There's a book out called The Mental Calculator's Handbook by Jan van Koningsveld and Robert Fountain, which naturally piqued my interest just by the title.
How does this book compare to existing books on mental math? Check out this review and find out!
First, you'll probably want to know what kind of mental math expertise the authors have. Robert Fountain is a British calculating prodigy who was the first of only 3 current International Grandmasters of Mental Calculation.
German mental calculation champion Jan van Koningsveld has held several world records relating to mental calculation, including taking only 3 minutes and 6 seconds to solve 10 problems, each of which involved multiplying two 5-digit numbers. You can find several videos online of his performances, and even if you don't speak German, they're easy to follow due to the numbers and his results being displayed.
At first glance of the contents, The Mental Calculator's Handbook doesn't seem to be much different than, say Arthur Benjamin's The Secrets of Mental Math. The first few chapters cover addition, subtraction, multiplication, division, and fractions.
Once you delve into the chapters themselves, they do begin with basic techniques similar to other books. I was pleased to discover, however, that they do take even these basic techniques farther than most books. The exercises at the end of each section, and the detail given about the techniques is written very clearly, so it's easy to understand.
This early attention to detail and emphasizing the finer points really begins to pay off when you begin learning the techniques in the later chapters, which include working out classic feats such as finding roots, our old friend calendar calculation, and the rarely-discussed factoring of numbers into their prime components.
The section on prime factorization was an especially interesting eye-opener. I was familiar with the basic techniques from my own work on primes in mental math, but the techniques here went much farther. Testing for divisibility by 2, 3, 5, and 9 are simple enough, but when many primes provide a challenge for divisibility tests, such as 7, 11, 13, and 37. The authors turn these into almost trivial challenges by showing how working with much larger numbers, such as 999 and 1,001.
Regular Grey Matters readers won't be surprised to know that I enjoy reading about and working out calendar-related challenges, and even here I was surprised! Besides just the basics of working out the day of the week for any date, you learn how to handle questions such in which years between 2000 and 2099 will Halloween fall on a weekend, and in which months of 1961 the 29th fell on a Sunday.
The Mental Calculator's Handbook winds up with brief biographies of various past mental calculators and their performances. This section especially was a very enjoyable read, and gives you an idea of just what can happen when such feats are demonstrated, and learn the sometimes sad and often amazing ways in which these performer's lives were affected.
If you're not sure of your own interest in mental calculation, I suggest starting with a more basic book, such as The Secrets of Mental Math and see if it's something you'll enjoy. Once you're ready to pursue it further, then you're ready for The Mental Calculator's Handbook, and it's greater attention to detail and mastery of the field. As a matter of fact, this book is a great bridge between the simpler mental math books, and the far more advanced ones, such as Ronald W. Doerfler's Dead Reckoning: Calculating Without Instruments.
Overall, if you're interested in mental math, and want to go beyond the basics, The Mental Calculator's Handbook is an excellent resource to take you to those next steps.
There's a book out called The Mental Calculator's Handbook by Jan van Koningsveld and Robert Fountain, which naturally piqued my interest just by the title.
Sometimes, you run across an old idea again and again, and you don't pay much attention to it. Then, someone comes along and shows you a new application of that old principle, and it seems like something completely new!
That's what happened to me recently, when a Magic Café user shared an idea I'm about to share with you.
The basic idea is to have someone choose a number from 1 to 100, then add that number together with the two numbers next to it. If the answer is a two- or three-digit number, then those digits are added together, and this prcoess is repeated until the answer is a one-digit number.
For example, if the total was 157, they would add 1 + 5 + 7 = 13. Since this resulted in a two-digit number, the digits in this answer would be added together again, 1 + 3 = 4. Since 4 is a one-digit number, they'd stop here.
While this seems fair, you can know, even before the number is chosen by the audience, that the total will be 6!
This idea was developed by a Magic Café user whose screen name is Pixelated. But how is this possible?
First, when you ask for a number from 1 to 100, you mention that it's going to be added together with the two numbers next to it. “Next to it” is a deliberately vague phrase that could apply to the two numbers immediately above and below their choice, the two numbers after their choice, or the two numbers before their choice. Until you know their number, you don't know which of these 3 groups you'll need to use.
Once they name a number, you have to ensure that the largest of the 3 is a multiple of 3 (naturally, you should be familiar with all the multiples of 3 from 1 to 100). Pixelated suggests having a row of 3 boxes on a board or piece of paper.
Let's say the person names the number 57. Since 57 is a multiple of 3, you would write it in the rightmost box, and then write the two previous numbers, 55 in the leftmost box, and 56 in the middle box.
What if they'd named 56? You will hopefully recognize that 56 is 1 less than a multiple of 3, so you'd write that in the middle box, so that 57 would naturally go in the rightmost box, and 55 would go in the leftmost box.
If they name 55, you should recognize it as a number that's 1 greater than a multiple of 3 (54, in this case), so you know to write it in the leftmost box, writing 56 in the middle box and 57 in the rightmost box.
Note that, even though the above represents 3 different cases, you always wound up with 55, 56, and 57 in that order. Also the reference to the “two numbers next to the chosen number” makes sense in all 3 cases.
Probably the easiest way to think about the process is to take the chosen number, and count up to the next multiple of 3, unless, of course, the chosen number is already a multiple of 3. Whatever the nearest multiple of 3 is, you think of that as a “ceiling” you cannot go beyond. If someone were to name 47 as their number, you can think: 47...48...Hey! 48 is a multiple of 3...48 has to go in the rightmost box, so 47 must go in the middle. That leaves 46 for the leftmost box.
In short, although you seem to specify how the three consecutive numbers will be chosen, it's your ability to specify the order which gives you control over the total.
Once you have the three numbers, have an audience member add them up as explained above. If the numbers are 55, 56, and 57, they'd add them together to get 168. Next, they'll add 1 + 6 + 8 to get 15, and then add 1 + 5 to get 6.
What about 46 + 47 + 48? That's 141, and 1 + 4 + 1 = 6. No matter which number is chosen, you can always make the resulting 1-digit total equal 6.
WHY DOES THIS WORK?: First, you'll need to be familiar with the concept of digital roots. Here's a quick introduction, or refresher course, on digital roots via video:
Since you're arranging the numbers so that the largest of the 3 numbers is a multiple of 3, that largest number can be written algebraically as 3x (in other words, 3 times some number). The two numbers immediately prior to it can then be described as 3x - 1 and 3x - 2. Add these three numbers together, you get 3x + 3x - 1 + 3x - 2 = 6x - 1 + 3x - 2 = 9x - 3.
As regular Grey Matters fans and mental math wizards already know, Any time you multiply a number by 9, the answer will have a digital root of 9. We can also be sure, then, that 9x - 3 means that the digital root will be equal to 9 - 3, or 6.
VARIATIONS: Using a similar approach, you could always make sure that the multiple of 3 winds up in the middle. In this case, the numbers would be 3x, 3x - 1, and 3x + 1. Added together, this comes out to simple 9x, so the total would have a digital root of 9.
You could also always ensure that the multiple of 3 is the leftmost number, making the numbers turn out to be 3x + 3x + 1 + 3x + 2 = 6x + 1 + 3x + 2 = 9x + 3. In this case, the number would have a digital root of 3.
Besides simply adding up the numbers themselves, you could also have the people add up the individual digits in each number. Going back to our previous example using 55, 56, and 57, you could have them add 5 + 5 + 5 + 6 + 5 + 7 = 15 + 6 + 5 + 7 = 21 + 5 + 7 = 26 + 7 = 33. Of course, 33 becomes 3 + 3 = 6, so it still works out to the same digital root.
THOUGHTS: The simplest presentation, of course, is to somehow show that you predicted the number 6 (or 3, or 9, depending on which approach you used). However, 6 could also mean something besides just a simple number. It might be a page, or a word on a page (the 6th word), or a time on a clock, or a month (6th month is June), or any of a million other possibilities.
As was once said about the laser, this is currently a solution looking for a problem. Somewhere out there is the perfect application for this numeric force. Try playing around with this idea, and let me know in the comments about any ideas you develop!
NOTE: This post originally appeared on Grey Matters back in July 2011. I'm reprinting it today because Werner Miller's mathematical magic really is worth a second look.
Grey Matters favorite Werner Miller is back, and he's brought more of his amazing mathematical wizardry with him!
If you're not already familiar with him, he's a retired mathematics teacher in Germany who has created some of the most original and compelling magic routines I've ever come across. He is the author of several magic books, including Ear-Marked, which is available in the Grey Matters store.
Starting off, we have a couple of good routines that are perfect as promotional tools, since you can print them on your business cards or brochures, and have people perform them for themselves or others without understanding how they work. There's Vive Le Roi!, which includes several variations of routines where you move your finger from card to card, eventually winding up on a predicted card. You can have two people do this together, as they'll be on different cards until the last card.
The other trick along this line is Magic Patchwork, a similar trick with a magic square. He mentions that it was inspired by Pedro Alegria’s El cuadro de colores, but the link given is no longer functioning. Fortunately, it was captured by the web archive. The original is here, with a translation to English via Google Translate available here.
Werner Miller also created a very sneaky calculator trick, called You Push the Button... that seems to be a mathematical trick, but isn't. The use of the calculator helps conceal the outright sneaky method.
Getting back to his specialty of mathematical magic, he offers a great routine with dice. It's called Lined Up, and has two different phases, both of which begin with different-colored dice arranged with the numbers 1 through 6 in numerical order. In Phase 1, you have someone choose a die and turn that number face down. After getting the new total of these dice, you announce which color die has been turned over. Phase 2 is similar, except that you have someone choose a die and turn over every die EXCEPT for the chosen one!
I've saved my favorite for last! It's called Ghost Rider, and uses a chess knight and some file cards. One of the file cards is signed, then mixed into the pile and dealt out into a 3 by 3 square. The spectator then uses the knight and their own free choices to find their own signed card! Part of the principle is taught here on Grey Matters in my Knight Shift post, as mentioned generously in Werner Miller's article. His added touches, however, make this a very impressive trick.
If you like Werner Miller's style and would like to see more, check out the rest of Werner Miller's work here on Grey Matters!
August's snippets are here!
This month, our snippets return to their original roots, and are just a mixed bag of goodies I thought might be of interest to Grey Matters readers.
• While reading Numericana, I learned about a trick dubbed Enigma. The video for the performance is shown below:
The explanation video can be found here. However, Grey Matters readers can use their knowledge of quick binary conversion to speed up the needed calculations!
• Speaking of base conversions, here's an unusual fact. For any integers x and y, xy in base x will always be 1 followed by y zeros! For example, 68 in base 6 is 1 followed by 8 zeros. Here are the number 2 through 6 raised to the second through the tenth power in Wolfram|Alpha, as an example.
• Futility Closet features a simple way to calculate the probability of any number from 2 to 12, when rolling 2 six-sided dice.
• While playing around with memorizing the speed of light in meters per second (299,792,458 m/s), I was originally using the method of using words of a given length to represent a given number (3-letter word to represent 3, etc.). I noticed that it was taking longer to count the longer words, and thought it might be better to combine that technique with words that rhyme.
In other words, use 9-letter words that rhyme with the word NINE, 8-letter words that rhyme with the word EIGHT, and so on. Using sites like WordHippo and RhymeZone, here's what I came up with for the speed of light: “To recombine, valentine, shorten storyline. Do more alive, soulmate!
That's all for this month's snippets. I hope you enjoyed them!
Have you practiced your multiplication by 63 skills? Good! Have you practiced your multiplication by 72 skills? Great!
Today, you'll learn to put these skills to use in a surprising and amazing way!
How many seconds are there in a 365-day year? To find out, we'd multiply 365 days per year by 24 hours per day by 60 minutes per hour by 60 seconds per minute, for a grand total of 31,536,000 seconds per year.
How about the number of seconds in two consecutive 365-day years? We just multiply the previous total times 2, and we get the rather interesting total of 63,072,000 seconds in 2 years!
Since we've learned to multiply by 63 and 72 quickly, we can now use this knowledge to quickly estimate how old someone was in seconds on their last birthday!
As you've seen in the previous posts, writing down the answers as you develop them is very helpful, and is an important part of working through the feat of estimating someone's age in seconds. You should write down the answer as you go, using a writing instrument and surface which allows you to erase, such as a pencil and paper, or a dry-erase board and marker. This is because you may need to make minor corrections as you proceed.
When you start, it's important that you ask for the person's age only, and have no idea of their birthday. Since the above calculations involve multiplying by 365, leap years are not taken into consideration. As you'll see later, the emphasis should be on the speed of doing an estimate in your head, not the exactness of the answer with every detail taken into consideration.
Once you have someone's age, the first step is to divide it by 2. This is important because 63,072,000 seconds is the amount of seconds in two 365-day years. You're effectively figuring out the number of seconds in how many pairs of years there are in their age.
The person's age divided by 2 will be treated as the given number, as used in the 2 previous posts. There are 3 different kinds of challenges you'll face in this feat, and I'll explain them below from easier to more difficult.
EVEN AGES UP TO 26: If the person's age is an even number of 26 or less, this is the simplest calculation. When their age is divided by 2, the given number will also be an integer (whole number), and the skills you've developed so far will be all you need.
As an example, let's assume you ask someone their age, and the reply that their age is 26. As you write their age down at the top of the page (or board), you work out that 26 ÷d; 2 = 13, so your given number is 13.
Now, you'll be effectively multiplying 13 by 63 million instead of 63, so I start by putting down a comma. To the immediate left of this comma will be the millions, and to the immediate right will be the hundred-thousands place. So far, the writing surface would look like this (not including the age at the top):
Now, we work through our process for multiplying by 63: “13 tripled is 39...*write down the 9 to the immediate left of the comma*...39 doubled is 78...78 plus 3 is 81...*write down the 81 to the immediate left of the 9*...” Before we move on to the multiplication by 72, let's take a look at the writing surface as it would look now:
Remember, this represents 756 million at this point. The places are important, of course. The next step is to multiply by 72,000, which means putting down another comma to the right of the above answers, and putting dow the comma between the thousands and hundreds place (dashes used to represent empty spaces):
From here, you continue your calculations to multiply by 72, picking up from where you left off: “78 doubled is 156...*write down the 6 to the immediate left of the comma*...78 plus 15 is 93...*write down the 93 to the immediate left of the 6*...” At this point, you should have an answer which appears like this:
The final step is always to write 3 zeroes to the right of the rightmost comma:
Sure enough, if you have a person multiply their age, 26 in this example, by 365 by 24 by 60 by 60, they'll find that 819,936,000 is the right answer!
Even better, that's the process they'll assume you went through in your head to get your answer, so you're getting credit for doing the full work, when you're really only using a short cut.
EVEN AGES FROM 28 to 66: Ages ranging from 28 to 66 will result in given numbers being integers ranging from 14 to 33. When multiplied by 72, any of these numbers are over 1,000. This means, of course, that when multiplied by 72,000, these numbers will result in answers of 1 million or more, so you'll have to deal with carrying that 1 over into the answer you received when multiplying by 63.
For this example, let's say the person's age is 42, which makes the given number 21.
Start running through the process as before, thinking and writing as you go: “21 tripled is 63...*write down the 3 to the immediate left of the comma*...63 doubled is 126...126 plus 6 is 132...*write down 132 to the immediate left of the 3, include comma to denote 1 billion, add in comma before thousands place*...” So far, you would have this:
Continuing on to multiply by 72, you'd think and write, “126 doubled is 252...*write down the 2 to the immediate left of the comma*...126 plus 25 is 151...*write down 151 to the left of the 2*...” Here's where you run into the challenge. As you start to write the 151 to the left of the 2, you realize there's going to be overlap. Write the tens and ones digit of 151 (the 5 and rightmost 1) as you would normally, like this:
Next, erase the 3 and mentally add 1 to it, giving 4. So, you replace the 3 with a 4, and the answer will look like this:
Don't forget to add the 3 zeroes to the right of the rightmost comma, so you have the full answer:
Dealing with carrying seems tricky at first, but once you get the hang of it, it's not much of an obstacle.
ODD AGES UP TO 65: Odd ages will result in a given number ending in .5, as you've seen in the previous tutorials. When dealing with these numbers, you may have to deal with carrying, as above, but you'll have to make 2 more simple adjustments.
The first, of course, is writing “&,5#148; instead of “.5”. When you're only multiplying by 63, the number just represents .5, but when multiplying by 63 million, that .5 starts representing 500,000! I'll discuss the other adjustment when we come to it.
For this example, we'll say that the person is 35, which, when divided by 2, makes your given number 17.5.
You would start just as before: “17.5 tripled is 51...52.5...*write down the 2,5*...52.5 doubled is 105...105 plus 5 is 110...*write down 110 to the immediate left of the 2,5, add comma for thousands*...” So far, you have:
At this point, you can probably see the other problem coming. Not only will you have to carry as before, but you now have to deal with that 500,000. Don't worry, this is easy to solve. When you're working out the final addition for the multiplication by 72, mentally add 50 to it. Once you have this number, erase the 5, and write the total as you would before. If needed, use the carrying technique taught earlier.
Continuing with our example from where we left off: “105 doubled is 210...*write down the 0 to the immediate left of the comma*...105 plus 21 is 126...126 plus 50 (to deal with the 500,000) is 176...*erase the 5, write down the 76, erase the 2, add 1 for the carry, write 3 and final zeroes*” Your final answer, with the 3 zeroes should look like this:
Yes, this is the correct answer for 35 × 365 × 24 × 60 × 60.
Once you become familiar with both carrying and dealing with ages of an odd number, you're ready to handle any age up to 66 in this feat!
TIPS:• You can use this link to practice with even ages only, and then use this link to practice with any age from 1 to 66. Don't forget to check your answer with this link, setting x to the age you were given (default age at the link is 42).
• After going through the calculation, I like to explain, “I only asked for the age, so I have no idea if the birthday they're talking about was last year, this year, or even next year. So, instead of trying to take every last detail such as leap years or seconds since their birthday into account, this is just a quick mental estimate.” They'll be startled to find out that this estimate is so exact, when you explain that you're multiplying their age by 365 days in a year, by 24 hours in a day, by 60 minutes in an hour, and by 60 seconds in a minute!
• You can grab interest at the beginning by talking about how people hear the terms million and billion, but never really think of the difference. Explain that 1 million seconds is only about 111⁄2 days, while 1 billion seconds is roughly 312⁄3 years!
If you know you're going to be performing this feat ahead of time, you can use Wolfram|Alpha's calculations and Wikipedia's year and month pages to find historical events you could mention. It's one thing to say that 1 billion seconds ago was December 1981, but more vivid to say something like 1 billion seconds ago, Britney Spears was born.
Practice this feat and have fun using it to amaze people!
Hopefully, you've been practicing your skills at multiplying by 63, and are ready to move on to the next level.
In this post, you'll learn how to quickly multiply by 72. This builds on your skills at multiplying by 63, so this post will actually combine both of them.
BASIC STEPS: For multiplying by 63, I listed 5 steps in the last post. By the time you've completed step 3, you've tripled and doubled the original given number. In other words, you've multiplied the given number by 6 at this point. For multiplying by 72, 6 happens to be the starting point, so this list will continue from that earlier list:
6) Recall the number calculated in step 3.
7) Double the number from step 6. Do not forget the number from step 6, however.
8) Write down the ones digit of the result you calculated in step 7.
9) To the hundreds and tens digits ONLY of the number you calculated in step 7, add the entire number from step 6.
10) Write this new total to the immediate left of the digit written in step 8.
As you can see, this is nearly identical to the steps for 63, with some minor rearrangement. Let's turn to some practical examples to help clear up any confusion.
WHOLE NUMBERS: For our first example, let's start with 13. First, we'll multiply it by 63, and then multiply by 72 after that. We'll use this form to write our answers:
13 × 3 = 39, and we write down the 9 next to the 63 (leaving room to place numbers to its left):
Now, we double 39 to get 78, trying to keep both 39 and 78 in mind. We add 78 to the tens digit of the first answer, which is 3. 78 + 3 = 81, so we write 81 down to the left of the 6, and we're done with the first part:
That's all for 63, now how about 72? The number we calculated in step 3 was 78, so we'll start from there. We double this number to get 156. As instructed in step 8, we write down the ones digit of 156, which is 6:
Now, we add 78 to just the hundreds and tens digit of 156. What does that mean? The hundreds digit of 156 is 1 and the tens digit is 5, so we're going to add 78 + 15 to get 93. Finally, as instructed in step 10, we write this total down to the immediate left of the number we wrote previously:
There we go! We've calculated 13 times both 63 and 72 fairly quickly. Notice how the last doubled number from the calculation for 63 becomes the first step for 72? Once you practice calculating in this way, you'll probably develop an apreciation for the efficiency of this approach.
I'll run through another number to help lock in the idea. This time, 19 will be the given number. 3 × 19 = 57, so we write down the 7:
57 doubled is 114, so we add 114 plus the ones digit, 5, to get 119:
Not forgetting the 114 at this point, we double that to get 228, and write down the ones digit, 8:
Now we add 114 to 22 to get 136, and we write that down to the left of the 8:
Quicker than you may have thought possible, we've multiplied 19 by 63 and 72.
Just as before, practice your skills for multiplication of both 63 and 72 by random whole numbers up to 33. Don't forget to check your answers for both number by using this link, and setting x to the appropriate given number.
QUICK EXPLANATION: Note that the ones digit from only the first multiplication and last multiplication is written down. The ones digit of the second number you calculate is never used in this way. I'll quickly explain why.
When you start with a given number, which we'll dub N, you multiply it by 3 to get 3N, then by 2 to get 6N, and finally 2 again to get 12N. Now, 60N is just 6N × 10.
What you're really doing in this method is calculating 3N and 6N, then adding 6N to all the digits except the rightmost digit, which effectively treats it like 60N. Adding 60N + 3N will result in 63N, which is why the first trick works.
Still thinking of 6N, you double that to 12N. Once again, you treat 6N like 60N by adding 6N to all but the rightmost digit. Not surprisingly, 60N + 12N = 72N, which is how we get our second answer.
Let's go back to that 13 example above to get a better idea of how this works. 13 × 3 = 39. Next, 39 is double to get 78. As mentioned, this should really be 780, since we're trying to figure 60N + 3N. 780 + 39 = 780 + 30 + 9 = 810 + 9 = 819. See what happened there?
Because of the multiplication by 10, we know the last digit of 60N will end in a 0, so we know the last digit won't change. All that remains then, is to add 780 + 30. This is much easier to do if we don't attach any important to the zeroes, so we simply do 78 + 3 = 81, and put the 81 in its proper place.
The explanation for 72 is similar. We start with 78 (6N), and double that to 156 (12N). To work out 60N + 12N, we'd be doing 780 + 156 = 780 + 150 + 6 = 930 + 6 = 936. Again, the rightmost digit of 156 (12N) won't change, so we can just write it down immediately. By performing 78 + 15 instead of 780 + 150, the whole thing is simplified for mental math.
If you're worried about adding two digits to two digits, just add the tens digit, followed by the ones digit. To simplify 78 + 15, perform 78 + 10 to get 88 first, and then add 5 to get 93. To better understand the advantages of left-to-right addition in mental math, check out this video.
NUMBERS ENDING IN .5: Here's some good news. By the time you've calculated 6N, as discussed above, you're dealing with a whole number, so as far as 72 goes, there's little difference at this stage. To demonstrate, let's use a given number of 14.5.
14.5 × 3 is 43.5 (Remember how to get here quickly? It was discussed in the previous post!), so we write down 3.5:
43.5 doubled is 87, so we add 87 + 4 (the tens digit of 43.5) to get 91, we write down 91 to the left of the previous numbers:
Starting from 87, we double that to get 174, so we immediately write down that 4 (the rightmost digit of 174):
87 + 17 = 104, so now we put down that 104 to the left of the previous number, and now we have both complete answers:
See? With a number ending in .5, the multiple of 63 will always end in .5, but the multiple of 72 will always be an even number.
To familiarize yourself with multiplying 63 and 72 by all possible numbers you'll need to handle, you can use this link, which will occasionally generate random numbers ending in .5. Again, don't forget to check your answers for both 63 and 72 by using this link, and setting x to the appropriate given number.
When you've mastered these calculations, you're almost ready to handle the full mystery feat. The nature of this feat will be revealed in Thursday's post!
I'm going to take an unusual approach for my next several posts. I'm going to help you develop an unusual mathematical skill, but I'm going to save the nature of that skill as s surprise.
I'll start today by teaching you a particular math shortcut you'll need to develop. On Sunday, I'll teach you a similar, related math shortcut. In Next Thursday's post, I'll show you how to put these two simple shortcuts together in a way that gives you an impressive and surprising skill.
The first thing you'll need to learn is how to multiply by 63 quickly.
The first thing that will make this feat easier is that you're going to write parts of it down as you go. This will take pressure of your memory, as well as make the whole process more visual for your audience.
Before learning this skill, you should already be comfortable multiplying numbers up to 33 by 3 in your head. You should also be able to quickly multiply numbers up to 200 by 2 in your head, as well.
WHOLE NUMBERS: Here's the basic process, which will seem abstract at first. Don't worry, the practical examples that follow should help clear up any confusion. We start with a given number from 1 to 33:
1) Multiply the given number by 3.
2) Write down the ones digit of this answer on paper.
3) Take the amount you calculated in step 1, and double that number. Do not forget the number from step 1, however.
4) Take the number from step 3, and add ONLY the tens digit of the number from step 1 to it.
5) Write the total from step 4 on the paper, to the immediate left of the digit written in step 2.
If the references to the ones place and tens place confuse you, here's a quick refresher on place value.
Let's try running through this process with 12 as the given number.
The first step is to mentally multiply 12 by 3, which is 36. For the second step, we simply write down the number 6 on the paper, because the ones digit of 36 is 6.
In step 3, we're going to mentally double 36 to get 72, so now we're thinking of 36 AND 72, as we're not supposed to forget the number calculated in step 1.
Step 4 tells us to add the number from step 3, and add it to the tens digit of the number from step 1. In our example, the tens digit of 36 is 3, so we add 72 (the number calculated in step 3) plus 3 (the tens digit of 36) to get 75.
The final step is easy, we write that 75 down to the immediate left of the 6 we wrote down earlier. At this point, the answer should look like this: 756.
So, 12 × 63 should be 756, and Wolfram|Alpha verifies 756 as the correct answer!
The process does seem new and long, but it's not hard when you become familiar with it. As you practice, the steps will merge together, and you'll flow more easily from step to step.
Let's run through the process again, this time working out the answer to 17 × 63.
We start by mentally figuring 17 × 3 is 51, and we write down the 1 (the ones digit of 51). After figuring out the first answer like this, writing the ones digit down will quickly become habit.
51 doubled becomes 102, and then we add 102 (our newly doubled number) to 5 (the tens digit) to get 107. Write down this 107 to the immediate left of 1 already down on paper, and we have 1,071, which is the correct answer!
When writing down the ones digit, such as the 1 from 51 in the previous example, I like to imagine that the tens digit is also there, as if it's written in a special ink which only I can see. That way, when I double that number, I can look at the paper later and add this “invisible” number.
Eventually, the whole process should flow in a continuous, easy manner. Using 23 in the following example, I'll write it out as you should actually think of it in performance. The text in regular characters is what you would think, and the text in italics are actions:
“23 tripled is 69...*write down 9*...69 doubled is 138...138 plus 6 is 144...*write down 144 to the left*...DONE!”
At this point, you'd have 1449 on the paper, which is 23 × 63!
From the earlier examples, were you able to follow this “stream-of-conciousness” version? If not, go back and make sure all the steps are clear!
As I mentioned earlier, you want to become comfortable performing this with numbers up to 33. I suggest practicing by having Wolfram|Alpha generate random numbers with which to practice. You can then verify your answers with a calculator, or by having Wolfram|Alpha work out the answer (just change x to whatever number you used).
NUMBERS ENDING IN .5: To truly master the skill I'm eventually going to teach, you should also be able to handle numbers ending in .5. This might seem much more challenging, but it's really not.
Let's start with just the multiplication by 3. Take the number 10 and 10.5, for example. The former is easy to multiply by 3, as it's 30. 10.5 multiplied by 3 works out to 31.5. All you're really doing when there's a .5 involved is adding 1.5. A simple way to think of this is just to initially ignore the .5, and multiply the rest of the number by 3, then think of the next number followed by “.5”.
Let's try multiplying 12.5 by 3 mentally. Start by just multiplying 12 × 3 to get 36. The next number is 37, and we throw the .5 on the end to get 37.5! Get the idea?
In step 2, it says that you should only write the ones digit of the number down. This needs to be amended to include the possible .5 at the end. In our previous example, step 1 gave us 37.5, so this modified step 2 means you should write down 7.5 on the paper.
What is 17.5 × 3? 17 tripled is 51, the next number is 52 and .5, so it's 52.5, and then you'd write down 2.5.
Let's try multiplying 23.5 by 3 in a stream-of-consciousness approach similar to the 23 example we used earlier:
“23 tripled is 69...70...point 5...*write down 0.5*”
Step 3 is, of course, doubling the number. Fortunately, doubling any number which ends in .5 will always result in a whole number.
When doubling a number which ends in .5, just double the whole part, then mentally jump to the next whole number. What is 37.5 × 2? The whole part is 37, which doubled is 74. The next whole number is 75, so 75 is your answer!
How about 52.5? 52 doubled is 104, and the next whole number, and the answer, is 105! By now, doubling 70.5 should seem to be almost no problem at all. 70 doubled is 140, and the next whole number is 141.
Now, for the first time, let's try running through the full process starting with 28.5 as the given number. I'll run through it in the now-familiar stream-of-consciousness style:
“28.5...28 tripled is 84...85 point 5...*write down 5.5*...85 doubled is 170...171...171 plus 8 is 179...*write down 179 to the left*...DONE!”
What's written on the paper now is 1795.5, which is 28.5 × 63!
Once you've got the full process down, including how to handle numbers ending in .5, it's time to practice. Use this link to half Wolfram|Alpha generate random numbers from 0 to 33, which may or may not end in .5. You can check your answers with the same link used above, or a pocket calculator.
PART 1 WRAP-UP: Practice this skill between now and this coming Sunday, and then you'll be ready to handle the next mental math shortcut. The doubled number you calculate in step 3 will also be useful as this next shortcut's starting point, so you're already a little bit ahead of the game!
Perhaps out of necessity, science fiction authors tend to have a better grasp of math than your average fiction author.
I thought it would be fun to take a close look at some of the mathemtical understanding of sci-fi authors, so we can see what happens when these worlds collide.
I'll start with a rather amusing anecdote about John Taine, as posted over on futility closet:
Eric Temple Bell led two lives. By day he was a mathematician at Caltech; by night he wrote science fiction as John Taine.Speaking of math books, no less than sci-fi author Isaac Asimov wrote a book on mental calculation titled Quick & Easy Math. It's out of print now, but used copies are still available.
By a happy chance the two personalities met in 1951, when the Pasadena Star-News asked Taine to review Bell’s book Mathematics, Queen and Servant of Science.
Not one to lose an opportunity, he accepted. “The last flap of the jacket says Bell ‘is perhaps mathematics’ greatest interpreter,’” Taine wrote. “Knowing the author well, the reviewer agrees.”
Popular Science magazine published a condensed version of that book in its December 1964 issue, which is available online for free now, courtesy of Google books. It runs from pages 77-83, and cover mainly the basic arithmetic operations. The style is very conversational, so the concepts are easy to grasp.
Even when sci-fi authors aren't writing math books, they'll try and sneak in math lessons into their sci-fi works themselves. For example, Arthur C. Clarke snuck an unusual lesson about primes in his book The Garden of Rama. The formula's meaning and importance is explained in the following numberphile video:
If you enjoy sci-fi suthors and their taste of math, the best place to indulge yourself is OMNI magazine, and you can find its full run online for free, courtesy of the Internet Archive. Between their stories, their Games column, and the whole futuristic attitude of the magazine, it's easy to get lost in these pages.
Have fun, and enjoy exploring these resources!