THE COLORED SOCKS & WEIGHTY PROBLEM
THE COLORED SOCKS
TEN RED SOCKS and ten blue socks are all mixed up in
a dresser drawer. The twenty socks are exactly alike
except for their color. The room is in pitch darkness
and you want two matching socks. What is the smallest
number of socks you must take out of the drawer in
order to be certain that you have a pair that match?
SOLUTION
Many people, trying to solve this puzzle, say to
themselves, «Suppose the first sock that I remove is red.
I need another red one to match it, but the next sock
might be blue, and the next one, and the next one, and
so on until all ten blue socks are taken from the drawer.
The next sock has to be red, so the answer must be
twelve socks."
But something is overlooked in this reasoning. The
socks do not have to be a red pair. It is only necessary
that they match. If the first two fail to match, then the
third is sure to match one of the other two, so the correct
answer is three socks.
WEIGHTY PROBLEM
IF A BASKETBALL weighs 10% ounces plus hall its own
weight, how much does it weigh?
SOLUTION
Before answering this puzzle, it is necessary to
know exactly what the words mean. One might, for example,
approach it this way: «The basketball weighs
1012 ounces. Hall its weight would then be 51A ounces.
We add these values together to get an answer of 15%
ounces. "
But the problem is to find the weight of the basketball,
and if this turns out to be 15% ounces, then it
cannot also be 10% ounces as first assumed. There
clearly is a contradiction here, so we must have misinterpreted
the language of the question.
There is only one interpretation that makes sense.
The basketball's weight is equal to the sum of two
values: 1O~ ounces and an unknown value that is half
the basketball's weight. This can be pictured on a balance
scale as shown in the illustration on the opposite
page.
If half a basketball is taken from each side of the
scale, the pans will still balance. An lO~-ounce weight
will be on one side and half a basketball on the other,
so half a basketball must weigh 10Y2 Ol.mces and
the whole basketball must weigh twice this, or 21
ounces.
Actually, without know it, we have solved the
problem by simple algebra! Instead of pictures, let us
represent half a basketball by the letter x. And instead
of showing two sides of a scale in balance, let us use
the algebraic sign of equality. We can now write the
Simple equation:
lOY2 + x=x+ x
If the same amount is taken from each side of this
equation it will still "balance." So we remove x from
each side and are left with:
lOY2 =x
You remember that x represented half a basketball.
If half a basketball weighs 10Y2 ounces, then the entire
basketball must weigh 21 ounces.
TEN RED SOCKS and ten blue socks are all mixed up in
a dresser drawer. The twenty socks are exactly alike
except for their color. The room is in pitch darkness
and you want two matching socks. What is the smallest
number of socks you must take out of the drawer in
order to be certain that you have a pair that match?
SOLUTION
Many people, trying to solve this puzzle, say to
themselves, «Suppose the first sock that I remove is red.
I need another red one to match it, but the next sock
might be blue, and the next one, and the next one, and
so on until all ten blue socks are taken from the drawer.
The next sock has to be red, so the answer must be
twelve socks."
But something is overlooked in this reasoning. The
socks do not have to be a red pair. It is only necessary
that they match. If the first two fail to match, then the
third is sure to match one of the other two, so the correct
answer is three socks.
WEIGHTY PROBLEM
IF A BASKETBALL weighs 10% ounces plus hall its own
weight, how much does it weigh?
SOLUTION
Before answering this puzzle, it is necessary to
know exactly what the words mean. One might, for example,
approach it this way: «The basketball weighs
1012 ounces. Hall its weight would then be 51A ounces.
We add these values together to get an answer of 15%
ounces. "
But the problem is to find the weight of the basketball,
and if this turns out to be 15% ounces, then it
cannot also be 10% ounces as first assumed. There
clearly is a contradiction here, so we must have misinterpreted
the language of the question.
There is only one interpretation that makes sense.
The basketball's weight is equal to the sum of two
values: 1O~ ounces and an unknown value that is half
the basketball's weight. This can be pictured on a balance
scale as shown in the illustration on the opposite
page.
If half a basketball is taken from each side of the
scale, the pans will still balance. An lO~-ounce weight
will be on one side and half a basketball on the other,
so half a basketball must weigh 10Y2 Ol.mces and
the whole basketball must weigh twice this, or 21
ounces.
Actually, without know it, we have solved the
problem by simple algebra! Instead of pictures, let us
represent half a basketball by the letter x. And instead
of showing two sides of a scale in balance, let us use
the algebraic sign of equality. We can now write the
Simple equation:
lOY2 + x=x+ x
If the same amount is taken from each side of this
equation it will still "balance." So we remove x from
each side and are left with:
lOY2 =x
You remember that x represented half a basketball.
If half a basketball weighs 10Y2 ounces, then the entire
basketball must weigh 21 ounces.
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