3,628,800

This post is inspired by a recent puzzle published by Alex Bellos in the Guardian: Did you solve it? Puzzles you can do in the pub | Mathematics | The Guardian

The title of the post is 10! = 1x2x3x4x5x6x7x8x9x10 = 3,628,800. Interestingly, this value is the same number of seconds in six weeks. The challenge in Bellos's puzzle was to make the comparison without using a calculator.

One can confirm this without a calculator. Write out the number of seconds in six weeks as:

6 (weeks) x 7 (days) x 24 (hours) x (60 minutes) x (60 seconds)

Now one can cancel factors in the above equation by matching up with terms in the equation for 10!.

Cancel the 6 in both equations.

Cancel the 7 in both equations.

Cancel the 24 by matching with the 3 and 8 in the 10! equation

Reduce the 60 to 6 in the seconds equation by cancelling the 10 in the 10! equation.

Reduce the other 60 to 6 in the seconds equation by canceling the the 2 and the 5 in the 10! equation.

These canceling operations leaves us with 6x6=36 in the seconds equations and 4x9=36 in the factorial equation, therefore the number of seconds in six weeks equals 10!.

Alex Bellos is also the author of one of my favorite math books: Math Vacation: My Favorite Math Websites (jamesmacmath.blogspot.com)