Puzzle Contest #3: Grades 5 and
up Answers
1. This problem was featured on an episode of Columbus,
but it is much older than that. You see 10 sacks of gold
coins, all in a row. Only one of the sacks contains true
gold
coins;
the
other
9
sacks
are
counterfeit.
Counterfeit coins are gold plated, and look the exactly
the same as real coins, but are almost worthless.
You
must identify the sack of pure gold coins. A pure gold
coin weighs 2 ounces, while a gold plated copper coin only
weighs 1 ounce.
You have a scale, and are permitted
exactly one weighing.
You might take a coin from bag
number 7 and weigh it. If it reads 2 ounces you've found
the gold, but if it reads just 1 ounce, the gold coins are
in one of the other 9 sacks. How can you find the sack of
gold coins with just one weighing? (50 points)
Solution: Take 1 coin from the first sack, 2 coins from the second, 3 coins from the third,
and so on up to 10 coins from the tenth sack. Place them all on the scale and determine the
weight of all 55 coins. If they were all counterfeit, the weight would be 55 ounces. If the first
sack holds gold, one coin will be pure gold, and the weight will be 55 + 1 = 56 ounces. If the two
coins from the second sack are gold, the weight will be 55 + 2 = 57 ounces. Similar reasoning
holds for the other sacks. Subtract 55 from the total weight. This gives the number of the sack
that holds the gold coins.
2. What sign could you put between fractions
c
d
a
b
and
to get a + c ? (20 points)
b+d
Solution: You could put the sign “=” because
a c
a a+c d
= is a proportion. Using the property of proportions =
=
b d
b b+d c
(use the Cross Product Property for proportions to check it).
3. Two trains are traveling at 50 miles an hour. They are
100 miles apart, and headed straight towards each other.
A fly can travel at 70 miles an hour. He starts at one
train and flies towards the other. When he reaches the
second train he turns around and heads back to the first.
He keeps flying back and forth until the trains meet. How
far has the fly traveled? (30 points)
Solution: A fly keeps flying until the trains meet. It takes 1 hour for the trains to meet
because in 1 hour each train covers 50 miles each, so the two of them together will cover the
distance of the 100mi. In that 2 hour the fly covers 70 × 1=70 miles. That’s the answer.