A Triangle Calculator

Image: Wikimedia Commons

Edric Cane came up with a simple way to establish any row in Pascal’s triangle, creating a simple sequence of fractions that, when multiplied successively, will produce the numbers in any desired row. Here’s an example for Row 7, giving the coefficients for (a + b)7 = a7 + 7a6b + 21a5b2 + 35a4b3 + 35a3b4 + 21a2b5 + 7ab6 + b7:

triangle calculator - row 7

Another example, for Row 10:

triangle calculator - row 10

The same can be done for any desired row.

(Thanks, Alex.)

Theme and Variations

Image: Wikimedia Commons

All of Johann Sebastian Bach’s surviving brothers were named Johann: Johann Rudolf, Johann Christoph, Johann Balthasar, Johannes Jonas, and Johann Jacob. His father was Johann Ambrosius Bach, and his sister was Johanna Juditha.

By contrast, his other sister, Marie Salome, “stuck out like a sore thumb,” writes Jeremy Siepmann in Bach: Life and Works. “And they all had grandparents and uncles and cousins whose names were also Johann, something. Johann Sebastian’s own children included Johann Gottfried, Johann Christoph, Johann August, Johann Christian, and Johanna Carolina.”

(Thanks, Charlie.)

Some Odd Words

Doubtful but entertaining:

Several sources define vacansopapurosophobia as “fear of blank paper” — it’s not in the Oxford English Dictionary, but it’s certainly a useful word.

I’ve also seen artiformologicalintactitudinarianisminist, “one who studies 4-5-letter Latin prefixes and suffixes.” I don’t have a source for that; it’s not in the OED either.

In Say It My Way, Willard R. Espy defines a cypripareuniaphile as “one who takes special pleasure in sexual intercourse with prostitutes” and acyanoblepsianite as “one who cannot distinguish the color blue.”

In By the Sword, his history of swordsmen, Richard Cohen defines tsujigiri as “to try out a new sword on a chance passerby.” Apparently that’s a real practice.

And one that is in the OED: mallemaroking is “the boisterous and drunken exchange of hospitality between sailors in extreme northern waters.”

(Thanks, Dave.)



Miss Alice E. Lewis sent this curiosity to the Strand in 1903:

These false horseshoes were found in the moat at Birtsmorton Court, near Tewkesbury. It is supposed that they were used in the time of the Civil Wars, so as to deceive any person tracking the marks. The one on the left is supposed to leave the mark of a cow’s hoof, the one on the right that of a child’s foot.

The same idea has been used by moonshiners and patented at least twice. Does this really work?

An Odd Fact


Mentioned in James Tanton’s Mathematics Galore!:

In 1740 the French mathematician Philippe Naudé sent a letter to Leonhard Euler asking how many ways a positive integer could be written as a sum of distinct positive integers (regardless of their order). In considering the problem Euler found something remarkable.

Let D(n) be the number of ways to write n as a sum of distinct positive integers. So, for example, D(6) is 4 because there are four ways to do this for 6: 6, 5 + 1, 4 + 2, and 3 + 2 + 1.

And let O(n) be the number of ways to write n as a sum of odd integers. So O(6) is 4 because 6 can be written as 5 + 1, 3 + 3, 3 + 1 + 1 + 1, or 1 + 1 + 1 + 1 + 1 + 1.

Euler showed that O(N) always equals D(N).



In early 1919, under the headline “The Great Indian Rope Trick Photographed for the First Time,” the Strand published this image by Lieutenant F.W. Holmes, VC, MM. He said he’d taken it at Kirkee, near Poona, in 1917. An old man had begun “by unwinding from about his waist a long rope, which he threw upwards in the air, where it remained erect. The boy climbed to the top, where he balanced himself, as seen in the photograph, which I took at that moment. He then descended … I offer no explanation.”

London’s Magic Circle invited Holmes to present his photo at a special meeting open to the public, who were asked to wear evening dress “to give a good impression.” Holmes repeated his story, which seemed to challenge the position that the trick had never been performed or was the effect of hallucination or hypnosis.

The editor of the Magic Circular, S.W. Clarke, charged that the photo showed a boy “balanced on top of a rigid rope or pole.” Holmes had already stated that the juggler “had no pole — a thing that would have been impossible of concealment.” But under questioning he admitted that there had been no rope — he’d merely seen a boy balancing atop a bamboo pole and had taken a photo of it.

That should have disposed of the story. But, as often happens, news of the debunking was much less interesting than news of the “proof,” and few newspapers published it. “If the question of the rope trick’s existence arose, and it arose many times,” writes Peter Lamont in The Rise of the Indian Rope Trick, “somebody regularly pointed out that the camera never lied, but nobody ever suspected the photographer. As a result, the Holmes photograph remained for many definitive proof that the rope trick was real.”

Hope and Change

Just stumbled across this in an 1889 newspaper:

To those who love mathematics, here is a simple problem for you to figure out: A man purchased groceries to the amount of 34 cents. When he came to pay for the goods he found that he had only a $1 bill, a 3-cent piece and a 2-cent piece. The grocer, on his side, had only a 50-cent piece and a quarter. They appealed to a bystander for change, but he, although willing to oblige them, had only two dimes, a 5-cent piece, a 2-cent piece and a 1-cent piece. After some perplexity, however, change was made to the satisfaction of everyone concerned. What was the simplest way of accomplishing this?

($1 is worth 100 cents, a quarter 25 cents, and a dime 10 cents.)

Click for Answer