In Pascal’s triangle, each number is the sum of the two above it. Obviously, the infinite pyramid contains an infinite number of 1s, but most numbers appear surprisingly seldom:

• 2 appears just once.
• 3, 4, 5, and all odd primes appear exactly twice.
• 6 appears three times.
• Infinitely many numbers appear exactly six times, but we don’t know whether any appear exactly five or seven times.
• 3003 appears eight times, possibly the only such specimen.

In 1971, Berkeley mathematician David Singmaster suggested that there may be a finite upper bound on the number of times that any number can appear (apart from 1). But that remains an unsolved problem.

I’d missed this: In 2006 a geneticist, a philosopher, and a chicken farmer all agreed that the egg came before the chicken.

Nottingham University geneticist John Brookfield pointed out that the first chicken (the first creature bearing chicken DNA) must have begun as an embryo in an egg. “The first living thing which we could say unequivocally was a member of the species would be this first egg, so I would conclude that the egg came first.”

David Papineau, philosopher of science at King’s College, London, agreed. “I would argue it is a chicken egg if it has a chicken in it. … If a kangaroo laid an egg from which an ostrich hatched, that would surely be an ostrich egg, not a kangaroo egg.”

And Charles Bournes, chair of trade body Great British Chicken, said, “Eggs were around long before the first chicken arrived. Of course they may not have been chicken eggs as we see them today, but they were eggs.”

According to the BBC, “Professor Brooke added the debate could finally be laid to rest.”

The second star in the Big Dipper’s handle is actually two stars, Mizar and Alcor. Distinguishing the two with the naked eye has been used as a test of vision for hundreds of years. Arabic tradition held that only those with the sharpest eyesight could see Mizar’s companion, and the 13th-century Persian astronomy writer Zakariya al-Qazwini wrote that “people tested their eyesight by this star.” In Japan, it was said that being unable to see Alcor with the naked eye foretold an impending death of old age, and Alexander von Humboldt and François Arago both noted that Alcor can be seen only with difficulty.

In The Herschels and Modern Astronomy (1901), Irish astronomer Agnes Mary Clerke wrote, “The Arabs in the desert regarded it as a test of penetrating vision; … Vidit Alcor, at non lunam plenam (Latin for ‘he saw Alcor, but not the full moon’), came to be a proverbial description of one keenly alive to trifles, but dull of apprehension for broad facts.”

(Thanks, Sharon.)