Eleven Coins

[8v10un30sun]

Hahaha...Okay so here's the fix to the algorithm...He he!!  Good catch!

Procedure using a balance:

1) Measure 1 coin and call this W1.
2) Measure all 11 and call it W2.  If W1*11/W2 < 1 then the coin is lighter else heavier.

I almost thought my procedure failed...then I did a little math and woola!  It works!  Ta-da!  Sankyou sir!

Let me know if it's broken because I tried a VERY small dataset with very small variability!  

Btw, I like your divide and conquor approach.  I was thinking about something along the same line if the scale did not have a way to measure a single unit but it was late like it is right now and my brain was in a big mush.  I got to the part where you weight 3 on each...if they are equal then it's in the other 5...then my mind started to branch all over and then I was like what if it wasn't equal then it's in one of those two pile but WHICH!?!?!?  Then I said screw it...Enuff with this game.   

Your algorithm is off.
For it to work, the coin you pick for W1 necessarily must be a real coin all the time.

Let's run one example. Say 10 real coins weigh 1lb each, and the 11th fake coin weighs 2 lb.
1) By chance you picked the fake coin (1/11%). W1 = 2 lb.
2) W2 = 12 lb
    2*11 > 12  does not imply the fake one is lighter.  

[NeoSamurai]

Hahaha...Okay so here's the fix to the algorithm...He he!!  Good catch!

Procedure using a balance:

1) Measure 1 coin and call this W1.
2) Measure all 11 and call it W2.  If W1*11/W2 < 1 then the coin is lighter else heavier.

I almost thought my procedure failed...then I did a little math and woola!  It works!  Ta-da!  Sankyou sir!

Let me know if it's broken because I tried a VERY small dataset with very small variability! 

Btw, I like your divide and conquor approach.  I was thinking about something along the same line if the scale did not have a way to measure a single unit but it was late like it is right now and my brain was in a big mush.  I got to the part where you weight 3 on each...if they are equal then it's in the other 5...then my mind started to branch all over and then I was like what if it wasn't equal then it's in one of those two pile but WHICH!?!?!?  Then I said screw it...Enuff with this game.   

Algorithm's still broken.

Using the same assumption as before with 10 real coins at 1lb each, and 11th fake coin at 2 lb.

1) W1 = 1 lb.
2) 1*11/12 < 1 does not imply fake is lighter.

I see what you're trying to do but I don't think it'll work for this problem. Your step 2 requires a necessary condition in step 1. This version of your algorithm will work in the case when W1 is the fake coin, but not when it's the real coin. And your first version will work when W1 is real, but not fake. But how do you tell if it's real or fake in step 1?? Your step 2, as is, necessitates that you know this fact. It isn't robust enough to cover all scenarios.

[LuckyState98]

Are we still using a simple balance or pan scale?

[NeoSamurai]

Are we still using a simple balance or pan scale?

I think OP probably meant only simple 2-pan balance and not any sort of weighing scale. I'm just humoring 8v.

I don't think you can solve this problelm with an arithmetic algorithm anyway, but if you can, I would be interested in seeing it.

[8v10un30sun]

 

Algorithm's still broken.

Using the same assumption as before with 10 real coins at 1lb each, and 11th fake coin at 2 lb.

1) W1 = 1 lb.
2) 1*11/12 < 1 does not imply fake is lighter.

I see what you're trying to do but I don't think it'll work for this problem. Your step 2 requires a necessary condition in step 1. This version of your algorithm will work in the case when W1 is the fake coin, but not when it's the real coin. And your first version will work when W1 is real, but not fake. But how do you tell if it's real or fake in step 1?? Your step 2, as is, necessitates that you know this fact. It isn't robust enough to cover all scenarios.

[8v10un30sun]

I think OP probably meant only simple 2-pan balance and not any sort of weighing scale. I'm just humoring 8v.

I don't think you can solve this problelm with an arithmetic algorithm anyway, but if you can, I would be interested in seeing it.

I'll think about it but I'm starting to think it's not possible.    

I will think about this later...Maybe measure 2 coins and measure 11 coins or measure 10 coins and measure 11 coins.  I think the 10 and 11 might provide insights...I'll ponder it more to see if there are gaps.  Hahaha.

[NeoSamurai]

It looks like you can solve this arithmetically . I'm splitting the coins into group of 5 and 6, but this method will work for any sort of grouping as long as you use all the coins.

Let A be the weight of the group of 5 coins. Let B be the weight of the group of 6 coins. Let x = real coin, and y = fake coin.

Then there's only two possibilities, either the fake is in the group of 5 or 6:

5x = A,  5x + y = B,  C = A + B = 10x + y

or    4x + y = A,   6x = B,   C = A + B = 10x + y

Clearly the solution is unique.

At the start of the algorithm, simply let x = A/5. Solve for y,  y = B - 5x.  If 10x + y = C, then you have guess correctly that the group of 5 consists of only real coins.

Else, your initial guess was incorrect and its actually the second case. So set x = B/6, y = A - 4x.

[distressKing]

Among 11 coins of the same denomination, 1 is know to be counterfeit. Using a simple 2-pan scale or balance, how would you determine in 2 weighings if the fake coin is heavier or lighter than a real coin?

OK I think I got it... Has to be two scenerios...

1st... you do 5 and 5 if it balance means the 11th coin is fake so switch with one of the coins to determine is it goes up or down..

2nd scenerio.. if it is uneven the first weighing. than on the scale you see how much each side weighs. and weigh the 11th coin because that's the real one and do the math to see if it is lighter or heavier. so.. still two weighings..

Right or wrong? 

[YAX]

you can't weigh the coins because you don't have a scale.  But you should be able to

step 1: remove two coins at a time from either side of the balance and this won't affect the balance until you get to the point where you remove one real coin and one fake coin.  
step 2: After that point, the scale would balance out, then
step 3: you simply put back one of the two coins you removed and
step 4: replace the other removed coin with the 11th coin.  

Then if the scales balance, that means the one in your hand that you replaced in step 4 is the fake one and if it came from the lighter side of the scale, then the fake coin is lighter and if it came from the heavier side, then the fake coin is heavier.  

If the scale don't balance then that means the one you put back was the fake one.

[distressKing]

you can't weigh the coins because you don't have a scale.  But you should be able to

step 1: remove two coins at a time from either side of the balance and this won't affect the balance until you get to the point where you remove one real coin and one fake coin.  
step 2: After that point, the scale would balance out, then
step 3: you simply put back one of the two coins you removed and
step 4: replace the other removed coin with the 11th coin.  

Then if the scales balance, that means the one in your hand that you replaced in step 4 is the fake one and if it came from the lighter side of the scale, then the fake coin is lighter and if it came from the heavier side, then the fake coin is heavier.  

If the scale don't balance then that means the one you put back was the fake one.

But you will be weighing more than 2 times... and thought it was either a balance or scale that you could use?

[YAX]

But you will be weighing more than 2 times... and thought it was either a balance or scale that you could use?

no, not really.  removing the coins doesn't change the balance so the scale hasn't moved at all.  besides, how else will you get the scale back to balance if you don't remove the coins?