## Saturday, September 2, 2017

### Number Sense

How good is your "number sense"? How many of the following can you answer without using a calculator or looking up a conversion factor?
1. Are there more inches in a mile, or Sundays in 1000 years?
2. Are there more seconds in a week, or feet in 100 miles?
3. Are there more millimeters in a mile, or seconds in a month?
4. Which is larger, multiplying all the numbers from 1 to 10, or multiplying just the even numbers from 1 to 16?
5. Which is longer, 666 days or 95 weeks?
6. Which is longer, 666 inches or 55 feet?
7. Which is longer, 666 hours or 28 days?
8. Are there more ounces in a ton or inches in a kilometer?
9. Which is hotter, $0^{\circ}C$ or $0^{\circ}F$?
10. Which is larger, $e^\pi$ or $\pi^e$?

Tim Kington said...

8/10. I messed up the ton one, and pi^e vs e^pi is brutal. I'm not sure I could get anywhere with that one given pencil and paper, and unlimited time.

Bill the Lizard said...

Tim,

I thought #5 was tricky, since they're so close. You didn't just use Newton's method for #10? ;)

Anonymous said...

For #10 you could use the Taylor series expansion for e^x = 1 + x + x^2/2! + ... and notice that e^x > 1 + x for x not 0. Substitute x -> pi/e - 1 to get e^(pi/e - 1) > pi/e. Multiplying by e gives e^(pi/e) > pi. Finally, raise both sides to the power e and you have e^pi > pi^e.