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.

76.—­THE BARREL OF BEER.

A man bought an odd lot of wine in barrels and one barrel containing beer.  These are shown in the illustration, marked with the number of gallons that each barrel contained.  He sold a quantity of the wine to one man and twice the quantity to another, but kept the beer to himself.  The puzzle is to point out which barrel contains beer.  Can you say which one it is?  Of course, the man sold the barrels just as he bought them, without manipulating in any way the contents.