Do there exist a right triangle and an isosceles triangle, each with sides of rational length, that have both the same perimeter and the same area?
Yes, there’s exactly one such pair!
The right triangle has sides of lengths (377, 135, 352) and the isosceles of lengths (366, 366, 132).
Mercyhurst College mathematician Charles Redmond used to get frantic phone calls at the end of each term from colleagues in other departments who’d written syllabi saying “Quiz 1 is 10% of your final grade” and now couldn’t figure out how to do the necessary calculation.
“There’s a good reason,” Redmond wrote. “They said something they didn’t mean to say, and they had only a vague notion of what they meant to say in the first place.”
He suggested that students might turn this confusion to their advantage. Suppose Quiz 1 is worth 10 or more points and your score is 9. If x is your final grade and “Quiz 1 is 10% of your final grade,” then 0.10x = 9 and suddenly your final grade is 90, an A.
“Not bad for about a week’s worth of work. Take the rest of the semester off.”
(Ed Barbeau, “Fallacies, Flaws, and Flimflam,” College Mathematics Journal 33:2 [March 2002], 137-139.)
Carnegie-Mellon mathematician Po-Shen Loh has offered a new, simple proof of the quadratic formula that provides a natural, intuitive algorithm for solving general quadratic equations.
More here. “May this story encourage the reader to think afresh about old things; seeing as how new progress was made on this 4,000 year old topic, more surprises certainly await the light of discovery.”
(Thanks, Jason.)
Matthew Scroggs has released this year’s Christmas card for Chalkdust magazine.
This year’s card contains nine mathematical puzzles. Finding the right answers will reveal a Christmas-themed picture in the image above.
As the Civil War fractured Washington D.C., socialite Rose O’Neal Greenhow coordinated a vital spy ring to funnel information to the Confederates. In this week’s episode of the Futility Closet podcast we’ll describe one of the war’s most unlikely spies, and her determination to aid the South.
We’ll also fragment the queen’s birthday and puzzle over a paid game of pinball.
In 1978, archaeologists excavating a late Paleolithic tomb in northern Israel uncovered the skeletons of an elderly human and a 5-month-old puppy. They had lain there together for 12,000 years.
“The most striking thing about these remains was the fact that whoever presided over the original burial had carefully arranged the dead person’s left hand so that it rested, in a timeless and eloquent gesture of attachment, on the puppy’s shoulder,” writes James Serpell in In the Company of Animals (1996).
“The contents of this tomb not only provide us with some of the earliest solid evidence of animal domestication, they also strongly imply that man’s primordial relationship with this particular species was an affectionate one. In other words, prehistoric man may have loved his dogs and his other domestic animals as pets long before he made use of them for any other purpose.”
(Simon J.M. Davis and François R. Valla, “Evidence for Domestication of the Dog 12,000 Years Ago in the Natufian of Israel,” Nature 276:5688 [1978], 608.)
Portuguese graffiti artist Odeith invested 30 spray cans and 10 hours to transform two concrete walls into a wrecked bus.
During a storm in January 1992, a container was swept overboard from a ship in the North Pacific. As it happened, it contained 28,800 children’s bath toys, and oceanographer Curtis Ebbesmeyer realized they offered the basis for a serendipitous study of surface currents. Working with his colleague James Ingraham, Ebbesmeyer began to track the toys as they drifted around the globe, accumulating reports from beachcombers, coastal workers, and local residents as they began to wash up on beaches. Using computer models, they were able to predict correctly that toys would make landfall in Washington state, Japan, and Alaska, and even become trapped in pack ice and spend years creeping across the top of the world before making an eventual reappearance in the North Atlantic. “Ultimately,” Ebbesmeyer wrote, “the toys will turn to dust, joining the scum of plastic powder which rides the global ocean.”
For some reason, media accounts of the story always carried the image of a solitary rubber duck, though the toys had also included beavers, turtles, and frogs. “Maybe it’s a kind of racism,” Ebbesmeyer speculated to journalist Donovan Hohn in 2007. “Speciesism.”
See Ambassador.
sitzmark
n. an impression made in snow by a skier falling backward onto his or her backside
The high-altitude glacial lake Roopkund, in the Indian state of Uttarakhand, contains a large number of human skeletons. Local legend tells of a royal party who were killed by a large hailstorm near here, and many of the skeletons show signs of blows by large round objects falling from above. Radiocarbon dating estimates that these people died around 850 CE.
But another set of victims seem to have succumbed much more recently, a group from the eastern Mediterranean who died only 200 years ago. So the casualties can’t all be attributed to a single catastrophic event, but the full truth is still emerging.
