Secrets of Mental Math - Arthur Benjamin (bill gates best books .TXT) 📗

5) = 89

Step 5. 8 + (8 × 7) = 64

Answer: 649,986

Notice that the number of multiplications at each step is 1, 2, 3, 2, and 1 respectively. The mathematics underlying the criss-cross method is nothing more that the distributive law. For instance, 853 × 762 = (800 + 50 + 3) × (700 + 60 + 2) = (3 × 2) + [(5 × 2) + (3 × 6)] × 10 + [(8 × 2) + (5 × 6) + (3 × 7)] × 100 + [(8 × 6) + (5 × 7)] × 1000 + (8 × 7) × 10,000, which are precisely the calculations of the criss-cross method.

You can check your answer with the mod sum method by multiplying the mod sums of the two numbers and computing the resulting number’s mod sum. Compare this number to the mod sum of the answer. If your answer is correct, they must match. For example:

If the mod sums don’t match, you made a mistake. This method will detect mistakes, on average, about 8 times out of 9.

In the case of 3-by-2 multiplication problems, the procedure is the same except you treat the hundreds digit of the second number as a 0:

Step 1. 7 × 6 = 42

Step 2. 4 + (7 × 4) + (3 × 6) = 50

Step 3. 5 + (7 × 8) + (0 × 6) + (3 × 4) = 73

Step 4. 7 + (3 × 8) + (0 × 4) = 31

Step 5. 3 + (0 × 8) = 3

Answer: 31,302

Of course, you would normally just ignore the multiplication by zero, in practice.

You can use the criss-cross to do any size multiplication problem. To answer the 5-by-5 problem below will require nine steps. The number of multiplications in each step is 1, 2, 3, 4, 5, 4, 3, 2, 1, for twenty-five multiplications altogether.

Step 1. 9 × 7= 63

Step 2. 6 + (9 × 6) + (4 × 7) = 88

Step 3. 8 + (9 × 8) + (0 × 7) + (4 × 6) = 104

Step 4. 10 + (9 × 2) + (2 × 7) + (4 × 8) + (0 × 6) = 74

Step 5. 7 + (9 × 4) + (5 × 7) + (4 × 2) + (2 × 6) + (0 × 8) = 98

Step 6. 9 + (4 × 4) + (5 × 6) + (0 × 2) + (2 × 8) = 71

Step 7. 7 + (0 × 4) + (5 × 8) + (2 × 2) = 51

Step 8. 5 + (2 × 4) + (5 × 2) = 23

Step 9. (5 × 4) + 2 = 22

Answer: 2,231,184,483

You can check your answer by using the mod sums method.

CASTING OUT ELEVENS

To double-check your answer another way, you can use the method known as casting out elevens. It’s similar to casting out nines, except you reduce the numbers by alternately subtracting and adding the digits from right to left, ignoring any decimal point. If the result is negative, then add eleven to it. (It may be tempting to do the addition and subtraction from left to right, as you do with mod sums, but in this case you must do it from right to left for it to work.) For example:

The same method works for subtraction problems:

Shakuntala Devi: That’s Incalculable!

In 1976 the New York Times reported that an Indian woman named Shakuntala Devi (b. 1939) added 25,842 + 111,201,721 + 370,247,830 + 55,511,315, and then multiplied that sum by 9,878, for a correct total of 5,559,369,456,432, all in less than twenty seconds. Hard to believe, though this uneducated daughter of impoverished parents has made a name for herself in the United States and Europe as a lightning calculator.

Unfortunately, most of Devi’s truly amazing feats not done by obvious “tricks of the trade” are poorly documented. Her greatest claimed accomplishment—the timed multiplication of two thirteen-digit numbers on paper—has appeared in the Guinness Book of World Records as an example of a “Human Computer.” The time of the calculation, however, is questionable at best. Devi, a master of the criss-cross method, reportedly multiplied 7,686,369,774,870 × 2,465,099,745,799, numbers randomly generated at the computer department of Imperial College, London, on June 18, 1980. Her correct answer of 18,947,668,177,995,426,773,730 was allegedly generated in an incredible twenty seconds. Guinness offers this disclaimer: “Some eminent mathematical writers have questioned the conditions under which this was apparently achieved and predict that it would be impossible for her to replicate such a feat under highly rigorous surveillance.” Since she had to calculate 169 multiplication problems and 167 addition problems, for a total of 336 operations, she would have had to do each calculation in under a tenth of a second, with no mistakes, taking the time to write down all 26 digits of the answer. Reaction time alone makes this record truly in the category of “that’s incalculable!”

Despite this, Devi has proven her abilities doing rapid calculation and has even written her own book on the subject.

It even works with multiplication:

If the numbers disagree, you made a mistake somewhere. But if they match, it’s still possible for a mistake to exist. On average, this method will detect an error 10 times out of 11. Thus a mistake has a 1 in 11 chance of sneaking past the elevens check, a 1 in 9 chance of sneaking past the nines check, and only a 1 in 99 chance of being undetected if both checks are used. For more information about this and other fascinating mathemagical topics, I would encourage you to read any of Martin Gardner’s recreational math books.

You are now ready for the ultimate pencil-and-paper multiplication problem in the book, a 10-by-10! This has no practical value whatsoever except for showing off! (And personally I think multiplying five-digit numbers is impressive enough since the answer will be at the capacity of most calculators.) We present it here just to prove that it can be done. The criss-crosses follow the same basic pattern as that in the 5-by-5 problem. There will be nineteen computation steps and at the tenth step there will be ten criss-crosses! Here you go:

Here’s how we do it:

Step 1.    6 × 1=6

Step 2.    (6 × 5) + (1 × 1) = 31

Step 3.    3 + (6 × 4) + (2 × 1) + (1 × 5) = 34

Step 4.    3 + (6 × 9) + (5 × 1) + (1 × 4) + (2 × 5) = 76

Step 5.    7 + (6 × 2) + (7 × 1) + (1 × 9) + (5 × 5) + (2 × 4) = 68

Step 6.    6 + (6 × 8) + (5 × 1) + (1 × 2) + (7 × 5) + (2 × 9) + (5 × 4) = 134

Step 7.    13 + (6 × 6) + (5 × 1) + (1 × 8) + (5 × 5) + (2 × 2) + (7 × 4) + (5 × 9) = 164

Step 8.    16 + (6 × 6) + (2 × 1) + (1 × 6) + (5 × 5) + (2 × 8) + (5 × 4) + (5 × 2) + (7 × 9) = 194

Step 9.    19 + (6 × 7) + (4 × 1) + (1 × 6) + (2 × 5) + (2 × 6) + (5 × 4) + (5 × 8) + (5 × 9) + (7 × 2) = 212

Step 10.  21 + (6 × 2) + (4 × 1) + (1 × 7) + (4 × 5) + (2 × 6) + (2 × 4) + (5 × 6) + (5 × 9) + (7 × 8) + (5 × 2) = 225

Step 11.  22 + (1 × 2) + (4 × 5) + (2 × 7) + (4 × 4) + (5 × 6) + (2 × 9) + (7 × 6) + (5 × 2) + (5 × 8) = 214

Step 12.  21 + (2 × 2) + (4 × 4) + (5 × 7) + (4 × 9) + (7 × 6) + (2 × 2) + (5 × 6) + (5 × 8) = 228

Step 13.  22 + (5 × 2) + (4 × 9) + (7 × 7) + (4 × 2) + (5 × 6) + (2 × 8) + (5 × 6) = 201

Step 14.  20 + (7 × 2) + (4 × 2) + (5 × 7) + (4 × 8) + (5 × 6) + (2 × 6) = 151

Step 15.  15 + (5 × 2) + (4 × 8) + (5 × 7) + (4 × 6) + (2 × 6) = 128

Step 16.  12 + (5 × 2) + (4 × 6) + (2 × 7) + (4 × 6) = 84

Step 17.  8 + (2 × 2) + (4 × 6) + (4 × 7) = 64

Step 18.  6 + (4 × 2) + (4 × 7) = 42

Step 19.  4 + (4 × 2) = 12

If you were able to negotiate this extremely difficult problem successfully the first time through, you are on the verge of graduating from apprentice to master mathemagician!

PENCIL-AND-PAPER MATHEMATICS EXERCISES

Sum the following column of numbers, checking your answer by reading the column from bottom to top, and then by the mod sums method. If the two mod sums do not match, recheck your addition:

Sum this column of dollars and cents:

Subtract the following numbers, using mod sums to check your answer and then by adding the bottom two numbers to get the top number:

For the following numbers, compute the exact square root using the doubling and dividing technique.

To wrap up this exercise session, use the criss-cross method to compute exact multiplication problems of any size. Place the answer below the problems on one line, from right to left. Check your answers using mod sums.

Chapter 7 A Memorable Chapter: Memorizing Numbers

Chapter 7

A Memorable Chapter:
Memorizing Numbers

One of the most common questions I am asked pertains to my memory. No, I don’t have an extraordinary memory. Rather, I apply a memory system that can be learned by anyone and is described in the following pages. In fact, experiments have shown that almost anyone of average intelligence can be taught to vastly improve their ability to memorize numbers.

In this chapter we’ll teach you a way to improve your memory for numbers. Not only does this have obvious practical benefits, such as remembering dates or recalling phone numbers, but it also allows the mathemagician to solve much larger problems mentally. In the next chapter we’ll show you how to apply the techniques of this chapter to multiply five-digit numbers in your head!

USING MNEMONICS

The method presented here is an example of a mnemonic—that is, a tool to improve memory encoding and retrieval. Mnemonics work by converting incomprehensible data (such as digit sequences) to something more meaningful. For example, take just a moment to memorize the sentence below:

“My turtle Pancho will, my love, pick up my new mover, Ginger.”

Read it over several times. Look away from the page and say it to yourself over and