20170628, 01:19  #1 
"Matthew Anderson"
Dec 2010
Oregon, USA
3^{2}·71 Posts 
random walk
Hi Mersennneforum,
Here is a puzzle that some of you will enjoy. It is a modification of the 'drunkard's walk'. A man leaves a bar and flips a coin. If it is heads, he walks one mile north. If it is tails, he goes one mile east. Then, given two possible locations, he flips a coin again. Heads, sends him only half a mile north and tails he goes half a mile east. The third flip of the coin only sends him a quarter of a mile in either of the two directions. Assume the drunkard can filp a coin an infinite number of times. The questions are what is the longest distance and the shortest distance that he travels as measured 'as the crow flies from the bar? Also, what is the distribution of distances he travels? Regards, Matt Last fiddled with by MattcAnderson on 20170628 at 01:20 Reason: forgot about infinite flipping. 
20170628, 01:54  #2  
"Forget I exist"
Jul 2009
Dumbassville
2^{6}×131 Posts 
Quote:
Last fiddled with by science_man_88 on 20170628 at 02:09 

20170628, 02:35  #3 
Undefined
"The unspeakable one"
Jun 2006
My evil lair
3^{2}×23×29 Posts 
And what if the bar is one mile South of the North Pole?

20170628, 03:17  #4 
"Matthew Anderson"
Dec 2010
Oregon, USA
3^{2}×71 Posts 
Hi Again,
I don't have a full solution, but the attached file has some answers. Regards, Matt 
20170628, 05:08  #5 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
10010000101110_{2} Posts 

20170628, 05:15  #6  
Undefined
"The unspeakable one"
Jun 2006
My evil lair
1011101110011_{2} Posts 
Quote:
I gave a hint already above. But the location (in both space and time) of the bar is important. Different terrains, different altitudes, even different days will all change the distance calculations. 

20170628, 05:50  #7 
6809 > 6502
"""""""""""""""""""
Aug 2003
101Γ103 Posts
2^{2}×3×13×59 Posts 
I figured that 2.0 was the max and 1.414 was the min.
Ran 10000 simulations (and 8000 in a second run) and found that this is the case. Attached is a histogram of the values. The separate run of 8000 gave an average of 1.624 and Std Dev of 0.179 It would be interesting to plot the actual points, but the should fall on/near the (0,2) to (2,0) line. What if the first 2 were between north and east, then the next 2 are west and south? 
20170628, 11:29  #8 
"Matthew Anderson"
Dec 2010
Oregon, USA
3^{2}×71 Posts 
Hi again,
Okay, lets assume the bar is in Paris. There is no need to march to the north pole and then try to walk east. Regards, Matt 
20170628, 12:15  #9 
Undefined
"The unspeakable one"
Jun 2006
My evil lair
3^{2}×23×29 Posts 
The scenario as posted only shows two flips of the coin as actual actions. The third flip specifies a directional possibility but fails to say which flip maps to which direction, and also doesn't actually say if any action is taken, merely what would happen. And the following paragraph says the "drunkard can filp [sic] a coin an infinite number of times", but doesn't say how that relates to any previous actions, (or what a filp is).
So based upon that the maximal distance to two heads going North for 1.5 miles. And the minimal distance is tails then heads going East then North (from Paris) for marginally less than sqrt(5/4) miles, because there is a small curvature of the path when going East which I'm too lazy to work out just now. Plus lots of assumptions about how crows fly. 
20170628, 13:53  #10 
Aug 2006
3×1,987 Posts 
I think we're on an infinite plane with two distinguished orthogonal directions.

20170628, 17:44  #11 
Bamboozled!
"πΊππ·π·π"
May 2003
Down not across
24414_{8} Posts 

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