Part 65

26.--THE JUNIOR CLERKS' PUZZLE.

Although Snoggs's reason for wishing to take his rise at L2, 10s. half-yearly did not concern our puzzle, the fact that he was duping his employer into paying him more than was intended did concern it. Many readers will be surprised to find that, although Moggs only received L350 in five years, the artful Snoggs actually obtained L362, 10s. in the same time. The rest is simplicity itself. It is evident that if Moggs saved L87, 10s. and Snoggs L181, 5s., the latter would be saving twice as great a proportion of his salary as the former (namely, one-half as against one-quarter), and the two sums added together make L268, 15s.

27.--GIVING CHANGE.

The way to help the American tradesman out of his dilemma is this. Describing the coins by the number of cents that they represent, the tradesman puts on the counter 50 and 25; the buyer puts down 100, 3, and 2; the stranger adds his 10, 10, 5, 2, and 1. Now, considering that the cost of the purchase amounted to 34 cents, it is clear that out of this pooled money the tradesman has to receive 109, the buyer 71, and the stranger his 28 cents. Therefore it is obvious at a glance that the 100-piece must go to the tradesman, and it then follows that the 50-piece must go to the buyer, and then the 25-piece can only go to the stranger. Another glance will now make it clear that the two 10-cent pieces must go to the buyer, because the tradesman now only wants 9 and the stranger 3. Then it becomes obvious that the buyer must take the 1 cent, that the stranger must take the 3 cents, and the tradesman the 5, 2, and 2. To sum up, the tradesman takes 100, 5, 2, and 2; the buyer, 50, 10, 10, and 1; the stranger, 25 and 3. It will be seen that not one of the three persons retains any one of his own coins.

28.--DEFECTIVE OBSERVATION.

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.

32.--THE EXCURSION TICKET PUZZLE.

Nineteen shillings and ninepence may be paid in 458,908,622 different ways.

I do not propose to give my method of solution. Any such explanation would occupy an amount of space out of proportion to its interest or value. If I could give within reasonable limits a general solution for all money payments, I would strain a point to find room; but such a solution would be extremely complex and cumbersome, and I do not consider it worth the labour of working out.

Just to give an idea of what such a solution would involve, I will merely say that I find that, dealing only with those sums of money that are multiples of threepence, if we only use bronze coins any sum can be paid in (n + 1) squared ways where n always represents the number of pence. If threepenny-pieces are admitted, there are

2n cubed + 15n squared + 33n --------------------- + 1 ways. 18

If sixpences are also used there are

n^{4} + 22n cubed + 159n squared + 414n + 216 --------------------------------- 216

ways, when the sum is a multiple of sixpence, and the constant, 216, changes to 324 when the money is not such a multiple. And so the formulas increase in complexity in an accelerating ratio as we go on to the other coins.

I will, however, add an interesting little table of the possible ways of changing our current coins which I believe has never been given in a book before. Change may be given for a

Farthing in 0 way. Halfpenny in 1 way. Penny in 3 ways. Threepenny-piece in 16 ways. Sixpence in 66 ways. Shilling in 402 ways. Florin in 3,818 ways. Half-crown in 8,709 ways. Double florin in 60,239 ways. Crown in 166,651 ways. Half-sovereign in 6,261,622 ways. Sovereign in 500,291,833 ways.

It is a little surprising to find that a sovereign may be changed in over five hundred million different ways. But I have no doubt as to the correctness of my figures.

33.--A PUZZLE IN REVERSALS.

(i) L13. (2) L23, 19s. 11d. The words "the number of pounds exceeds that of the pence" exclude such sums of money as L2, 16s. 2d. and all sums under L1.