My Solutions to the Seven Bridges of Königsberg Math Problem

Tonight I was reading the life story of Leonhard Euler and came across the Seven Bridges of Königsberg mathematical problem. It involves trying to cross the seven bridges of Königsberg of Prussia in a path where each bridge is crossed only once. The problem was solved negatively in 1736, by Euler declaring that it was impossible. The rules are to cross each bridge only once and return to the starting point. I spent about fifteen minutes eXamining this problem and came up with a novel solution: 

1. Start in the Middle of a Bridge
2. Use Stilts  
3. Clarify the Definition of the Word 'Once' with Fractions

This is a picture that I obtained from wikipedia and reproduce through their Creative Commons Attribution-Share Alike 3.0 Unported license. I have modified the original picture with labels of the bridges and a path for my journey.

My journey starts in the very middle of bridge A, with the starting point highlighted in purple

Then I travel across bridges B, C and D and arrive at a point midway between the entrance for bridges E and F

Then I climb aboard a pair of gigantic stilts and cross bridges E and F at the same time. This route is highlighted with a green path.

Then I get off the gigantic stilts and go across bridge G and then cross the same half of bridge A to arrive back at my starting point, in the middle of bridge A.

So I have crossed half of bridge A twice, so one "half" plus one "half" equals one "once" and I have returned to my starting point. The only thing I regret is that my cup of hot coffee sat in the breeze of my air conditioner too long while I typed this and did the artwork and now it is cold. Luckily I don't have to use stilts to get to my microwave oven in the kitchen or cross any bridges.

Reference to the wikipedia article: Seven Bridges of Königsberg Math Problem

Update: I just now realized I could have walked with gigantic stilts the entire journey, but I am so used to my Vibram FiveFingers shoes that I prefer to walk in them as often as possible. I figure the city is quite old and there may be cobblestones involved, so I will have to practice stilting on that particular surface. I am imagining myself in a humongous Godzilla costume tromping through Königsberg destroying everything with my gigantic stilts.

Update 2011.07.17: This morning I thought of another route with stilts that doesn't involve starting and stopping in the middle of a bridge. I don't have access to draw another map at the moment so I'll describe it using the current map lettering scheme.

Start on the bottom part of the map at the letter B, not actually on the bridge B.
Cross bridges B and G at the same time on the stilts.
Cross bridge F with both feet.
Cross bridges E and D at the same time on stilts, then continue on with the right stilt to cross bridge C
Cross bridge A and return to the starting point at letter B.

Update 2011.08.05: This morning while demonstrating a Möbius Strip to my favorite restaurant owner, I thought of another route AND another means of crossing bridges. I thought to myself, why not go over AND under the bridge? Cross the ENTIRE bridge? In other words, walk on the UNDER side of the bridge, kinda like Batman and Robin, with suction cups on the bottom of your shoes!

So I drew another map that shows the path, with color coding of GREEN meaning that the bridge was crossed on the TOP side, and PURPLE showing that it was crossed on the bottom. I start with the GREEN square and cross bridge A first, then go to B then G on top, then return on those three bridges on the underside, via the PURPLE path. Then Bridge C on TOP then return on its underside, then the top side of bridge D then the top and bottom of bridge E, then the top and bottom of bridge F, then lastly on the bottom side of bridge D to return to the starting point.

Update 2011.08.14 - I figured out another path that crosses all seven bridges ONCE but doesn't take you back to the original starting point. The secret to this path is to go ALL the way around the world. Use the map (above) for my most recent under over solution and start at the green square. Travel across A, B, C, D, E, and then F. At this point it looks like you are trapped on the top side BUT all you have to do is think GLOBALLY - and then just go "north"/"up" over the earth by land and sea; north, then south then north again back around to reach bridge G, and then cross it.

Second Update 2011.08.14 - I decided to try hooking two Möbius strips together starting with a piece of paper in the shape of a cross. At the intersection of the horizontal and vertical pieces I drew the map of Königsberg. Then I drew the river so that it reconnected with itself on the "horizontal" Möbius strip. The "vertical" pieces were hooked into the horizontal pieces, also twisted a half turn to form another Mobius strip.

After traveling from bridges A through F, the path starts down the vertical piece. When it reaches the intersection of the two Möbius strips, the path takes a right turn because it is blocked by the river. So then the path simply continues then on the horizontal member until it makes it's way back around to the "other" side of the river at bridge G. The path then goes across the bridge G to complete the journey. Each bridge is only crossed once and the river is never crossed except by a bridge path.

I shot photos of my double Möbius Strip creation and published parts of the path: