2 • Old Wine in a Not-so-new Bottle
A bottle of wine cost ten dollars. The wine was worth nine
dollars more than the bottle. How much was the bottle
worth?
3 • How Much Profit?
The amazing thing about this puzzle is that people always
seem to fight over the answer! Yes, different people work it
out in different ways and come up with different answers,
and each insists his answer is correct. The puzzle is this:
A dealer bought an article for $7, sold it for $8, bought it
back for $9, and sold it for $10. How much profit did he
make?
.
.
.
'
'
SOLUTIONS:
1 • A common wrong answer is $10. Now, suppose we each
had, say, $50. If I gave you $10, you would then have $60 and
I would have only $40; hence you would have
· $20 more than
I, rather than $10.
The correct answer is $5.
2 • A common wrong answer is $1. Now, if the bottle were
really worth a dollar, then the wine, being worth $9 more
than the bottle, would be worth $10. Hence the wine and
bottle together would be worth $11. The correct answer is
that the bottle is worth 50¢ and the wine is worth $9.50.
Then the two add up to $10.
3 • Some argue as follows: After having bought the article
for $7 and sold it for $8, he has made a dollar profit. Then, by
buying the article back for $9 after having sold it for $8, he
loses a dollar; hence at this point he is even. But then (the argument
continues) by selling for $10 what he has bought for
$9, he has made a dollar again; therefore, his total profit is $1.
Another argument leads to the conclusion that the dealer
broke even: When he sold the article for $8 after having
bought it for $7, he made $1 profit. But then he loses $2 by
buying back for $9 the item for which he originally pai d $7,
an d so at this point he is $1 in the hole. Then he g ets back the
dollar by selling for $10 the article for which he last paid $9,
and so now he is even.
Both arguments are wrong; the correct answer is that the
dealer ma de $2. There are several ways to arrive at this--one
such goes as follows: First, after selling for $8 the article for
which he has paid $7, he has clearly made $1 . Now, suppose
that instead of buying back the same article for $9 and ,then
selling it for $10, he were to buy a different article for $9 and
sell it for $10. Woul d this really be any different from a
purely economic point of view? Of course not! He would obviously,
then, be making another dollar on the buying and
selling of this second article. Thus, he has made $2.
Another and ve ry simple proof: The de aler's total outlay is
$7 + $9 = $16, an d his total return is $8 + $10 = $18, giving
a profit of $2.
For those not convinced by these arguments, let us suppose
that the dealer has a certain amount of money -say, $10 0-
at the opening o f the day and that he makes just these four
deals. How much will he have at the close of the day? Well,
first he pays $7 for the article, leaving him with $93. Then he
sells the article for $8, giving him $10 1 . Next he buys the article
back for $9, bringing him down to $92. Finally, he sells
the article for $10 and thus winds up with $102. So he has
starte d the day with $100 an d en de d it with $102. How, then,
coul d his profit be anything other than $2?
From the book:
THE LADY
OR THE TIGER?
and Other Logic Puzzles
INCLUDING A MATHEMATICAL NOVEL
THAT FEATURES GODEL'S
GREAT DISCOVERY
BY
Raymond Smullyan
