Ponder This Challenge - May 2000 - Northwest to the north pole

Ponder This Challenge:

This month we have a multi-part puzzle, suggested by Brad Long.
Throughout, we will assume the earth is a sphere with a
circumference of 25,000 miles.

Part 1:

Start at the equator where it crosses the prime meridian.
Head northwest, and continue to travel northwest
until you reach the north pole. How far will you travel?

Part 2:

Where will you be when you first cross the prime meridian?
(How far from the north pole?)

Part 3:

Where would you first cross the prime meridian if you always headed north-northwest (NNW, i.e. 22.5 degrees away from north) instead of northwest?

Part 4:

We originally asked you to discuss the cinematographic implications. This was an attempt at humor, based on the Alfred Hitchcock film titled "North by Northwest". Since then we've found out that (a) the Hitchcock film has different, unrelated titles in different languages, so that the joke falls flat; and (b) the direction "North by Northwest" is not where we thought it was. So we're skipping Part 4.

We will post the names of those who submit a correct, original solution! If you don't want your name posted then please include such a statement in your submission!

We invite visitors to our website to submit an elegant solution. Send your submission to the ponder@il.ibm.com.

If you have any problems you think we might enjoy, please send them in. All replies should be sent to: ponder@il.ibm.com

Solution

• Click here to view the solution

  Part 1:

  Let y denote latitude in radians, so that
  y=0 at the equator and y=pi/2 at the pole.
  Let x denote longitude in radians, with x=0 at prime meridian.
  Let t be total distance travelled, measured in radii.

  Then dy/dt = cos(pi/4) = 1/sqrt(2).
  So t=y*sqrt(2)+C, and initial conditions give C=0.
  When we finish, y=pi/2, so t=pi/sqrt(2) radii.
  Use (25000 miles = 2 pi radii) to conclude
  t = 6250 sqrt(2) miles = 8839 miles.

  Part 2:

  dx/dt = (sin(pi/4))*(1/cos(y)).
  Then dx/dy=1/cos(y).
  x = log(|sec y + tan y|) + C.
  Again initial conditions imply C=0.
  We can solve to find y=(pi/2)-2*arctan(exp(-x)).
  Substituting x=2*pi we find
  y=(pi/2)-2*arctan(exp(-2*pi)),
  so that the distance from the pole is:
  (pi/2)-y=2*arctan(exp(-2*pi)) = 0.003735 radians = 14.86 miles.

  Part 3:

  If we change the heading to north by northwest, we find:
  dy/dt = cos(pi/8).
  dx/dt = (sin(pi/8))*(1/cos(y)).
  dx/dy = (tan(pi/8))*(1/cos(y)).
  x = (tan(pi/8))*(log(|sec y + tan y|)) + C.
  y = (pi/2)-2*arctan(exp(-x/tan(pi/8))).
  Recalling tan(pi/8)=sqrt(2)-1, and substituting x=2*pi, we find:

  y = (pi/2)-2*arctan(exp(-2*pi/(sqrt(2)-1))),
  so that the distance from the pole is:
  (pi/2)-y=2*arctan(exp(-2*pi/(sqrt(2)-1)))=0.0000005167 radians
  = 0.002056 miles = 10.855 feet.

  Part 4:

  I was going to make some lame joke about Alfred Hitchcock's movies, "North by Northwest" and "The Birds", but the punch line would have involved penguins at the north pole (erroneously).

  ---

  If you have any problems you think we might enjoy, please send them in. All replies should be sent to: ponder@il.ibm.com

Solvers

• Jimmy Hu (5.1.2000 @ 3:10 PM EDT)
• Tao Lin (5.1.2000 @ 8:33 PM EDT)
• Stephan Voss (5.2.2000 @ 9:59 AM EDT)
• Robert Galejs (5.2.2000 @ 3:51 PM EDT)
• Dr. Ludger Weber (5.3.2000 @ 5:07 AM EDT)
• Manikantan Srinivasan (5.4.2000 @ 10:42 AM EDT)
• Ether Jones (5.4.2000 @ 8:06 PM EDT)
• Frankie Dintino (5.4.2000 @ 10:41 PM EDT)
• Tim Edmonds (5.5.2000 @ 7:04 AM EDT)
• Dave Biggar (5.5.2000 @ 7:34 AM EDT)
• Prithu Barthi Tiwari (5.5.2000 @ 10:44 AM EDT)
• K.V.Manjunath (5.5.2000 @ 10:44 AM EDT)
• Dan Peleg (5.5.2000 @ 12:41 PM EDT)
• Jacques Willekens (5.7.2000 @ 6:56 AM EDT)
• (5.7.2000 @ 6:56 AM EDT)
• Mohit Sauhta (5.8.2000 @ 5:38 PM EDT)
• (5.8.2000 @ 5:38 PM EDT)
• Bob Delaney (5.11.2000 @ 4:25 PM EDT)
• (5.11.2000 @ 4:25 PM EDT)
• Stephen M Hou (5.11.2000 @ 5:44 PM EDT)
• (5.11.2000 @ 5:44 PM EDT)
• Suman Datta (5.14.2000 @ 7:35 AM EDT)
• (5.14.2000 @ 7:35 AM EDT)
• Ariel Arad (5.17.2000 @ 12:58 PM EST)
• (5.17.2000 @ 12:58 PM EST)
• Bill Schwennicke (5.18.2000 @ 5:18 PM EDT)
• (5.18.2000 @ 5:18 PM EDT)
• Dean Flatt (5.20.2000 @ 6:45 PM EDT)
• (5.20.2000 @ 6:45 PM EDT)
• Yevgeniy Vladimirovich Kalinin (5.21.2000 @ 3:45 PM EDT)
• David Radcliffe (5.23.2000 @ 6:21 PM EDT)
• James Chan (5.30.2000 @ 2:34 AM EDT)