If squares are drawn on the sides of a triangle and external to it, then the areas of the triangles formed between the squares all equal the area of the triangle itself.

(Roger Webster, “Bride’s Chair Revisited,” Mathematical Gazette 78:483 [November 1994], 345-346.)

The red-crested tree rat hadn’t been seen since 1898 when one turned up at the front door of a Colombian ecolodge in 2011. It posed for photos for two hours and then disappeared again. “He just shuffled up the handrail near where we were sitting and seemed totally unperturbed by all the excitement he was causing,” said volunteer researcher Lizzie Noble. “We are absolutely delighted to have rediscovered such a wonderful creature after just a month of volunteering with ProAves.”

The Bermuda land snail had been thought extinct for 40 years when it turned up in a Hamilton alleyway. “The fact that there was so much concrete around them probably saved them from the predators that we believe killed the vast majority of the population island-wide,” ecologist Mark Outerbridge told the Royal Gazette. “People have been looking for these snails for decades and here they are surrounded by concrete and air conditioners living in a 100-square-foot alleyway in Hamilton.”

The mountain pygmy possum was first identified in a fossil in New South Wales in 1895 and thought to be extinct. Seventy years later, in August 1966, a live pygmy possum was found by chance in a ski hut in the Snowy Mountains of Victoria. R.M. Warneke telegraphed W.D.L. Ride, “BURRAMYS EXTANT STOP NOT REPEAT NOT EXTINCT STOP LIVE MALE CAPTURED MOUNT HOTHAM STOP AM TRYING FOR FEMALE.” The lonely possum died before a companion could be found, but four isolated populations of pygmy possums are now known to persist in the Snowy Mountains.

(Joseph F. Merritt, The Biology of Small Mammals, 2010.)

26072323311568661931
43744839742282591947
118132654413675138222
186378732807587076747
519650114814905002347

Any three of these numbers add up to a perfect square.

(Discovered by Stan Wagon.)

For what it’s worth: In 2015 University of Wisconsin psychologists Megan Savage and Charles Snowdon considered that music might appeal better to other species if it used the same tempos and frequency ranges that they use to communicate.

Accordingly they got musician David Teie to compose three songs that ought to appeal to felines and tried them out on 47 domestic cats, comparing their reactions to Bach’s “Air on a G String” and Fauré’s “Elegie.” The cat music was pitched about an octave higher than human voices, and its tempos replicated purring and suckling rather than a human heartbeat.

The cats showed no interest in the music intended for humans, but they showed a “significant preference for and interest in” Teie’s cat-targeted songs, approaching the speakers and often rubbing their scent glands on them. Also, for some reason young and old cats seemed to like the cat music better than middle-aged ones.

Savage and Snowdon conclude that these results “suggest novel and more appropriate ways for using music as auditory enrichment for nonhuman animals.” Here’s a sample to try on your own cat:

(Charles T. Snowdon, David Teie, and Megan Savage, “Cats Prefer Species-Appropriate Music,” Applied Animal Behaviour Science 166 [May 2015], 106–111.) (Thanks, Noah.)

These are called Borwein integrals, after David and Jonathan Borwein, the father-and-son mathematicians who first presented them in 2001.

Engineer Hanspeter Schmid writes, “[W]hen this fact was recently verified by a researcher using a computer algebra package, he concluded that there must be a ‘bug’ in the software. It is not a bug, though; this series of integrals really only results in π/2 up to a certain point, and then breaks down. This astonishes most mathematically educated readers, as especially those readers mentally extrapolate the sequence shown above and find it surprising that something fundamental should change when the factor sinc(x/15) is introduced.” He gives a graphic explanation of what’s happening.

(David Borwein and Jonathan M. Borwein, “Some Remarkable Properties of Sinc and Related Integrals,” Ramanujan Journal 5:1 [March 2001], 73–89.) (Thanks, Dan.)

In his 2008 book Impossible?, Julian Havil presents an argument offered in Massachusetts in 1854 contending that foreigners were more likely to be insane than native-born Americans. These figures were offered:

The probability that a foreign-born person was deemed insane was 625/230000 = 2.7 × 10^-3, and for a native-born person the probability was 2007/894676 = 2.2 × 10^-3, which seems to support the claim.

But we get a different story when we divide the data by social hierarchy, into what were called the pauper and independent classes:

In the pauper class the probability of a foreign-born person being deemed insane is 182/9272 = 0.02, which is the same as that for a native-born person (250/12763 = 0.02). And the same is true in the independent class, where both probabilities are 2.0 × 10^-3. Havil writes, “So, if an adjustment is made for the status of the individuals we see that there is no relationship at all between sanity and origin” (an example of Simpson’s paradox).

“Squaring the square” refers to tiling a square with other squares, each with sides of integer length.

In a “perfect” squared square, like the one above, each smaller square is of a different size. The Cambridge University team that first sought perfect squares found a novel way to go about it — they transformed the square tiling into an electrical circuit in which each square is a resistor that connects to its neighbors above and below, and then applied Kirchhoff’s circuit laws to that circuit.

The example below isn’t perfect, but the technique did succeed — the smallest perfect square they found is 69 units on a side.

(Rowland Leonard Brooks, et al., “The Dissection of Rectangles into Squares,” Duke Mathematical Journal 7:1 [1940], 312-340.)