A Curious Weighing Problem
You have a bag with 101 coins identical in appearance, but 50 of these coins are fake, and the other 51 coins are genuine.
You know that a fake coin has a different weight from a genuine coin. The difference is only 1 gram, but you don't know if the fake coins are heavier or lighter than the genuine coins.
You received a curious scale to measure the weight of these coins. This scale has two pans, and instead of compare weighs, this is a digital scale that will show to you the modulus of the difference between the weights in the two pans. For example, if a pan has 67 grams, and the other pan have 55, the scale will show to you or (the same value). Also, the scale only works if there are something in both pans.
The problem is:
If you take at random a coin from this bag, it is possible to know if this coin is fake or not, using the scale only 1 time to weigh the coins?
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1 solution
Let's suppose that any genuine coin weigh 2 gram, and any fake coin weigh 1 grams. It can be reversed and the fake and genuine can weigh 2 and 1 gram respectively, but the scale will show to you the modulus of the difference, which one will always positive, then not matter what way to figure out the weighs.
If you take at random a Genuine Coin , you will have left 50 fake coins and 50 genuine coins. Now you need to weigh 50 coins in a pan and 50 coins in the other.
Suppose that, you have the luck to put the fake coins in a pan and the genuine coins in the other. Then you will have a pan with 100 grams (genuine coins), and a pan with 50 grams (fake coins). The scale will show to you the modulus of the difference which is 50.
Now, if is there 30 genuine coins and 20 fake coins in a pan, and 30 fake coins and 20 genuine coins in the other, you will have a pan with 80 grams and other pan with 70 grams, and the scale will show 10.
If you have a number x of genuine coins in a pan, you will also have 5 0 − x fake coins in this pan. Then, this pan will have 2 x + 5 0 − x = 5 0 + x grams. If in one pan you have 5 0 − x fake coins, in the other pan you will have x fake coins and 5 0 − x genuine coins. Then, this pan will have x + 1 0 0 − 2 x = 1 0 0 − x grams.
The modulus of the difference will be ∣ 1 0 0 − x − 5 0 − x ∣ = ∣ 5 0 − 2 x ∣ . This value will be positive and even .
If you take at random a Fake coin , will be left 49 fake coins and 51 genuine coins. If you use the same logic as above:
You will have a number x of genuine coins in a pan, you will also have y fake coins in this pan. Then, this pan will have 2 x + y grams. If in one pan you have y fake coins, in the other pan you will have 4 9 − y fake coins and 5 1 − x genuine coins. Then, this pan will have 4 9 − y + 1 0 2 − 2 x = 1 5 1 − y − 2 x grams.
The modulus of the difference will be ∣ 1 5 1 − y − 2 x − 2 x − y ∣ = ∣ 1 5 1 − 2 y − 4 x ∣ . This value will be positive and odd .
Then we can conclude that if you take a genuine coin and weigh the other coins, 50 by 50, the scale will show you an even number, and if you take a fake coin and weigh the other coins, 50 by 50, the scale will show you an odd number. I means that you can know if a coin is genuine or not using the conditions of the problems.