Of course the date on a penny is on the same side as Britannia—­the “tail” side.  Six pennies may be laid around another penny, all flat on the table, so that every one of them touches the central one.  The number of threepenny-pieces that may be laid on the surface of a half-crown, so that no piece lies on another or overlaps the edge of the half-crown, is one.  A second threepenny-piece will overlap the edge of the larger coin.  Few people guess fewer than three, and many persons give an absurdly high number.

29.—­THE BROKEN COINS.

If the three broken coins when perfect were worth 253 pence, and are now in their broken condition worth 240 pence, it should be obvious that 13/253 of the original value has been lost.  And as the same fraction of each coin has been broken away, each coin has lost 13/253 of its original bulk.

30.—­TWO QUESTIONS IN PROBABILITIES.

In tossing with the five pennies all at the same time, it is obvious that there are 32 different ways in which the coins may fall, because the first coin may fall in either of two ways, then the second coin may also fall in either of two ways, and so on.  Therefore five 2’s multiplied together make 32.  Now, how are these 32 ways made up?  Here they are:—­

(a) 5 heads 1 way (b) 5 tails 1 way (c) 4 heads and 1 tail 5 ways (d) 4 tails and 1 head 5 ways (e) 3 heads and 2 tails 10 ways (f) 3 tails and 2 heads 10 ways

Now, it will be seen that the only favourable cases are a, b, c, and d—­12 cases.  The remaining 20 cases are unfavourable, because they do not give at least four heads or four tails.  Therefore the chances are only 12 to 20 in your favour, or (which is the same thing) 3 to 5.  Put another way, you have only 3 chances out of 8.

The amount that should be paid for a draw from the bag that contains three sovereigns and one shilling is 15s. 3d.  Many persons will say that, as one’s chances of drawing a sovereign were 3 out of 4, one should pay three-fourths of a pound, or 15s., overlooking the fact that one must draw at least a shilling—­there being no blanks.

31.—­DOMESTIC ECONOMY.

Without the hint that I gave, my readers would probably have been unanimous in deciding that Mr. Perkins’s income must have been L1,710.  But this is quite wrong.  Mrs. Perkins says, “We have spent a third of his yearly income in rent,” etc., etc.—­that is, in two years they have spent an amount in rent, etc., equal to one-third of his yearly income.  Note that she does not say that they have spent each year this sum, whatever it is, but that during the two years that amount has been spent.  The only possible answer, according to the exact reading of her words, is, therefore, that his income was L180 per annum.  Thus the amount spent in two years, during which his income has amounted to L360, will be L60 in rent, etc., L90 in domestic expenses, L20 in other ways, leaving the balance of L190 in the bank as stated.