Unit 5 Sum Like It Not

Do you really need your computer (to do mathematics, I mean) or would you be better off settling down with paper and pencil and a pot of strong coffee? The question is prompted by the flood of mail I receive whenever this column includes problems for readers to tackle for themselves.

Inevitably a large proportion of correspondents say they were able to solve the problems without recourse to any technology higher than, perhaps, a coffee-maker. Indeed, some go so far as to suggest it might be quicker to adopt a paper-and-pencil approach. (Usually I am in complete agreement, though I would never dream of admitting this in a column entitled Micromaths.)

One drawback with any new technology is the tendency to overuse it - especially if that technology represents a considerable financial investment. But even when a computer is necessary to solve a parti­cular problem, some initial thought (before sitting at the keyboard) will often result in a quicker, better solution when the time comes to hit the RUN button. Indeed, in many cases success depends on first giving careful thought to how your program should be written.

A spectacular illustration is provided by testing a given whole number to see if it is prime (i.e. cannot be exactly divided by any smaller number other than 1.) If you try to do this by the obvious method of checking all possible smaller numbers one at a time, even the fastest computer in the world (one which can perform 400 million arithmetical operations in a single second on numbers with 50 digits or more) could take up to a billion centuries to come up with the answer for a 50 digit number.

Smaller prime numbers still take about the same length of time. But by using some clever mathematics the same computer could come up with the result in under 15 seconds, such is the power of mathematics.

The difficulty with primarily testing is that the mathematics needed to produce the fast computer program is extremely deep, and only accessible to the expert. On a more mundane level, however, there is a marvelous little problem due to the English mathematician D. E. Littlewood. This asks for the smallest number N with the property that if you shift the first digit in N to the end, the resulting number is exactly half as big again as N. This could clearly be attacked using a microcomputer, but if you try it you are unlikely to succeed. Far better to use a little ingenuity instead.

1) Translate the following sentences or parts of the sentences.

1. ...to solve the problems without recourse to any technology higher .than, perhaps, a coffee-maker.

2. ...it might be quicker to adopt a paper-and-pencil approach.

3. ...some initial thought will often result in a quicker, better solu­tion...

4. It cannot be divided by any smaller number...

5. Even the fastest computer in the world...

6. On a more mundane level, however, there is a marvelous little problem...

7. This asks for the smallest number N with the property that if you shift the first digit in N to the end, the resulting number is exactly half as big again as N.

8. Far better to use a little ingenuity instead.

2) Answer the following questions.

1. Do you need a computer to solve difficult mathematical problems?

2. Can you adopt a paper-and-pencil approach?

3. What drawback with new technology does Keith Devlin speak about?

4. What does success of solving mathematical problems with computer depend on?

5. What illustration does he give?

6. What column of the paper does Keith Devlin write in?

7. Does he usually get many letters?

3) Keith Devlin gives the following two problems for those of the readers who really want to, use a micro. Imagine you want to. Let him know you get on. Write a. letter to him using the information given after the problems.

(1) How many pairs of two-digit numbers can you find with the property that their product consists of the same four digits in some order or other? For example, 21 multiplied by 87 gives 1827. In each case, all four digits should be different.

(2) The number 51,249,876 is made up of all the digits from 1 to 9 except 3. If you multiply this number by the missing 3 the result is 153,749,628, which consists of all nine digits (each used only once). How many other examples can you find of this phenomenon ? First look for eight-digit numbers multiplied by their missing digit. Then seven-digit numbers multiplied by the missing two in some order. And so on. And the best of luck!

Address

Keith Devlin

Computer Guardian    

Date

Dear Sir

I live   

Explain who you are

1 I am writing in connection with   and why you are writing

2 Describe your problems (how you have solved problems)

Yours sincerely

Name

4) Translate the following advertisement.