OT - Travel distance puzzle

QuikSand · 05-21-2003, 10:58 AM

Two friends are walking through a big city, and were trying to decide which of two fairly nearby restaurants to visit for lunch. After some discussion, they decided to go to A_____ rather than B____, because A_____ was a two blocks shorter trip.

However, as they were talking, two crows were listening on their conversation from overhead. (Suspension of disbelief needed here) To them, the decision made no sense - from the crows' perspective, A____ and B_____ were exactly the same distance away.

Assuming city blocks to be perfectly square, and assuming that neither restaurant was on the street or avenue that our two friends were on at the time, how far away were each of the two restaurants?

WSUCougar · 05-21-2003, 11:14 AM

albionmoonlight · 05-21-2003, 12:45 PM

A is 7 blocks up and 1 block over (8 blocks).

B is 5 blocks up and 5 blocks over (10 blocks).

The "crow flies" distance to each one is the same: Sqrt(50) blocks.

QuikSand · 05-21-2003, 12:48 PM

Very nice... I was starting to think that I needed to spell out some things more clearly (like the desire for integer answers).

Well done.

Maple Leafs · 05-21-2003, 12:48 PM

Albino's answer works for me.

I wound up with:

A is 11 blocks up, 3 over (14 block trip)
B is 9 blocks up, 7 over (16 block trip)

To the crows, each is sqrt(130)

albionmoonlight · 05-21-2003, 12:49 PM

dola--

nothing elegant about my solution (assuming that it is right). Once I figured that it was A^2 + B^2 = C^2, I just started writing down perfect squares until I found a combination that fit (49+1 = 25+25).

QuikSand · 05-21-2003, 12:50 PM

Yes, I realize that the answer is not unique... I inserted the term "fairly nearby" to try to preclude absurd answers... but both of these two work for me. I had alb's in mind, but Leafs' fits, too.

BishopMVP · 05-21-2003, 02:19 PM

A 5 blocks away

and

B 3 blocks up/4 to the side

works as well.

QuikSand · 05-21-2003, 02:25 PM

It works fine, except or this tidbit from the puzzle:

Quote:

Originally posted by QuikSand
and assuming that neither restaurant was on the street or avenue that our two friends were on at the time

QuikSand · 05-21-2003, 02:30 PM

So, maybe this is the method to find a whole series of solutions:

You should get my drift here...

5,0 and 4,3
7,1 and 5,5
9,2 and 6,7
11,3 and 7,9
...

Looks to me like for any integer N, this form will work:

(2N+3,N-1) and (N+3,2N+1)

Carrying out the algebra ought to prove this to be so... but the several cases above (including the one I just imputed on my own from the space left) strongly suggest we have a solid pattern.

05-21-2003, 11:14 AM	#2
WSUCougar Rider Of Rohan Join Date: Jul 2001 Location: Port Angeles, WA or Helm's Deep	2/3? __________________ It's not the years...it's the mileage.

05-21-2003, 12:45 PM	#3
albionmoonlight Head Coach Join Date: Oct 2000 Location: North Carolina	A is 7 blocks up and 1 block over (8 blocks). B is 5 blocks up and 5 blocks over (10 blocks). The "crow flies" distance to each one is the same: Sqrt(50) blocks.

05-21-2003, 12:48 PM	#4
QuikSand lolzcat Join Date: Oct 2000 Location: Annapolis, Md	Very nice... I was starting to think that I needed to spell out some things more clearly (like the desire for integer answers). Well done.

05-21-2003, 12:48 PM	#5
Maple Leafs Coordinator Join Date: Jan 2002	Albino's answer works for me. I wound up with: A is 11 blocks up, 3 over (14 block trip) B is 9 blocks up, 7 over (16 block trip) To the crows, each is sqrt(130)

05-21-2003, 12:49 PM	#6
albionmoonlight Head Coach Join Date: Oct 2000 Location: North Carolina	dola-- nothing elegant about my solution (assuming that it is right). Once I figured that it was A^2 + B^2 = C^2, I just started writing down perfect squares until I found a combination that fit (49+1 = 25+25).

05-21-2003, 12:50 PM	#7
QuikSand lolzcat Join Date: Oct 2000 Location: Annapolis, Md	Yes, I realize that the answer is not unique... I inserted the term "fairly nearby" to try to preclude absurd answers... but both of these two work for me. I had alb's in mind, but Leafs' fits, too.

05-21-2003, 02:19 PM	#8
BishopMVP Coordinator Join Date: Oct 2000 Location: Concord, MA/UMass	A 5 blocks away and B 3 blocks up/4 to the side works as well.

05-21-2003, 02:30 PM	#10
QuikSand lolzcat Join Date: Oct 2000 Location: Annapolis, Md	So, maybe this is the method to find a whole series of solutions: You should get my drift here... 5,0 and 4,3 7,1 and 5,5 9,2 and 6,7 11,3 and 7,9 ... Looks to me like for any integer N, this form will work: (2N+3,N-1) and (N+3,2N+1) Carrying out the algebra ought to prove this to be so... but the several cases above (including the one I just imputed on my own from the space left) strongly suggest we have a solid pattern.

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