手机vr是什么:趣闻数学，答对了！

Arthur T. Benjamin has two passions: magic and math. When not amazing audiences around the country — squaring five-digit numbers in his head or guessing your number, any number — the Mathemagician is a professor of math at Harvey Mudd College in Claremont, Calif. There’s even whimsy to his Ph.D. dissertation, at Johns Hopkins, titled “Turnpike Structures for Optimal Maneuvers”: the maneuvers were inspired by a way of arranging Chinese checkers to move expeditiously across the board. Below, Dr. Benjamin shares some of the concepts from his DVD course “The Joy of Mathematics” and his book “Secrets of Mental Math: The Mathemagician’s Guide to Lightning Calculation and Amazing Math Tricks.”

亚瑟是哈维玛德学院的数学老师，平生两爱好：魔术，数学。一个五位数的平方可以心算得出来，你心里想的任何数字可以给你猜出来。亚瑟出过一张叫《趣味数学》的光盘，还有一本书叫《心算的秘密》，这里截选一部分内容。

Follow the instructions below. I bet I can predict your answers. How do I do it? Explanations follow.

下面每道题我都保证能猜出各位的答案。如何猜到的？看随后的解析。

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1. Choose a number from 1 to 70 and then divide it by 7. (I’ll be nice and let you use a calculator, but you’ll need one that has at least seven decimal places.) If your total is a whole number (that is, no digits after the decimal point) divide the answer by 7 again. Is there a 1 somewhere after the decimal point? I predict that the number after the 1 is 4. Am I right? Now add up the first six digits after the decimal point.

1.

从1到70中任选一个数，除以7，如得到整数，再除以7。小数点后的数字中是不是有1？我猜1后面是4，对不对？现在把小数点后的前六位数字相加。

Your answer is 27.

How do I do it?

The six fractions 1/7 to 6/7 have the same repeating sequence of six numbers after the decimal point, each fraction starting with a different number in the sequence 142857. Think of these numbers as in a circle (Diagram A), and going around that circle forever.

1/7,2/7,3/7,4/7,5/7,6/7这六个分数的小数点后六位数字是相同的，都是142857，而排列顺序不同。请看A表，这是个无限循环数列，不同的分数，小数点后的首位数字也不同。

The first number after the decimal point in the decimal version of 1/7 is 1 (.142857); of 2/7, 2 (.285714); of 3/7, 4 (.428571); of 4/7, 5 (.571428); of 5/7, 7 (.714285); and of 6/7, 8 (.857142). Adding 1, 4, 2, 8, 5 and 7, you always get 27.

1/7小数点后首位是1（.142857）；2/7小数点后首位是2（.285714）；3/7小数点后首位是4（.428571）；4/7小数点后首位是5（.571428）；5/7小数点后首位是7（.714285)；6/7小数点后首位是8（.857142）。1加4加2加8加5加7，和为27。

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2. Choose your favorite number from 1 to 100. Double it, then add 5. Now multiply by 50, then add 1,759. If you have had your birthday already in 2010: happy birthday and add 1 to the total. Subtract the year you were born.

2.

从1到100中选一个最喜爱的数，乘以2，加上5，乘以50，加上1760。如果你2011年的生日已经过了，生日快乐！得出的数再加上1吧，然后减去你的出生年份。

Your answer begins with your favorite number, followed by your age.

How do I do it?

As with most of this mathematical magic, the secret is elementary algebra. Suppose your favorite number is x. Double it, for 2x; add 5, for 2x + 5; then multiply by 50, for 100x + 250. Adding 1,759 gets you 100x + 2009. Subtracting the year you were born (and adding the number 1 if you had your birthday already this year) will produce your favorite number followed by your age.

Note: This trick will fail if you are more than 99 years old.

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3. Choose a number from 1 to 20. Double it, add 10, divide by 2, and then subtract the number you started with. (Based on what you learned with Problem 2, you should be able to predict this number along with me.)

3.

从1到20中任选一个数，乘以2，加上10，除以2，减去你开头选的数。

Your number is 5.

How do I do it?

After doubling x, you have 2x; add 10 and it becomes 2x + 10. Then divide by 2 and it becomes x + 5. After subtracting the original number x, you have 5.

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4. Use any A.T.M. card or credit card with 16 digits. Write the number in a grid of boxes like the one shown in Diagram B, in alternating fashion. For example, if your card number is 3141592653589793, then your numbers would be 34525599 and 11963873.

4.

随便拿一张信用卡或储蓄卡。在B表里填写卡上的号码，顺序随机。比如卡号是3141592653589793，你可以写成34525599和11963873。

Add up the digits of the top number, then double it. Write that number down.

Add up the digits of the bottom number. Write that number down.

How many of the digits of the top number are 5 or greater? Write that down.

Now add the three numbers you wrote down, and look at the last digit of your answer.

Is the last digit 0?

How do I do it?

There’s no magic here. This is the Luhn system used for credit and A.T.M. cards. The steps you just took are what happens automatically when you use your card. A wrong digit or transposing nearly any two consecutive digits in your 16-digit card number can be detected, without using a database, because the final digit of the total won’t add up to 0. See for yourself. Do the math and your last digit will be 0. Change any number on your card and it won’t.

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5. Choose a four-digit number, with each digit different. Scramble the digits to create a new number. (For instance, if your first number is 2,357, your second number might be 7,325.) Subtract the smaller number from the larger, and add together the digits of the resulting number. If your sum is one digit, go no further. If your sum is a two-digit number, then add the two digits together.

5.

选一个四位数，其中四个数字各不相同，然后打乱顺序生成一个新数，（如2357，打乱后成7325）用大数减去小数，把这个新数的各数字相加。结果如是个位数，到此为止；结果如是两位数，再把它的十位数与个位数相加。

Your number is 9.

How do I do it?

Here’s a two-part explanation for how this trick works, thanks to the amazing properties of 9.

First, when you subtract the smaller number from the larger one, the result is always a multiple of 9. This happens because, when you divide any number and any scrambled variation of it by 9, the remainder — what’s left over after dividing — is always the same. (You'll have to trust us on why. It's based on "casting out nines.") For example, 28 divided by 9 is 3 with a remainder of 1. Swapping the digits, for 82, and dividing by 9 gives 9, and again the remainder is 1.

首先，大数减去小数得出的新数永远是9的倍数。根据“舍九检验”，任何一个数除以9得出的余数，与打乱后的新数除以9得出的余数永远相同。例如，28除以9等于3，余数为1，打乱后的新数82除以9等于9，余数为1。

Hence, when subtracting the smaller number from the larger one, the identical remainders will cancel each other out, and the result is — poof! — a multiple of 9.

Algebraically, what you have is (9x + r) - (9y + r) = 9(x - y).

Second, every multiple of 9 (9, 18, 27, 36, etc.) has the property that its digits will always add up to a multiple of 9. For example, the number 4,968 is a multiple of 9 (552 × 9), and its digits add up to 27 (3 × 9).

其次，(9,18,27,36,etc.)每一个9的倍数皆符合定理：各数位的数字相加永远是9的倍数。如9的552倍是4968，它各数位的数字相加等于27，是9的3倍。

Because of how the problem is stated, with four-digit numbers, the sum of the digits will always be 9 or 18 or 27. Adding 1 + 8 or adding 2 + 7 (as specified in the problem) again gives 9.

这道题的限制条件是一个四位数，所以各数位的数字相加永远是9或18或27。而1加8等于9，2加7也等于9。

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6. In the highlighted grid on the calendar (Diagram C), circle four dates so there is one circled in each row and each column. Add the four circled numbers.

6.

看C表中的高亮部分，从中挑4天出来，要保证每排每列都有数字被选到。四个数相加。

Your total is 64.

How do I do it?

Look at the Sunday dates, to the left of the grid (Diagram E).

Now think of the numbers for Monday through Thursday as simply those Sunday numbers plus 1, 2, 3 or 4.

In choosing four numbers from different rows and columns, you are just adding the Sunday numbers (54) plus 1 + 2 + 3 + 4 (10), which equals 64.

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7. In a table like the one shown in Diagram D, write any number in Rows 1 and 2. Add those numbers, and put the total in Row 3. Add the numbers in Rows 2 and 3, and put the answer in Row 4. Continue this process until you have numbers in all 10 rows. Now add up the 10 numbers and divide by the number in Row 7.

7.

在D表的第一排和第二排写下任意两个数，两数之和写到第三排，第二排和第三排的数字之和写到第四排，以此类推，然后把10个数字相加，除以第七排的数。

Your answer is 11.

How do I do it?

Suppose you start with the number x in Row 1 and number y in Row 2. Then Row 3 will be x + y, which leads to Row 4 being x + 2y, and so on. Diagram F shows what the final table looks like.

假设第一排的数是x，第二排是y，所以第三排是(x+y)，第四排(x+2y)，以此类推。结果请看F表。

The grand total is 11 times 5x + 8y. Note that 5x + 8y is in Row 7. Hence, dividing the total by what’s in Row 7 will always yield the answer 11.

10个数字之和是(55x+88y)，即11*(5x+8y)，而（5x+8y)是第七排的数。所以总数除以第七排的数等于11。

Tip: To look like a human calculator, ask somebody to write numbers in the table, as before. When he shows you the list, make a great show of “mind reading” and challenge him to add all the numbers with a calculator faster than you can in your head. Just multiply Row 7 by 11.

提示：想当超级心算王吗？请你的对手按以上方式写满10排数目，然后他用计算器，你用超级心算，看谁能更快算出10个数的总和。你只须用第七排的数乘以11，帅吧？

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10. Without a calculator, determine 2,317 divided by 25. (Hint: multiply, don’t divide.) Compute to two decimal places.

8.

Because 1/25 is 4/100, to divide any number by 25 you simply multiply it by 4, then divide by 100. Here, we multiply 2,317 by 4 (getting 9,268), then divide by 100 to get 92.68.