Each of these pairs of numbers contains the 10 digits:
57321 60984
35172 60984
58413 96702
59403 76182
Square any one of them and it will grow into its own 10-digit pandigital number.
Science & Math
In the Dark
“A universe simple enough to be understood is too simple to produce a mind capable of understanding it.” — Cambridge cosmologist John Barrow
Homework
One day while teaching a class at Yale, Shizuo Kakutani wrote a lemma on the blackboard and remarked that the proof was obvious. A student timidly raised his hand and said that it wasn’t obvious to him. Kakutani stared at the lemma for some moments and realized that he couldn’t prove it himself. He apologized and said he would report back at the next class meeting.
After class he went straight to his office and worked for some time further on the proof. Still unsuccessful, he skipped lunch, went to the library, and tracked down the original paper. It stated the lemma clearly but left the proof as an “exercise for the reader.”
The author was Shizuo Kakutani.
Who’s On First?
Stigler’s Law of Eponymy states that “no scientific discovery is named after its original discoverer.” Examples:
- Arabic numerals were invented in India.
- Darwin lists 18 predecessors who had advanced the idea of evolution by natural selection.
- Freeman Dyson credited the idea of the Dyson sphere to Olaf Stapledon.
- Salmonella was discovered by Theobald Smith but named after Daniel Elmer Salmon.
- Copernicus propounded Gresham’s Law.
- Pell’s equation was first solved by William Brouncker.
- Euler’s number was discovered by Jacob Bernoulli.
- The Gaussian distribution was introduced by Abraham de Moivre.
- The Mandelbrot set was discovered by Pierre Fatou and Gaston Julia.
University of Chicago statistics professor Stephen Stigler advanced the idea in 1980.
Delightfully, he attributes it to Robert Merton.
Infallible
[Bertrand] Russell is reputed at a dinner party once to have said, ‘Oh, it is useless talking about inconsistent things, from an inconsistent proposition you can prove anything you like.’ Well, it is very easy to show this by mathematical means. But, as usual, Russell was much cleverer than this. Somebody at the dinner table said, ‘Oh, come on!’ He said, ‘Well, name an inconsistent proposition,’ and the man said, ‘Well, what shall we say, 2 = 1.’ ‘All right,’ said Russell, ‘what do you want me to prove?’ The man said, ‘I want you to prove that you are the pope.’ ‘Why,’ said Russell, ‘the pope and I are two, but two equals one, therefore the pope and I are one.’
— Jacob Bronowski, The Origins of Knowledge and Imagination, 1979
Hidden Order
Erect squares on the sides of any parallelogram and their centers will always form a square.
In any triangle, the midpoints of the sides and the feet of the altitudes always fall on a circle.
Stubborn
Write down any natural number, reverse its digits to form a new number, and add the two:
In most cases, repeating this procedure eventually yields a palindrome:
With 196, perversely, it does not — or, at least, it hasn’t in computer trials, which have repeated the process until it produced numbers 300 million digits long.
Is 196 somehow immune to producing palindromes? No one’s yet offered a conclusive proof — so we don’t know.
Fair Point
One threatening morning as Einstein was about to leave his house in Princeton, Mrs. Einstein advised him to take along a hat.
Einstein, who rarely used a hat, refused.
‘But it might rain!’ cautioned Mrs. Einstein.
‘So?’ replied the mathematician. ‘My hair will dry faster than my hat.’
– Howard Whitley Eves, In Mathematical Circles: Quadrants III and IV, 1969
Math Notes
12 = 3 × 4; 56 = 7 × 8
Words and Numbers
In English, the name of each integer shares a letter with each of its neighbors. ONE shares an O with TWO, TWO shares a T with THREE … and so on to infinity.