THE MIDAS TOUCH
King Midas was a very wealthy and powerful king who ruled over 10 states. Each
year each of the states would have to send him, as taxes, a bag fi lled with 100 gold coins. Each
gold coin would have to weigh exactly 4 ounces.
One year after the King received all the bags of gold, with each clearly labeled with the
name of the state it came from, a messenger came running in to the King. He shouted, “Your
Majesty, one of the Governors tricked you. His bag of gold is fi lled with coins that are only
3 ounces per coin instead of 4.” But, before he could utter the name of the Governor or the
state that produced the counterfeit coins, an arrow fl ew through the window and struck him in
the throat. He fell over, dead. So, the King had 10 bags of gold in front of him and no way to
know which contains the counterfeit coins.
Now the King had a scale which is similar to a regular bathroom scale in that it tells you the
total weight placed on it (i.e. 5 ounces, 3 lbs, etc.). And, because he was a very cheap kind of
guy you needed to put a penny into the scale to get a single weighing. Th e King only had one
penny in the whole kingdom so he knew he had to solve this in one weighing.
Question: How was the King going to fi gure out who is the cheating Governor with just one
weighing of the scale?
Put all ten bags on the scale and then remove them one by one. Watch for a drop of 300 oz instead of 400. That would be the bag of 3 oz coins. The "penny" scale will count that as one weighing.
To figure out which bag contains the counterfeit coins with just one weighing of the scale, King Midas can follow these steps:
1. First, he needs to number the bags from 1 to 10 to keep track of them during the weighing.
2. Next, he needs to take 1 coin from bag 1, 2 coins from bag 2, 3 coins from bag 3, and so on, until taking 10 coins from bag 10. This means he will have a total of 55 coins (1 + 2 + 3 + ... + 10 = 55 coins).
3. Now, King Midas places all these coins together on one side of the scale and the other side of the scale remains empty.
4. Here's where the "penny trick" comes into play. The King needs to put a penny on the empty side of the scale to balance the weight.
Now, let's analyze the possible outcomes and what they reveal:
- If all the coins were genuine (4 ounces each), the total weight would be 55 coins * 4 ounces/coin = 220 ounces.
- However, if one of the bags contains the counterfeit coins (3 ounces each), the total weight would be 55 coins * 3 ounces/coin + X ounces, where X is the number of ounces the counterfeit coins weigh less than the genuine ones.
- Since King Midas only has one penny to use, the scale will balance when the total weight on both sides is the same. Therefore, if one bag contains counterfeit coins, it will weigh less overall.
5. Based on the outcome, the King can determine which bag contains the counterfeit coins:
- If the total weight is 220 ounces, then all the bags contain genuine coins, and no bag is the culprit.
- If the total weight is less than 220 ounces, the bag from which he took the fewer weight coins is the one that is counterfeit. For example, if the total weight is 218 ounces, then the bag from which he took 7 coins instead of 8 is the one with the counterfeit coins.
By using this weighing strategy, King Midas can identify the cheating governor with just one weighing of the scale.