# Gold coin problem. Itâ€™s tricky and confusing.?

Suppose You have 5 stacks with 20 supposedly gold coins in each stack. Each authentic gold coin weighs 10 grams, but two of the stacks are composed of only counterfeit coins weighing 11 grams each. You are given a scale the weighs grams. Figure out a way to determine the counterfeit stacks in one weighing using the scale.

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label the stacks with 1, 2, 3, 4 and 5. take out 2^(n-1) coins from each stack with n = label number.

so we took out 1, 2, 4, 8 and 16 coins from 5 stacks. you should get weighing of 31*10 = 310 grams if there were no counterfeits. to know which 2 stacks have fakes, subtract your weighing by 310 and find the matching sum.

letâ€™s say your weighing = 322 g.

the difference = 322 - 310 = 12 g.

12 = 4 + 8, meaning fakes are from stacks 3 and 4.

but you also could do it this way :

take none from S1, 1 from S2, 2 from S3, 4 from S4, and 8 from S5.

youâ€™ll be weighing 15 coins, meaning if all of them are not fake, they should weight 150 grams. iâ€™ll list the difference the actual weight made with 150 g and the matching fake stacks.

S1 + S2 = 1 g

S1 + S3 = 2 g

S1 + S4 = 4 g

S1 + S5 = 8 g

S2 + S3 = 3 g

S2 + S4 = 5 g

S2 + S5 = 9 g

S3 + S4 = 6 g

S3 + S5 = 10 g

S4 + S5 = 12 g

the counterfeit coins stack would weight 220 grams compared to a gold coin stack that weights 200 grams.

you couldnâ€™t do it in one weighing

note

make your observations as you are putting the stacks on

put stack one on side A

put stack two on side B

If they are ballanced you know they have equal measurement

if it tilts one way then you know one is counterfeit

then put stack 3 on side A

stack 4 on side B and make further evaluations

ect

old problem with a new twist. take:

1 coin from stack 1

2 coins from stack 2

4 coins from stack 3

8 coins from stack 4

16 coins from stack 5

weigh the 31 coins.

If they are all good, weight would be 310g

if stack 1 only is bad, weight would be 311g

if stack 2 only is bad, weight would be 312g

if stack 3 only is bad, weight would be 314g

if stack 4 only is bad, weight would be 318g

if stack 5 only is bad, weight would be 326g

if stack 1 & 2 bad, weight would be 312g

if stack 1 & 3 bad, weight would be 315g

if stack 1 & 4 bad, weight would be 319g

if stack 1 & 5 bad, weight would be 327g

if stack 2 & 3 bad, weight would be 316g

if stack 2 & 4 bad, weight would be 320g

if stack 2 & 5 bad, weight would be 332g

if stack 3 & 4 bad, weight would be 322g

if stack 3 & 5 bad, weight would be 330g

if stack 4 & 5 bad, weight would be 334g