124.—­A QUESTION OF DEFINITION.

“My property is exactly a mile square,” said one landowner to another.

“Curiously enough, mine is a square mile,” was the reply.

“Then there is no difference?”

Is this last statement correct?

125.—­THE MINERS’ HOLIDAY.

Seven coal-miners took a holiday at the seaside during a big strike.  Six of the party spent exactly half a sovereign each, but Bill Harris was more extravagant.  Bill spent three shillings more than the average of the party.  What was the actual amount of Bill’s expenditure?

126.—­SIMPLE MULTIPLICATION.

If we number six cards 1, 2, 4, 5, 7, and 8, and arrange them on the table in this order:—­

    1 4 2 8 5 7

We can demonstrate that in order to multiply by 3 all that is necessary is to remove the 1 to the other end of the row, and the thing is done.  The answer is 428571.  Can you find a number that, when multiplied by 3 and divided by 2, the answer will be the same as if we removed the first card (which in this case is to be a 3) From the beginning of the row to the end?

127.—­SIMPLE DIVISION.

Sometimes a very simple question in elementary arithmetic will cause a good deal of perplexity.  For example, I want to divide the four numbers, 701, 1,059, 1,417, and 2,312, by the largest number possible that will leave the same remainder in every case.  How am I to set to work Of course, by a laborious system of trial one can in time discover the answer, but there is quite a simple method of doing it if you can only find it.

128.—­A PROBLEM IN SQUARES.

We possess three square boards.  The surface of the first contains five square feet more than the second, and the second contains five square feet more than the third.  Can you give exact measurements for the sides of the boards?  If you can solve this little puzzle, then try to find three squares in arithmetical progression, with a common difference of 7 and also of 13.

129.—­THE BATTLE OF HASTINGS.

All historians know that there is a great deal of mystery and uncertainty concerning the details of the ever-memorable battle on that fatal day, October 14, 1066.  My puzzle deals with a curious passage in an ancient monkish chronicle that may never receive the attention that it deserves, and if I am unable to vouch for the authenticity of the document it will none the less serve to furnish us with a problem that can hardly fail to interest those of my readers who have arithmetical predilections.  Here is the passage in question.

“The men of Harold stood well together, as their wont was, and formed sixty and one squares, with a like number of men in every square thereof, and woe to the hardy Norman who ventured to enter their redoubts; for a single blow of a Saxon war-hatchet would break his lance and cut through his coat of mail....  When Harold threw himself into the fray the Saxons were one mighty square of men, shouting the battle-cries, ‘Ut!’ ‘Olicrosse!’ ‘Godemite!’”