(no subject)

[hugme]

You are hired by your favorite 3 letter government agency to find a bag of fake gold. They place you in a room with many bags of gold coins, one of the bags, however, is full of fake coins. To help you with this endeavor you have been given a penny scale, you can drop a penny in to get one measurement. Along with this you are handed a small scrap of paper which has written on it the weight of a gold coin and the weight of a fake gold coin.

As with most projects, there was an unfortunate lack of forsite in planning and you were only allocated one penny.

Two questions...

1. How do you figure out which bag has the fake gold?

2. At what point in an increasing number of either bags or coins can you no longer solve this out with only one penny?

Comments

[stnuke.livejournal.com]

Let me rephrase to clarify for myself. There are a bunch of bags containing items, some number of these bags are 100% genuine and another number of which are 100% fake, and they all contain identical numbers of the appropriate items. The weight of a single fake item is known and constant, and the weight of a single real item is known and constant. You can make but a single weighing, not on a balance scale but on a scale with some kind of readout, in order to determine which of these bags has fake items in it.

Is this a fair restatement of the problem?

I have the answers, I think, for this, as well as a set of bounds.

[hugs.livejournal.com]

"some number of these bags are 100% genuine and another number of which are 100% fake"

yes

"and they all contain identical numbers of the appropriate items."

maybe... maybe not...

"The weight of a single fake item is known and constant, and the weight of a single real item is known and constant."

yes

"You can make but a single weighing, not on a balance scale but on a scale with some kind of readout, in order to determine which of these bags has fake items in it."

yes... one measurement...

"Is this a fair restatement of the problem?"

all but the one part yes...

[girlvinyl.livejournal.com]

Can't you just bite gold to tell if its real?

[weswilson]

you take the first bag and remove all but one coin... the second, all but two... the third, all but three.

You take a measurement of all the bags, noting the deviation between the predicted 100% gold weight, and figuring out through simple algebra how many fake coins were in the mix.

the number of fake coins is the number of the bag... that, and all it's contents were fake.

[weswilson]

oh... and as long as you don't have any bags overflowing with coins, this should work.

[weswilson]

I correct this... if there are more bags than numbers of coins in each bag, then this breaks down.

And if you wanted to do this problem with more than one bag of fake gold, you could just use arithmatically prime values, such that no two bags add up to the total of another bag.

[airdamien.livejournal.com]

well, if there's ONLY one bag of fake, you can cut your losses, stack all the bags on each side of the scale, assuming the the fake's weigh less (reverse this is they weigh more), and run off with the 51% of the gold.