## THE PASSENGER'S FARE.

(

Money Puzzles)

At first sight you would hardly think there was matter for dispute in

the question involved in the following little incident, yet it took the

two persons concerned some little time to come to an agreement. Mr.

Smithers hired a motor-car to take him from Addleford to Clinkerville

and back again for L3. At Bakenham, just midway, he picked up an

acquaintance, Mr. Tompkins, and agreed to take him on to Clinkerville

and bring him back to Bakenham on the return journey. How much should he

have charged the passenger? That is the question. What was a reasonable

fare for Mr. Tompkins?

DIGITAL PUZZLES.

"Nine worthies were they called."

DRYDEN: _The Flower and the Leaf._

I give these puzzles, dealing with the nine digits, a class to

themselves, because I have always thought that they deserve more

consideration than they usually receive. Beyond the mere trick of

"casting out nines," very little seems to be generally known of the laws

involved in these problems, and yet an acquaintance with the properties

of the digits often supplies, among other uses, a certain number of

arithmetical checks that are of real value in the saving of labour. Let

me give just one example--the first that occurs to me.

If the reader were required to determine whether or not

15,763,530,163,289 is a square number, how would he proceed? If the

number had ended with a 2, 3, 7, or 8 in the digits place, of course he

would know that it could not be a square, but there is nothing in its

apparent form to prevent its being one. I suspect that in such a case he

would set to work, with a sigh or a groan, at the laborious task of

extracting the square root. Yet if he had given a little attention to

the study of the digital properties of numbers, he would settle the

question in this simple way. The sum of the digits is 59, the sum of

which is 14, the sum of which is 5 (which I call the "digital root"),

and therefore I know that the number cannot be a square, and for this

reason. The digital root of successive square numbers from 1 upwards is

always 1, 4, 7, or 9, and can never be anything else. In fact, the

series, 1, 4, 9, 7, 7, 9, 4, 1, 9, is repeated into infinity. The

analogous series for triangular numbers is 1, 3, 6, 1, 6, 3, 1, 9, 9. So

here we have a similar negative check, for a number cannot be triangular

(that is, (n squared + n)/2) if its digital root be 2, 4, 5, 7, or 8.

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