In the first century BC, the country of Crout (we’ll say) was under the rule of the Roman empire. Crouton engineers needed a good way to measure a 90⁰ angle. They did so by joining straight rods of lengths 3, 4 and 5 into a triangle. As shown on the left in Figure 1, this arrangement forms a 90⁰ degree angle. The Greek mathematician Pythagoras had proven this fact centuries earlier with his famous theorem.

Dimius Minimus, a Roman mathematician visiting Crout, disagreed. On a large stone wall, he drew a 3×4 cubit rectangle shown on the right in Figure 1. He picked up red chalk and began lecturing a group of Crouton engineers.

“Look,” he said. “If I take the red path from the lower left to the upper right, I travel 3+4=7 cubits.”

He switched to the green chalk.

”If I take the green path, I also travel 7 cubits.”

The Crouton engineers, best known for creation of salad garnishments, nodded their heads in agreement. Then, with purple chalk, Dimius Minimus drew the more jagged  path.

“Even here, on this jagged staircase path, I travel 7 cubits.”

He paused dramatically.

“I can make the stairs as small as I want to and get as close to the diagonal line as I wish. Teeny tiny stairs reach the diagonal in the limit. So gentlemen…”

He dramatically dropped the purple chalk and declared,

“… the diagonal to this triangle is not five. It is seven!”

 None of the Croutons could find the flaw in Dimius Minimus’s argument.

Can you? Watch for the answer next Monday.

Solution to Micro Softy 25:  The Fishing Rod Blues

Last week’s Micro Softy concerned a fishing rod, 5 ft long, that couldn’t be carried on the bus, whose linear limit was 4 feet. After it was packaged, the rod was allowed on the bus, even though it had neither been bent nor disassembled. But how?

The diagonal in this week’s Micro Softy hints at the answer. As shown in Figure 2, the five foot fishing pole can be placed as a diagonal in a 3×4 foot square box. The maximum linear dimension, as measured, is now four feet. Thus the fishing rod is allowed on the bus.

Links to all the Monday Micro Softies

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.

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.

and

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