Hidden in π

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The Feynman point and other weird places in π

Six 9s in a row at decimal 762, a zero that is late for its own party, and why streaks that look impossible are exactly what randomness looks like.

· 5 min read

Count along the decimals of π, past the 3, and something odd happens at decimal 762. The digits go …1349999998372… and there they are: six 9s in a row, sitting at decimals 762 to 767. This spot has a name. It is called the Feynman point.

The story, and the catch

Richard Feynman was a Nobel Prize winning physicist and a famous joker. The story goes that he once said he would like to learn π by heart up to decimal 762, so that he could recite the digits, reach the six 9s, say "nine, nine, nine, nine, nine, nine, and so on" and leave his listeners thinking π had stopped.

It is a great story. Unfortunately, it may not be his. Nobody has found the remark in Feynman's own books, and it isn't in James Gleick's biography of him either. The earliest written version comes from the scientist and writer Douglas Hofstadter, who in a 1985 book described wanting to memorise π up to that very spot himself. The name stuck to Feynman anyway, which is probably what he would have found funniest.

How surprising is it?

Six 9s in a row feels like a message. Is it?

Think of π's digits as rolls of a ten-sided dice. The chance that six particular rolls all come up 9 is one in a million (ten times ten times ten times ten times ten times ten). There are about 762 places where a run of six could have started before the Feynman point, so the chance of seeing six 9s that early is roughly 762 in a million: less than one chance in a thousand.

That sounds very rare. But it is a bit of a trick, because we only noticed the six 9s after they happened. If π had six 3s or six 0s that early, we would have been just as amazed, and there are plenty of other patterns that would have caught our eye too: 123456, 314159, 000000. Once you count every pattern that would have made you say "whoa", surprises like this one become much less surprising. Somewhere, something odd is bound to happen.

When does it happen again?

Six 9s don't come back for a long time. The next run of six 9s starts at decimal 193,034. The first run of six identical digits that are not 9s is six 8s, at decimal 222,299. Seven 9s in a row first appear at decimal 1,722,776, eight 9s at 36,356,642 and nine 9s at 564,665,206. Each extra 9 sends the run roughly ten times deeper, which is exactly what the dice model says should happen.

Other weird places near the start

You don't have to go far to find oddities. A few from the first 2,000 decimals:

  • The missing zero. The digits 1 to 9 all show up within the first 13 decimals, but the first 0 doesn't appear until decimal 32. On average a zero should show up about every ten digits, so that is a long wait.
  • Triples. The first 111 starts at decimal 153, the first 000 at decimal 601 and the first 777 at decimal 1589.
  • The answer to everything. Fans of The Hitchhiker's Guide to the Galaxy will be pleased to know that 42 first turns up at decimal 92.
  • Round numbers. Decimal 10 is a 5, while decimal 100 and decimal 1,000 are both 9s. And the millionth decimal is a 1.

None of these mean anything. That is the point. A truly random string of digits is full of runs, repeats and coincidences. People who are asked to write down "random" digits by hand usually avoid repeats, and their fake digits end up looking less random than the real thing.

Streaks in your own life

The same idea explains a lot of everyday surprises. Flip a coin a hundred times and you will very likely see a run of five or six heads in a row somewhere. Put twenty-three people in a room and there is a better than even chance that two of them share a birthday. Random things clump. Our brains, which are brilliant at spotting patterns, then shout "that can't be chance!" when it is exactly what chance does.

Explore the weird bits

Sources

Filed under feynman-pointpatternsprobability

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