After thousands of years of trying, mathematicians are still working out the number known as pi or "π". We typically think of pi as approximately 3.14 but the most successful attempt to calculate it more precisely worked out its value to over 13 trillion digits after the decimal point. We have known since the 18th century that we will never be able to calculate all the digits of pi because it is an irrational number, one that continues forever without any repeating pattern.
In 1888, the logician John Venn, who also invented the Venn diagram, attempted to visually show that the digits of pi were random by drawing a graph showing the first 707 decimal places. He assigned a compass point to the digits 0 to 7 and then drew lines to show the path indicated by each digit.
Venn did this work using pen and paper but this is still used today with modern technology to create even more detailed and beautiful patterns.
But, despite the endless string of unpredictable digits that make up pi, it’s not what we call a truly random number. And it actually contains all sorts of surprising patterns.
Normal not random
The reason we can’t call pi random is because the digits it comprises are precisely determined and fixed. For example, the second decimal place in pi is always 4. So you can’t ask what the probability would be of a different number taking this position. It isn’t randomly positioned.
But we can ask the related question: “Is pi a normal number?” A decimal number is said to be normal when every sequence of possible digits is equally likely to appear in it, making the numbers look random even if they technically aren’t. By looking at the digits of pi and applying statistical tests you can try to determine if it is normal. From the tests performed so far, it is still an open question whether pi is normal or not.
For example in 2003, Yasumasa Kanada published the distribution of the number of times different digits appear in the first trillion digits of pi:
Digit Occurrences 0 99,999,485,134 1 99,999,945,664 2 100,000,480,057 3 99,999,787,805 4 100,000,357,857 5 99,999,671,008 6 99,999,807,503 7 99,999,818,723 8 100,000,791,469 9 99,999,854,780 Total 1,000,000,000,000
His results imply that these digits seem to be fairly evenly distributed, but it is not enough to prove that all of pi would be normal.
We need to remember the surprising fact that if pi was normal then any finite sequence of digits you could name could be found in it. For example, at position 768 in the pi digits there are six 9s in succession. The chance of this happening if pi is normal and every sequence of n digits is equally likely to occur, is 0.08%.
This block of nines is famously called the “Feynman Point” after the Nobel Prize-winner Richard Feynman. He once jokingly claimed that if he had to recite pi digits he would name them up to this point and then say “and so on”.
8 of the very hardest maths puzzles
8 of the very hardest maths puzzles
1/8 Crossing the bridge
Four people need to cross a rickety bridge at night. Unfortunately, they have only one torch and the bridge is too dangerous to cross without one. The bridge is only strong enough to support two people at a time. Not all people take the same time to cross the bridge. Times for each person: 1 minute, 2 minutes, 7 minutes, and 10 minutes. What is the shortest time needed for all four of them to cross the bridge?
Claire Backhouse/flickr/Creative Commons
2/8 Number magic
If you multiply me by 2, subtract 1, and read the reverse the result you’ll find me. Which numbers can I be?
Dustin Liebenow/flickr/Creative Commons
3/8 One thousand monkeys
A very big building in which one thousand monkeys are living is lighted by one thousand lamps. Every lamp is connected to a unique on/off switch, which are numbered from 1 to 1000. At some moment, all lamps are switched off. But because it is becoming darker, the monkeys would like to switch on the lights. They will do this in the following way: Monkey 1 presses all switches that are a multiple of 1 Monkey 2 presses all switches that are a multiple of 2 Monkey 3 presses all switches that are a multiple of 3 Monkey 4 presses all switches that are a multiple of 4 Etc., etc. How many lamps are switched on after monkey 1000 pressed his switches? And which lamps are switched on?
Buddhika Weerasinghe/Getty Images
4/8 School lockers
A high school has a strange principal. On the first day, he has his students perform an odd opening day ceremony: There are one thousand lockers and one thousand students in the school. The principal asks the first student to go to every locker and open it. Then he has the second student go to every second locker and close it. The third goes to every third locker and, if it is closed, he opens it, and if it is open, he closes it. The fourth student does this to every fourth locker, and so on. After the process is completed with the thousandth student, how many lockers are open?
Brett Levin/flickr/Creative Commons
5/8 One bulb, three switches
You have three switches in a room. One of them is for a bulb in next room. You cannot see whether the bulb is on or off until you enter the room. What is the minimum number of times you need to go in to the room to determine which switch corresponds to the bulb in next room?
JOEL SAGET/AFP/Getty Images
6/8 Cheryl's birthday
Albert and Bernard just become friends with Cheryl, and they want to know when her birthday is. Cheryl gives them a list of 10 possible dates: May 15, May 16, May 19, June 17, June 18, July 14, July 16, August 14, August 15, and August 17 Cheryl then tells Albert and Bernard separately the month and the day of her birthday respectively. Albert: I don’t know when Cheryl’s birthday is, but I know that Bernard does not know too. Bernard: At first I don’t know when Cheryl’s birthday is, but I know now. Albert: Then I also know when Cheryl's birthday is. So when is Cheryl’s birthday?
Jessica Diamond/flickr/Creative Commons
7/8 Sunday's child
Recently, somebody said: “My grandfather was born on the first Sunday of the year. His seventh birthday was also on a Sunday.” In which year was said grandfather born?
Will Clayton/flickr/Creative Commons
8/8 Probability of having boy
In a country where everyone wants a boy, each family continues having babies until they have a boy. After some time, what is the proportion of boys to girls in the country? (Assuming probability of having a boy or a girl is the same).
WALTRAUD GRUBITZSCH/AFP/Getty Images
Other interesting sequences of digits have also been found. At position 17,387,594,880 you find the sequence 0123456789, and surprisingly earlier at position 60 you find these ten digits in a scrambled order.
Pi-hunters search for dates of birth and other significant personal numbers in pi asking the question: “Where do I occur in the pi digits?” If you want to test to see where your own special numbers are in pi, then you can do so by using the free online software called Pi birthdays.