by Dudeney, Henry Ernest, 1857-1930
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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.? solution
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
(of) 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 c/?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 p 9 151 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? solution
Without the hint that I gave, my readers would probably have been unanimous in deciding that Mr. Perkins's income must have been £1,710. 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 tv\o years that amount has been spent. The only possible answer, according to the exact reading of her words, is, therefore, that his income was £180 per annum. Thus the amount spent in two years, during which his income has amounted to £360, will be £60 in rent, etc., £90 in domestic expenses, £20 in other ways, leaving the balance of £190 in the bank as stated.
32.?THE EXCURSION TICKET PUZZLE.? solution
Nineteen shillings and ninepence maybe 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 (fl+1 ) 2 ways where n always represents the number of pence. If threepenny-pieces are admitted, there are
2n 3 +15n 2 +33n
- +1
18 ways. If sixpences are also used there are
fl 4 +22n 3 +1 59a? 2 +41 4n+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 Oway.
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 maybe 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.? solution
(i)£13. (2) £23,19s. 11c/. The words "the number of pounds exceeds that of the pence" exclude such sums of money as £2,16s. 2d. and all sums under £1.
34.?THE GROCER AND DRAPER.? solution
