Today’s Micro Softy questions the law of conservation of money. (And, as always, the solution to last Monday’s poser is below.)

David is the name of a boy band. All five of the members of the band are named David. They are followed around the country during their concert tour by three groupies named Shirley, Goodness and Mercy. David’s next concert is in Toad Suck, Arkansas. On a budget, Shirley, Goodness and Mercy find a cheap hotel in Toad Suck that only charges $10 per room per night.

After they pay in cash and go to their rooms, the hotel clerk remembers that there is a discount special that week. Three rooms booked at the same time cost $25 dollars. So he calls the bell hop, Uriah, and gives him five one dollar bills, saying, “Take this to Shirley, Goodness, and Mercy in rooms 12, 13 and 14 and tell them about the special.”

On the way to the room, greedy Uriah thinks that $5 doesn’t distribute well among three people. So he pockets $2 and gives the three girls a dollar each.

Here is the accounting: After their refund, Shirley, Goodness, and Mercy paid $9 each for their rooms for a total of $27. Uriah kept the other two dollars. But 27 plus 2 is 29.

What happened to the other dollar? To solve this Micro Softy, you need to pinpoint the flaw in this logic. We’ll take it up next Monday.

Solution to Micro Softy 10:    The Monte Hall Problem

Image Credit: Mark - Adobe Stock

Here’s the problem:

Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what’s behind the doors, opens another door, say No. 3, which has a goat. He then says to you, ‘Do you want to pick door No. 2?’ Is it to your advantage to switch your choice of doors?

A few years ago, I wrote about the fact that pigeons can solve the Monte Hall Problem and some of the information for the solution is borrowed from that post:

You choose a door and game show host Monte Hall does not show you what’s behind it. Instead, he opens a door and shows you a goat. Since there are two goats, there is always an extra door with a goat behind it.

The winning strategy might not seem intuitive at first. At first it might not seem to make any difference whether you switch doors or not. But if you always switch your pick from the door you first chose, you win the car 2 times out of 3.

To see this, let’s look at all the ways the goats (G) and the cars (C) can be arranged, resulting in a win result (W) or a lose (L) result:

• GCG     W        
• GGC     W
• CGG      L        

For the first entry in the list, GCG, you choose a door hiding a goat. Monte Hall opens another door to show you the other goat — the last G in GCG. You switch doors from your initial choice and win the car. The “W” means “win.”

The second entry, GGC, is the same thing. You first choose a goat door — the first G in GGC. Monte Hall shows you the other goat so you switch from your first choice and win the car.

The third entry, CGG, is where you lose. The first door you choose — the C in CGG — hides the car. Monte Hall shows you a goat behind one of the other two doors. You switch doors from your first choice, see a goat, and lose.

Because all entries in the table are equally probable, you win the car 2 out of three times if you always switch from the first door you choose.

See you Monday for the next Micro Softy: Connect the Dots!

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Here are the Micro Softies from earlier Mondays, to date. The answer to one Monday’s puzzle is always given the next Monday:

Monday Micro Softy 1: Microsoft: What did you need to work there in the early days? They asked questions that were not about the details of computer technology. The questions made you think. Hard. For example, they would ask questions like, why are manhole covers round? Also, here’s the Round Trip puzzle.

Monday Micro Softy 2: The Dead Presidents’ Club The answer to Monday Micro Softy 1: The Round Trip puzzle is here too. Today’s puzzle: How would a girl who knew nothing of American history immediately know the name of at least one of three early Presidents who died on July 4?

Monday Micro Softy 3: The Wolverton Mountain Puzzle Here’s the answer to Dead President’s Club as well — and smart STEM people often DON’T get that one right. Today’s brain teaser is in honor of Claude King, Clifton Clowers, and Wolverton Mountain. It’s not high tech but it will surely test your thinking abilities.

Monday Micro Softy 4: Claude King bests Clifton Clowers Clowers offers Claude two slips of paper to choose from on a blind choice: marriage or death… Claude’s sweetheart Chloe warns him that, in reality, both slips say “death.” He says never mind. So how did he escape death and marry her?

Monday Micro Softy 5: The puzzle of Claude and Chloe’s two kids Puzzle: We learn that one of Claude and Chloe’s two children is a boy. With a 50–50 ratio, what chance is there that the other child is a boy too? Also, here’s the answer to the puzzle of how, twelve years earlier, Claude escaped the trap Clifton Clowers set for him, so he could marry Chloe.

Monday Micro Softy 6: Bad Adding? It looked to Claude like young Clay’s numbers didn’t add up but the boy seemed confident. What did he know that his father at first didn’t?And here’s the answer to the probability question: What are the chances that Claude and Chloe’s younger child is also a boy?

Monday Micro Softy 7: Who’s the Champ? A single number does not always determine which player is better. And here’s the answer to why young Clay’s arithmetic didn’t make sense at first to Claude — but then Claude realized that the boy was right.

Monday Micro Softy 8: Who’s the better barber? There are only two barbers to choose from and the choice may not be as easy as it looks. Also, here’s the answer to Who’s the Champ?
Micro Softy 7 is an illustration of Simpson’s Paradox, which says that averaging averages doesn’t always work.

Monday Micro Softy 9: To Flip or Not to Flip? Probability theory can sometimes help with seemingly impossible questions. But how? About the answer to Micro Softy #8, ask yourself a question, “Who shaves the barber?”

Monday Micro Softy 10:  The Monte Hall Problem In this case, it is a serious question: How badly do you want a goat in your life? Probability problems typically come down to analyzing the choices carefully and picking the most promising one.