Every one is familiar with the difficulties that frequently arise over the giving of change, and how the assistance of a third person with a few coins in his pocket will sometimes help us to set the matter right.  Here is an example.  An Englishman went into a shop in New York and bought goods at a cost of thirty-four cents.  The only money he had was a dollar, a three-cent piece, and a two-cent piece.  The tradesman had only a half-dollar and a quarter-dollar.  But another customer happened to be present, and when asked to help produced two dimes, a five-cent piece, a two-cent piece, and a one-cent piece.  How did the tradesman manage to give change?  For the benefit of those readers who are not familiar with the American coinage, it is only necessary to say that a dollar is a hundred cents and a dime ten cents.  A puzzle of this kind should rarely cause any difficulty if attacked in a proper manner.

28.—­Defective observation.

Our observation of little things is frequently defective, and our memories very liable to lapse.  A certain judge recently remarked in a case that he had no recollection whatever of putting the wedding-ring on his wife’s finger.  Can you correctly answer these questions without having the coins in sight?  On which side of a penny is the date given?  Some people are so unobservant that, although they are handling the coin nearly every day of their lives, they are at a loss to answer this simple question.  If I lay a penny flat on the table, how many other pennies can I place around it, every one also lying flat on the table, so that they all touch the first one?  The geometrician will, of course, give the answer at once, and not need to make any experiment.  He will also know that, since all circles are similar, the same answer will necessarily apply to any coin.  The next question is a most interesting one to ask a company, each person writing down his answer on a slip of paper, so that no one shall be helped by the answers of others.  What is the greatest number of three-penny-pieces that may be laid flat on the surface of a half-crown, so that no piece lies on another or overlaps the surface of the half-crown?  It is amazing what a variety of different answers one gets to this question.  Very few people will be found to give the correct number.  Of course the answer must be given without looking at the coins.

29.—­The broken coins.

A man had three coins—­a sovereign, a shilling, and a penny—­and he found that exactly the same fraction of each coin had been broken away.  Now, assuming that the original intrinsic value of these coins was the same as their nominal value—­that is, that the sovereign was worth a pound, the shilling worth a shilling, and the penny worth a penny—­what proportion of each coin has been lost if the value of the three remaining fragments is exactly one pound?

30.—­Two questions in probabilities.