[Guest]

I recently came across this very cool weighing problem. It's not your usual balance scale problem and it's quite hard, but doable.

There are 13 separate piles of coins. Each pile has 4 identical coins in it. 12 of the piles are all identical to each other, containing the genuine coins. A genuine coin weighs X grams, where X is an integer (important). The 1 pile of counterfeit coins look and feel identical to the genuine coins. But, they weigh X-delta, where delta can be any real number (important), such that 5>=delta>=-5. For example, if X=8, then the counterfeit coins can weigh anywhere from 3 to 13.

You have a digital readout scale that will display a single number for each weighing, accurate to any real number. Your task is to determine X, delta and the pile that contains the counterfeit coins by using the scale only 2 times.

You can return coins to their original piles after each weighing, and you can decide what your next weighing will be based on the result from the first weighing.

Edit on behalf of OP:

Of course if delta=0 the counterfeit pile cannot be identified by weighing.

Edited May 30, 2010 by bonanova
OP requested the delta=0 caveat

[Guest]

Wouldn't it be the case that if Delta=0, the counterfeit coins cannot be identified?

OP states there are counterfeits; their weight is within limits that include the weight of genuine coins.

Although it could be determined that Delta = 0; weighing would seem powerless to identify the counterfeit pile.

That is, the counterfeits could be struck from material of density equal to that of the genuine coins.

They would still be counterfeit.

I wrote a tongue-in-cheek a while back, after feeling it had been abused

by several posters, almost to the extent that it was not given proper status as a "genuine" number.

OP says only that Delta be any real number between 5 and -5.

Zero is a real number; OP says we should deal with it.

Ouch. One final request Mr. Moderator. I have a series of typos, in the middle of my second weighing results for case 3. 451.25, 454.5, 457.75 should be 441.25, 444.5, and 447.75 respectively. This too tough a problem to have the wrong solution coming from the OP, just lying around. Apologies.

[bushindo]

Ouch. One final request Mr. Moderator. I have a series of typos, in the middle of my second weighing results for case 3. 451.25, 454.5, 457.75 should be 441.25, 444.5, and 447.75 respectively. This too tough a problem to have the wrong solution coming from the OP, just lying around. Apologies.

That's a very clever solution. Excellent puzzle.

[Guest]

This method only seems to return integer values for delta, but delta is any real number on [-5, 5]. There are identifiability issues for certain values of delta. For instance, the four situations (delta = 1, pile = 1), (delta = 1/2, pile 4), (delta = 1/3, pile = 7), and (delta = 1/4, pile = A ) all give the same remainder 1.

D'oh! Reading comprehension FTL