Columbo's Minimum Information Problem: Can You Solve It?
Do you like Lieutenant Columbo? And do you remember the episode The Bye-bye Sky High I.Q. Murder Case? In it, Columbo investigates a murder in the Sigma club, a private international society. Every member of Sigma must pass tests and rank among the top 2% of people with the highest IQ. Columbo receives a riddle from one of the characters, Oliver Brandt, which is referred to as the Minimum Information Problem.
Introduction
Oliver Brandt asks Columbo if he likes riddles. He replies that he does, but he doesn't have much of a knack for them and that Mrs. Columbo is great at crosswords. The conversation and the riddle proceed as follows:
Brandt: "I'll give you a little riddle to test your logical thinking ability."
Columbo: "Go ahead, sir."
Brandt: "We call this riddle the minimum information problem."
Columbo: "Minimum information?"
Brand: "Basically none."
The Riddle – Which Bag Contains Fake Gold?
Brandt: "In the room, there are several bags with pieces of gold. As many bags as you want.
Each bag contains several pieces of gold. Again, as many as you want.
However, one bag is full of pieces of fake gold, which has a different weight.
So one piece of real gold weighs, let's say, 1 pound and a piece of fake gold weighs whatever you want."
Columbo: "How about 1 pound and 1 ounce."
Brandt: "Alright, why not?"
Brandt: "And you have a scale for coins."
Columbo: "That means I put in a penny and a ticket comes out that says how much that gold weighs."
Brandt: "Exactly, but you only have 1 penny."
Brandt: "So, which bag contains the pieces of fake gold?"
Columbo: "Is that all?"
Brandt: "That's all!"
Columbo: "Well, that's complicated."
Figure: Peter Falk as the Famous Lieutenant Columbo (source: Craiyon)
A Gentle Hint
If you're struggling with the solution, this gentle hint or clarification may help you, which will become clear at the end when the correct solution to the riddle is revealed.
At the first mention, Oliver Brandt talks about pieces of gold, but it's important to add that these are absolutely identical pieces of gold where one is like another and each piece weighs the same—you can imagine them as gold coins. The same applies to the fake pieces of gold—as if they were fake gold coins that are identical to each other, each weighing exactly the same, but each fake coin weighs slightly more than a real gold coin.
The Solution to the Riddle – Mrs. Columbo Figured It Out
At the end of the film, Columbo tells Brandt: "That riddle about the bags of gold? Mrs. Columbo figured it out."
- In the room are bags with gold and one of them is filled with fake gold, which is heavier than real gold.
- I have a scale for pennies and only 1 penny, thus just 1 chance to weigh anything.
- Let's say for simplicity there are only three bags.
- Each piece of real gold weighs 1 pound and each piece of fake gold weighs 1 pound and 1 ounce.
- I take 1 piece from bag 1, 2 pieces from bag 2, and 3 pieces from bag 3.
- That makes a total of 6 pieces of gold which we place on the scale.
- If all pieces were real gold, they would weigh 6 pounds.
- If the fake gold were in bag 1, it would weigh 6 pounds and 1 ounce because we took only 1 piece from bag 1.
- If it were in bag 2, it would weigh 6 pounds and 2 ounces since we took 2 pieces from bag 2.
- And if it were in bag 3, it would weigh 6 pounds and 3 ounces etc.
Sir, that's a great riddle!
One More Riddle for Columbo
Lieutenant, now I will tell you four words and you tell me which one does not belong:
Wood, faith, village, economy.
Columbo responds after a very brief contemplation that wood is the word that does not belong because we say Helvetian faith, Spanish village, and Turkish economy. All words except wood have an adjective associated with a place/country. Note: Helvetia refers to Switzerland.
You Might Be Also Interested
- Columbo: Death Hits the Jackpot;
- Two Beagles: The Probability Puzzle by a Woman with the Highest IQ;
- Chevalier de Mere's Probability Puzzle of the 17th Century;
- Monty Hall Problem aka Three Door Puzzle;
- An Argument for Divine Providence;
- All Articles on Probability.
Based on the original Czech article: Dva bíglové – pravděpodobnostní hádanka.
Based on the original Czech article: Hádanka ze seriálu Columbo – dokážete vyřešit „problém minima informací“?.