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We've got solutions of three
different types.
Solutions in the first group are "classic."  Way 1.
The only solution from the second group uses the same idea as those
from the first group, but it's a little bit tricky  Way 2.
Surprisingly, we've received two solutions, which are based on
weighing with a balance scale  Way 3.
We show some solutions to illustrate all these ways how to determine
which bag has counterfeit coins. 





Way 1  Solution by Jensen Lai 


Place the following on the
scale:
1 coin from bag 1
2 coin from bag 2
3 coin from bag 3
4 coin from bag 4
5 coin from bag 5
6 coin from bag 6
7 coin from bag 7
8 coin from bag 8
9 coin from bag 9
10 coin from bag 10
If all 55 coins were Gold, this would weigh 55 grams. However, each
counterfeit coin will increase the weight by .1 gram. The number of .1
grams over 55 grams this weighs, is the number of the bag with
counterfeit coins. For example, if the 55 coins weigh 55.6 grams,
there must be 6 counterfeit coins. These will have been from bag
number 6. 





Way 1  Solution by Mike Kurylo 


Take one coin out of the
first bag, two out of the second, ... ten out of the tenth bag. Weigh
all of them, your remainder will tell you which bag has the
counterfeit coins (.1=bag1, .2=bag2, etc). No remainder means the
tenth bag. 





Way 1  Solution by Nicole
Takahashi 


First label the bags 1
through 10. Then on the scale place 1 coin from bag 1, 2 coins from
bag 2, 3 coins from bag 3... and 10 coins from bag 10. Due to the
convenience of the numbers, the bag with the counterfeit coins will be
revealed by the tenths place of the scale reading: 55.1 means bag 1 is
counterfeit, 55.7 means bag 7 is, and 56.0 means bag 10 is. 





Way 2  Solution by Horst
Karaschewski 


Bring the bags in order. Take
no coins from the first, 1 coin from the second ... 9 coins from the
10th bags and weigh all these coins. The decimals plus 1 give the
number of the bag with counterfeit coins. 





Way 3  Solution by Sheila Hedger 


Here is the answer to Ten
Bags of Gold puzzles.
Mark each bag with a number one to ten.
Take out all but one coin each from bag one and two, making sure to
keep the coins marked and separate.
Take two coins each from bag three and four, making sure to keep the
coins marked and separate.
Take three coins each from bag five and six, making sure to keep the
coins marked and separate.
Take four coins each from bag seven and eight, making sure to keep the
coins marked and separate.
Take five coins each from bag nine and ten, making sure to keep the
coins marked and separate.
Place the bags 1, 3, 5, 7, and 9 on one side of the scale. Place the
bags 2, 4, 6, 8, and 10 on the other side of the scale. The
counterfeit coins weighs .1 gram more than the real coins. The scale
will determine which bag is counterfeit. If all coins were the same
weight the scales would be equal. The scale will show which side is
heavier and it is that side which has the counterfeit coin(s). The
amount of the difference in weight determines which bag is
counterfeit. If the scale shows .1 gram heavier, then the bag 1 or 2
that is on the heavier side is the counterfeit bag. If the scale shows
.2 grams heavier then the bag 3 or 4 that is on the heavier side is
the counterfeit bag. If the scale shows .3 grams heavier then the bag
5 or 6 that is on the heavier side is the counterfeit bag. If the
scale shows .4 grams heavier then the bag 7 or 8 that is on the
heavier side is the counterfeit bag. If the scale shows .5 grams
heavier then the bag 9 or 10 that is on the heavier side is the
counterfeit bag. 





Way 3  Solution by Jason Meyers 


I don't know if this solution
counts, as it depends on your definition of "one weighing" but if you
take:
10 coins from bag 1
20 coins from bag 2
30 coins from bag 3
40 coins from bag 4 and
50 coins from bag 5
on one side of the scale and
10 coins from bag 6
20 coins from bag 7
30 coins from bag 8
40 coins from bag 9 and
50 coins from bag 10
on the other side.
Now to the side which weighs less (so we know all of those coins are
authentic), add as many coins from any bag on that side as is
necessary to balance the scale. When that number of additional coins
is multiplied by 10, the bag on the heavier side which had that many
coins on the scale is the counterfeit bag. e.g. If the 15 side is
lighter, and you have to add 4 coins to that side, it's bag 9 (which
is heavier by 40*0.1=4 coins). 





Last Updated: September 30, 2007  Posted:
January 7, 2002 









