# you are given 5 coins and a balance scale. The coins look alike but one is counterfeit and either heavier ….?

you are given 5 coins and a balance scale. The coins look alike but one is counterfeit and either heavier or lighter than the others. Find the counterfeit coin using just 3 weighings on the scale, and tell if it is heavier or lighter than the others. Be sure to show all possibilities after each weighing.

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Let the coins be a,a,a,a,b where b is the counterfeit.

1. Select 4 coins and put 2 in one balance pan and 2 in the other balance pan.

a) if the balance is even, then your 5th coin is false, and you need one more weighing to see if it is lighter or heavier than one good coin that you have.

b) if the balance is off, then the counterfeit coin b is on one side or the other, so go to step 2.

2. The pans contain a,b and a,a. We pick a pan, anyone, not knowing whether it contains a,a or a,b since we donâ€™t know whether b is lighter or heavier than a.

Picking pan containing a,a

If you pick the pan containing a,a and remove one coin and replace it, it will still be a,a and your scale will not change. Then you know that the counterfeit coin is in the other pan and you have to do one more weighing (using one good coin from pan a a) and selecting one coin from the pan containing a,b. If you select a and weigh against a, they balance, then you know that b is counterfeit and you can tell if it is heavier or lighter by how the previous weighing was. If you select b and weigh against a, you can tell immediately whether it is lighter or heavier. Remember that the good coin is your 5th coin, so save it.

Picking pan containing a,b

If you pick the pan containing a,b and remove b and replace it with your remaining good coin a, the scales will balance and then you know that b is counterfeit. One more weighing of b against your good coin a is necessary

If you pick the pan containing a,b and remove a, and replace it with your remaining coin a, the scales wonâ€™t change, and so you know that the other coin is counterfeit and you only need 1 more weighing to see if it is lighter or heavier.

first weigh two coins apiece

if they are equal then you have your counterfeit coin

if that doesnâ€™t work then choose the lighter of the 4 coins that you weighed and then try it with those two.

hope this helps.