This week’s Micro Softy concerns Keith, who has a degree in probability, a field of mathematics.

Although mathematicians typically love things that don’t love back, Keith was an exception. He was in love and wanted to propose to his girlfriend Priscilla. But he needed an engagement ring. Luckily, Keith’s brother Nevil was a jeweler. Keith explored Nevil’s inventory and picked out a beautiful $15,000 ring. But Keith couldn’t afford it on a mathematician’s salary.

Nevil tried to help. “I’ve got any number of cubic zirconia rings in the back that look just like it. I could sell you one for $500.”

But Keith wanted the real diamond for his beloved Priscilla.

“I tell you what,” said Nevil sympathetically. “Since you’re my brother, I’ll put the diamond ring in a bag with 250 fake diamond rings and let Priscilla choose. She can keep the ring she picks out of the bag – even if it’s the real diamond. But no matter what ring she chooses, you have to pay me $600.”

Keith protested. “Wait! That’s more than the $500 you quoted to me for the fake diamond.”

“I know. But I‘m at risk here if she chooses the real diamond. Take it or leave it.”

Keith looked at the real diamond ring and the fakes. They looked and felt almost identical except for the price tag attached to each.

“Can we divide the rings into two bags?” asked Keith. “Priscilla can choose a bag and then pick a ring from the bag.” Nevil shrugged. “Sure.” He agreed that Keith could put the rings into the two bags however he wanted.

Was this a good bet? Should Keith accept the deal? Find out next Monday at MindMatters.ai

Solution to Micro Softy 26:   Arguing with Pythagoras

First, here’s last week’s Micro Softy: In Figure 1 below, which we saw last week, the distance from the bottom left of the rectangle to the upper right is five units. But the red, green and purple paths are all seven units. The steps in the purple path can be made smaller and smaller and approach the diagonal as close as you like. The distance is still seven units.

But distance, it turns out, can be measured in different ways.

The diagonal measuring 5 units uses Euclidean distance. When we use the term “as the bird flies,” we are thinking in terms of Euclidean distance. The paths sketched in red, green and purple use the so-called Manhattan distance so named because, in a planned city, streets run only north and south or east and west. Thus, a diagonal route is not possible.

There are many other ways to measure distance. Both Euclidean distance and Manhattan distance are special cases of the Minkowski distance. In engineering and math, the Euclidean distance is most useful. If you measure the diagonal with a ruler, you get a hypoteneuse of 5.

So here’s the answer to the Micro Softy: The distance between the lower left and the upper right of the triangle in Figure 1 is five units if you use Euclidean distance. It’s 7 units if you use Manhattan distance. Thus both answers are correct. And the distance will be different again if a different version of the Minkowski distance is used. To learn about all things Minkowski, check out my book: R.J. Marks II, Handbook of Fourier Analysis and Its Applications, Oxford University Press (2009).

Links to all the Monday Micro Softies

Monday Micro Softy 26: Arguing with Pythagoras. Will the diagonal of a triangle with sides of 3 and 4 feet be 5 feet or, as a visiting mathematician suggests, 7 feet? The answer to last week’s puzzle lies in remembering what happens when you put a rod on a diagonal inside a square box.

Monday Micro Softy 25: The Fishing Rod Blues The Memphis bus driver was sympathetic but he couldn’t let Johnny ride with his overlong fishing pole. Johnny solved the problem—but how? About last week’s Micro Softy: You CAN have a tie in 3D Tic Tac Toe. We illustrate it. And we show what the 4D game is like.

Monday Micro Softy 24: Have you ever tried 3D Tic Tac Toe? To liven up a predictable game, try doing it in three — or even four — dimensions! You won’t be bored. To solve the Barnum’s Circus ticket receipts puzzle, recall that X, Y and Z must all be whole numbers. Algebra then enables us to work out the solution.

Monday Micro Softy 23: Barnum’s Circus Receipts. Circus master Barnum’s ticket seller had not kept proper track of the tickets sold. Can the limited information — and some algebra — help Barnum figure it out? The solution to Micro Softy 22, given here, depends on the assumptions we make about Timmy’s surgeon.

and

Monday Micro Softy 22: Can there be two daddies? This week’s puzzler asks: Can a child’s father be dead and alive at the same time? Or is there another solution? The solution to last week’s puzzler lies in things we can do with binary numbers.

Monday Micro Softy 21 and previous links: Finding More of the Deadly Fentanyl Pills. There, you will also find links to Microsofties 11 through 20 as well.

At Monday Micro Softy 11: What Happened to That Other Dollar?, you will find links to the first ten Micro Softies. Have fun!