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> Once And Only Once
Hamlet
Posted: April 15, 2012 12:46 pm
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Hoi!

I have been thinking over mathematical conditional statements, eg. in The Seven Bridges of Köningsberg problem, from Wikipedia:
"The problem was to find a walk through the city that would cross each bridge once and only once."

and I'm thinking there must be a deeper philosophy behind saying "once and only once," instead of "once." I'm seeking insight, any opinions?


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CWB
Posted: April 15, 2012 01:13 pm
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it might delineate a "non conditional" definition such as :
bidirectional crossings count as two crossings ... basically , setting foot on a bridge takes it out of the picture for further use . thoughts of "well , i went this direction so the other direction is fair play" are negated .
one might consider the idea of going halfway across a bridge , jumping off into a boat and traveling to another bridge/area and then using "the other half-bridge crossing" left for later use .
ergo : "crossing" means use of the bridge (by any definition) , whether or not the "use" of that bridge is of the total span of that bridge .

think of it as a way around pages of "legalese" definitions of "crossing a bridge" .


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tekwiz
Posted: April 15, 2012 04:56 pm
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Yep, it means that there are other possible solutions based on interpretation of that phrase. Most likely through assuming that only one direction of crossing counts, ie crossing back over the same bridge doesn't count. The second part of the 'once' phrase limits the options to one.


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AwesomeMatt
Posted: April 15, 2012 07:38 pm
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"Once" might mean "at least once"
"Once and only once" is clear, a second passing is forbidden.

It makes the problem more interesting in its constraints. Else, you could do something stupid like, start at X, cross the first bridge, return home. Cross second bridge, return home. Etc.

But doing it once only requires finding a path that meanders about using all bridges but each bridge only once.
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CoulombMagician
Posted: April 16, 2012 02:10 am
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We can't ask Mr. Euler directly, however one would presume the phrasing is meant to imply at least once, but not more than once.in other words solutions which did not cross one or more bridges were excluded as well as solutions that crossed any bridge twice or more. The bridges joined two islands in the Pregel river and connected those islands to both riverbanks. Eric Bell briefly describes this early topology problem in his book "Men of Mathematics"
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MacFromOK
Posted: April 16, 2012 02:28 am
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QUOTE (CoulombMagician @ April 15, 2012 08:10 pm)
We can't ask Mr. Euler directly, however one would presume the phrasing is meant to imply at least once, but not more than once.

Agreed. beer.gif


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Colt45
Posted: April 16, 2012 07:24 am
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Yeah, it's just to show that it is only once, not at least once.


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