Version of the Same Problem
Strategy #6: Eliminate Incorrect Choices
Strategy #7: Perseverance is Key
Strategy #8: Second-Guess the Author
Strategy #9: Avoid Hard Work
Strategy #10: Go to Extremes
Have you ever been stumped by a math puzzle or problem?
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!
Even when you think you understand a concept, even one as simple as basic multiplication, you can come across a different perspective that makes you take a closer look.
In this post, we'll look at an almost magical way to multiply in a very visual manner!
Tipping Point Math recently posted this unusual multiplication technique in their Multiplying with a Parabola! video:
I had to try this technique out for myself, so I headed over to Desmos.com and created this interactive version (The image below also links to the interactive version).
The Desmos.com version lets you multiply any 2 whole numbers ranging from -15 to 15, using the sliders in boxes 2 and 3 on the right. Clicking on the dot where the line crosses the vertical (y) axis will display the coordinates, and the y-coordinate will be the answer to the multiplication problem. You can also click the arrows just to the left of boxes 2 and 3 to start and stop animation of the points.
Play around with it for a while, and discover the possibilities. By clicking on the wrench image on the upper right side of the screen, you can adjust the settings, including Projector Mode, which can make the graph less cluttered.
Working with this interactive version, you'll quickly find the answer to the challenge of multiplying 7 by -4 given in the video above. You may also find new questions, however!
For example, setting the sliders to multiply 5 by -5 puts the 2 lines in the same place, and gives the same point! With only 1 point, how does the computer know at what slope to draw the line? The short answer is that there's a bit of a cheat here. The computer will always draw a line through the coordinates (a, a2) through (0, ab), so the line is forced to give the right answer. More generally, though, there is a surprising way to mathematically define a line with a single point, as explained in this half-hour video about Galileo's laws of falling bodies.
Surprisingly, this basic idea can be expanded to handle a wide variety of calculations. For example, James Grime uses the graph of y = x3 - 3x to create his cubic curve calculator:
Graphical calculators such as this are known of nomograms, and are often both an art and a science. Ronald Doerfler's My Reckonings blog has some amazing examples of nomograms that are likely to boggle your mind. Even some of the simpler nomograms, such as this Educated Monkey tin toy, or this nomogram from Popular Mechanics for calculating the day of the week for any date, are still astounding to use and explore.
If you've run across any interesting nomograms yourself, feel free to share them in the comments below!
Money is a tough enough subject on its own. Compound interest seems difficult to wrap your head around, and nearly impossible to calculate without specialized tools.
In this post, however, you'll not only wrap your head around compound interest, but learn some amazing ways to estimate answers quickly in your head!
Compound interest is really all about the time value of money. OK, granted, that sounds like I just switched one buzzword for another. Perhaps having German Nande explain the time value of money in his TED-Ed video will help:
Perhaps figuring out that 10% added to $10,000 is $11,000, but wouldn't it be difficult to work out how long it would take $10,000 to turn into $110,000? Our first tool will begin to make calculations like this easy.
• The Rule of 72: This is one of the most well-known rules in finance. BetterExplained.com has an excellent article on the Rule of 72. In short, if you divide 72 by the interest rate in question, you'll get the number of years it will take your money to double at that interest rate.
For example, for the 10% example in the video, you'd work out 72 ÷ 10 = 7.2, which means it would take about 7.2 years to double your money at 10%. How long would it take at 6%? You work out 72 ÷ 6 = 12, so it would take 12 years to double your money at 6% interest.
To figure out the amount of time it would take to accumulate $110,000 at 10% compound interest, we could think about it in the following manner. In 7.2 years, the $10,000 would double to $20,000. In another 7.2 years (14.4 years total), the $20,000 would become $40,000. Another 7.2 years (21.6 years) would bring $80,000, and a final 7.2 years would take it to $160,000, so we can say that getting to $110,000 would take somewhere between 22 and 29 years.
That's accurate as far as it goes, but can we do better?
• The Rule of 114 and 144: As pointed out over in allfinancialmatters.com, there are similar rules for finding out how long it takes your money to triple and quadruple. For tripling, divide 114 by the interest rate, and for quadrupling, you divide 144 by the interest rate.
Let's see if we can't work out the $110,000 with these new tools. If we could have the original $10,000 triple, then quadruple (or vice-versa) at 10% interest, that would be 12 times our original amount. So, to determine the tripling time, we work out 114 ÷ 10 = 11.4 years. From there, the quadrupling time would be 144 ÷ 10 = 14.4 years. 11.4 + 14.4 = 25.8, or about 26 years. That's the same amount of time in the video!
That's not bad for a mental estimate. There's plenty that can and can't be done with these rules. For example, investopedia points out that using the long term inflation rate of 3%, you can compare prices from years ago to today's prices. At 3%, inflation should double prices every 24 years (72 ÷ 3 = 24), so prices should quadruple every 48 years, and so on.
The caveats explained in MindYourDecisions.com's post on the Rule of 72 should be understood. The rule of 72 doesn't apply when you're getting a variable return (such as stocks and bonds), the interest rate in question is too extreme, or when additional money is regularly added.
That last point is especially interesting. Just how do you calculate interest when regular amounts are included as you go?
• The Rule of 6: Fortunately, MindYourDecisions.com has an answer for that, as well. In that posts example, the author supposed that you add $100/month to an account at 5% interest for 1 year. The calculation shortcut simply involves multiplying the regular deposit amount by the interest rate and the number 6.
The answer given by this estimate is 6 × $100 × 0.05 = 600 × 0.05 = 30. $30 then is the estimate, which is pretty good compared to the actual calculated total of $32.26. If you want to see the accuracy of this formula for yourself, you can play with the numbers involved at this Wolfram|Alpha link. Simply set d to the regular deposit amount (d=100 in this example), and p to the percentage rate (5% is given by p=0.05); Wolfram|Alpha will then return two variables, u, which is the exact amount of dollars in interest you can expect, and v, which is the mental estimate.
Perhaps this rule should be called the rule of half, since you can apply this to any amount of months simply by halving the total number of months involved. How much, for example, could you expect in interest by putting in $150 per month at 4% interest per year, for 5 years (60 months)? We multiply 30 months (half of 60) × $150 per months × 0.04 = $4,500 * 0.04 = $180 in interest. The actual amount, as calculated here, is $181.82, using an additional variable, m, to represent the number of months in question (m=60 for 60 months).
Practice these financial tips, and be ready to astound your friends and family with your financial wizardry!