How good is your "number sense"? How many of the following can you answer without using a calculator or looking up a conversion factor?

Are there more inches in a mile, or Sundays in 1000 years?

Are there more seconds in a week, or feet in 100 miles?

Are there more millimeters in a mile, or seconds in a month?

Which is larger, multiplying all the numbers from 1 to 10, or multiplying just the even numbers from 1 to 16?

Which is longer, 666 days or 95 weeks?

Which is longer, 666 inches or 55 feet?

Which is longer, 666 hours or 28 days?

Are there more ounces in a ton or inches in a kilometer?

Which is hotter, $0^{\circ}C$ or $0^{\circ}F$?

Which is larger, $e^\pi$ or $\pi^e$?

Click below for the answers.

Inches in a mile. (63,360. There can be up to 52,178 Sundays in 1000 years.)

Seconds in a week. (604,800, compared to 528,000 feet in 100 miles.)

Seconds in a month. (Even if the month only has 28 days, that's 2,419,200 seconds, compared to only 1,609,340 millimeters in a mile.)

Just the even numbers from 1 to 16. (Multiplying all the numbers from 1 to 10 gives you 3,628,800. Multiplying the even numbers from 1 to 16 give you 10,321,920.)

666 days. (95 weeks is only 665 days.)

666 inches. (55 feet is 660 inches.)

28 days (which is 672 hours).

Inches in a kilometer. (39,370.1, compared to 35,840 ounces in a long ton, which is the heaviest ton.)

$0^{\circ}C$ is "hotter" since it is equal to $32^{\circ}F$

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.

## 3 comments:

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.

Tim,

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

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.

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