# False coins (digital balance edition)

forkedknight
Posers and Puzzles 26 Sep '11 16:23
1. forkedknight
Defend the Universe
26 Sep '11 16:23
Lots of people have heard the puzzle where you must identify a false coin using a balance, this is a slight twist on the puzzle:

You have a digital balance, i.e. it will give a digital reading of which side is heavier and by how much, accurate to 0.1g.

You have 10 bags of coins, each bag contains 10 coins. All of the normal coins are 1g each. In the false bags, the coins are 1.1g each.

How many weighings are required to identify which bag contains the false coins?
2. talzamir
Art, not a Toil
26 Sep '11 21:28
Refraining from giving the answer.. would love to know how to make the hidden text thingy. The solution I have in mind works for 11 bags of 10 coins too.
3. 27 Sep '11 08:17
[h idden] Like this [/h idden]

Like this
4. talzamir
Art, not a Toil
27 Sep '11 14:04
Thank you. Thus, my solution is -

Only one weighing, with a pile consisting of 1 coin from the 1st bag, 2 from the second etc, all the way to 10 from from the 10th. There are 55 coins, of which 1 .. 10 are false, making a total of 55 + n/10 grams of combined weight where n is the ordinal of the bag where the heavier false coins are. If there are 11 bags, no coins from the last bag mean that 55g total weight implies that the bags in the 11th coin are false.
5. forkedknight
Defend the Universe
27 Sep '11 17:392 edits
Very good, for 10 bags, the digital balance isn't actually required, a digital scale would work just as well.

However the follow on question is this:

What is the maximum number of bags of coins you could be given an still be able to identify the bag with the false coins in two weighings?

Each bag still contains 10 coins.
6. talzamir
Art, not a Toil
27 Sep '11 20:54
Hmm..
the best I can do with two uses of the digital scales is 121 bags.
7. forkedknight
Defend the Universe
28 Sep '11 02:14
Sounds like you're on the right track; remember, it's a digital balance, not just a digital scale.

What is your solution for 121 bags?
8. talzamir
Art, not a Toil
28 Sep '11 07:34
I formed a square of 11 rows and 11 lines of the bags, numbered 0...10. First weighing has 0 coins from each bag on line 0, 1 from each on line 1, etc, 10 from each on line 10. Second weighing does the same for row 0, row 1, etc. On each weighing, there are 11 x (0 + 1+ 2+ ... + 10) coins weighed, which is 565g of coins; but false coins cause the weight to be a bit over that. Subtract 565g from each value and multiply by 10 gives integer values m and n, and those are the coordinates of the bag with the false coins.

I wonder if we refer to a different kind of measuring device. Digital balance scales that I know of simply have stuff put on top and give an accurate measurement of the weigh on a display, usually a LED.
9. forkedknight
Defend the Universe
28 Sep '11 14:085 edits
Originally posted by talzamir
I wonder if we refer to a different kind of measuring device. Digital balance scales that I know of simply have stuff put on top and give an accurate measurement of the weigh on a display, usually a LED.
It's possible my terminology is wrong or nonsensical. I meant a two-pan balance, with a digital readout of which side is heavier and by how much.

except the slider part is digital.

This is very possibly a theoretical device; sorry for the confusion.
10. forkedknight
Defend the Universe
28 Sep '11 14:18
Originally posted by talzamir
I formed a square of 11 rows and 11 lines of the bags, numbered 0...10. First weighing has 0 coins from each bag on line 0, 1 from each on line 1, etc, 10 from each on line 10. Second weighing does the same for row 0, row 1, etc. On each weighing, there are 11 x (0 + 1+ 2+ ... + 10) coins weighed, which is 565g of coins; but false coins cause the weight ...[text shortened]... 0 gives integer values m and n, and those are the coordinates of the bag with the false coins.
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This solution seems sound. It took me a second to follow the bolded statement, since you can't include 0 coins from multiple bags in a weighing.

You can, however, set aside 11 bags and weigh 11x (1 + 2 + ... + 10) coins from 110 other bags, which is I think what you meant and provides the same conclusion of 121 bags.
11. talzamir
Art, not a Toil
28 Sep '11 15:40
I just liked the symmetry of it. n coins from each on the nth row / line. ^_^
12. forkedknight
Defend the Universe
28 Sep '11 16:05
Is it accurate to say that given N weighings on a digital scale, you can deduce the which bag is false for up to 11^N bags?

Looks like it works for N=3, and we've already proved N=1 and N=2.
13. talzamir
Art, not a Toil
28 Sep '11 18:13
And even more generally, if there are c coins per bag and n weighings, the total maximum number of bags is

(c+1)^n