Douglas Hofstadter's Innumeracy Exercises

The following fun exercises appeared on p. 134 of Hofstadter's Metamagical Themas, a collection of his Scientific American articles from the early 1980s (the title is a play on Martin Gardner's column, "Mathematical Games").

I have not sought permission to reprint them here, although I doubt he would be upset if more people explored—and became fluent with—really big numbers. I encourage you to buy the book, which has the same élan vitale that made GEB a classic.

Note: In case anyone is interested I have posted my "solutions." If you don't care about my thought process, you can grab my results as a list of tuples for easy processing; it might be interesting to aggregate people's answers.

I would suggest to interested readers that they attempt to build up their own numeracy in a very simple way. All they need to do is to get a sheet of paper and write down on it the numbers 1 to 20. Then they should proceed to think a bit about some large numbers that seem of interest to them, and try to estimate them within one order of magnitude (or two, for the larger ones). By "estimate" here, I mean actually do a back-of-the-envelope (or mental) calculation, ignoring all but factors of ten. Then they should attach the idea to the computed number. Here are some samples of large numbers:

  1. What's the gross state product of California?
  2. How many people die per day on Earth?
  3. How many traffic lights are there in New York City?
  4. How many Chinese restaurants are there in the U.S.?
  5. How many passenger-miles are flown each day in the U.S.?
  6. How many volumes are there in the Library of Congress?
  7. How many notes are played in the full career of a concert pianist?
  8. How many square miles are there in the U.S.? How many of them have you been in?
  9. How many syllables have been uttered by humans since 1400 A.D.?
  10. How many "300" games are bowled in the U.S. per year?
  11. How many stitches are there in a stocking?
  12. How many characters does one need to know to read a Chinese newspaper?
  13. How many sperms are there per ejaculate?
  14. How many condors remain in the U.S.?
  15. How many moving parts are in the Columbia space shuttle?
  16. How many people in the U.S. are called "Michael Jackson"? "Naomi Hunt"?
  17. What volume of oil is removed from the earth each year?
  18. How many barrels of oil are left in the world?
  19. How much carbon monoxide enters the atmosphere each year in auto exhaust fumes?
  20. How many meaningful, grammatical, ten-word sentences are there in English?
  21. How long did it take the 200-inch mirror of the Palomar telescope to cool down?
  22. What angle does the earth's orbit subtend, as seen from Sirius?
  23. What angle does the Andromeda galaxy subtend, as seen from earth?
  24. How many heartbeats does a typical creature live?
  25. How many insects (of how many species) are now alive?
  26. How many giraffes are now alive? Tigers? Ostriches? Horseshoe crabs? Jellyfish?
  27. What are the pressure and temperature at the bottom of the ocean?
  28. How many tons of garbage does New York City put out each week?
  29. How many letters did Oscar Wilde write in his lifetime?
  30. How many typefaces have been designed for the Latin alphabet?
  31. How fast do meteorites move through the atmosphere?
  32. How many digits are in 720 factorial?
  33. How much is a brick of gold worth?
  34. How many gold bricks are there in Fort Knox? How much is it worth?
  35. How fast do your wisdom teeth grow (in miles per hour, say)?
  36. How fast does your hair grow (again in miles per hour)?\
  37. How fast is Venice sinking?
  38. How far is a million feet? A billion inches?
  39. What is the weight of the Empire State Building? Of Hoover Dam? Of a fully loaded jumbo jet?
  40. How many commercial airline takeoffs occur each year in the world?

These or similar questions will do. The main thing is to attach some concreteness to those numbers from 1 to 20, seen as exponents. They are like dates in history. At first, a date like "1685" may be utterly meaningless to you, but if you love music and find out that Bach was born that year, all of a sudden it sticks. Likewise with the secondary meaning for small numbers. I can't guarantee it will work miracles, but you may increase your own numeracy and you may help to increase others'. Merry numbers!