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.