Scale puzzles typically present you with a situation where you have a certain
number of genuine objects, with one or more fake (counterfeit) objects that
are visually identical to the genuine article, but weigh either more or less
than the real item. Sometimes you will know if the fake is heavier or lighter.
Other times, you will have to figure that detail out, also.
Skip the rest of the lecture and scroll down to the
scale puzzles
Basically, there are two kinds of scales.
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A bathroom scale, a truck scale, and a postage scale are all examples of
the kind of scale that tells you how much something weighs. You stand on
a bathroom scale, and you see how much you weigh. Your bathroom scale is
calibrated to tell you your weight in pounds. A postage scale is calibrated
in ounces. A truck scale might be calibrated by hundreds or thousands of
pounds. This kind of balance is not usually very precise. They can go out
of adjustment over time. I don't need to explain this kind of scale. You
probably have one in your bathroom.
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The "scale of justice" (that some courthouse statues hold) and old-time pharmacy
scales are examples of balance scales. You put a combination of weights on
one side of the scale to find out how much something on the other side weighs.
When the sides of the scale balance, the "stuff" on one side weighs as much
as the sum of the weights on the other side. A balance scale is very precise.
As long as the scale balanced when you started, and when you finished, the
weight of your object will be exactly equal to the sum of weights on the
other side.
A balance scale can have three possible states:
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If the two sides balance, the weights are equal
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If one side goes down, it is heavier than the other side
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If one side goes up, it is lighter than the other side
Scale Puzzles
Let's start with two very simple scale puzzles:
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Scale Puzzles 101
You have three coins, A, B and C. Two are genuine, and one is fake. The fake
coin is heavier than the other two coins. Use a balance scale one time and
figure out which coin is fake.
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Scale Puzzles 102
You have two coins, A and B. One is genuine, the other is fake.
You also have one genuine coin G. You don't know if the
fake coin is heavier or lighter than the genuine coin. Use a balance scale
one time and figure out which coin is fake.
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Scale Puzzles 103
How many pre-measured weights (for the left pan of your balance scale) do
you need in order to measure the weight of an object in the right pan of
the balance scale (you should be able to measure any exact number of ounces
up to at least 120 ounces)?
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Binary Numbering System (not a puzzle, a
simple introduction to Base Two)
Okay, now you have finished your freshman courses in Scale Puzzles. Let's
do some more involved scale puzzles:
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Scale Puzzles 201
You have nine coins. One of them is fake. The counterfeit is identical to
the eight genuine coins in every way except weight. The fake is heavier than
a good coin. Use your balance scale exactly twice to figure out which coin
is the fake.
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Scale Puzzles 202
In SP103, you figured out that ten weights, when
added together, can measure the weight of any item from one ounce up to 1,023
ounces. In SP202, I challenge you to find a way
to use seven pre-measured weights to measure any amount from one ounce up
to 1,093 ounces.
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Scale Puzzles 203
There are ten bags of ten gold bars each. All gold bars weight 16 ounces
each. One bag is full of counterfeit 17-ounce bars. Use a regular (pounds
& ounces) scale one time only and find out what bag is counterfeit.
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Scale Puzzles 301
You have twelve identical coins. They all look the same, but one is a
counterfeit. It is either heavier or lighter than any of the others. Use
a balance scale three times to figure out which coin is counterfeit.
