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Posts by Nancy
Can you twist the above balloon using just one balloon? (Assume you can’t use scissors or pop the balloon into more than one segment.)
If you can figure out the above, you are solving an equivalent problem to one that the mathematician Euler resolved in 1735, laying the foundations of graph theory!
If you are familiar with graph theory, you may remember this as the “Seven Bridges of Königsberg” problem.
At the time, the city of Königsberg, Prussia had 7 bridges connecting its pieces of land that were separated by branches of the Pregel River. The question posed was whether you could walk through the city and cross all bridges exactly one time. Euler broke down this problem abstractly. In the below graph, the edges (lines) represent the bridges and the nodes (dots) represent the land the bridges connect.
If you compare a graph of the seven bridges of Königsberg with a graph of the balloon (I’ve blogged about graphing balloons before), you’ll see that they are the same. Asking whether you can cross all the bridges exactly once is the same as asking whether you can twist the above balloon using one balloon!
Pretty cool, huh?!
Balloon twisting has a lot to do with graph theory!
(Don’t read below this if you don’t want to know the solution yet.)
Some graph theory basics: if a node (aka vertex) has an odd number of lines coming from it, it is an odd node. If a node has an even number of lines coming from it, it is an even node. Euler proved that if a graph has exactly two odd nodes, you will be able to cross each edge exactly once (there is a Eulerian path), but you have to start at one of the odd nodes and end at the other odd node. If the graph has ALL even nodes, you can cross each edge exactly once AND end at the same point you started (the graph has a Eulerian path AND Eulerian circuit.)
In the graph above, all four nodes are of odd degree, so this graph does not have a Eulerian path or Eulerian circuit. The answer is: you cannot cross all bridges exactly once. And you cannot twist the balloon with exactly one balloon. (You’ll need two balloon segments.)
And for you history buffs, two of these bridges still exist in what was Königsberg, today Kaliningrad [Russia]. (Two of the bridges were damaged by Allied bombing during WWII, a couple were demolished and replaced with a new highway, and one was rebuilt.)
Now back to balloon twisting – I wonder how I can use this balloon – heart? head of a mouse or koala bear? base of a spaceship?
Happy Birthday to my super energetic and fun nephew! His favorite Ninja Turtle is Leonardo, the leader of the team!
To make this balloon, I used a green 350 for the head (with the same design as before), a brown and a goldenrod 260s for the shell, green 160s for the arms, a green 350 for the legs, two gray 160s for the swords, and brown 160s for the arm bands and leg bands. I had meant to take a picture of the new design of the shell, but forgot! I finished the balloon by using a a white paint marker and black marker for the face.
This birthday cake was made for a fun, energetic little girl who LOVES Minnie Mouse! Happy Birthday Kylen! It was so great to celebrate with you! Can’t believe you’re growing up so quickly!
This balloon is a birthday cake balloon with a Minnie Mouse balloon on top (the feet are squeezed in.) I had thought about twisting Minnie Mouse permanently in on top of the cake, but this way, you can easily take Minnie off and just hold the Minnie Mouse balloon if you want.
The balloon cake uses the same pentagonal base design you see in most of my balloon cakes. Minnie Mouse is made based on this design by Syan. I need to work on the cheeks a bit so Minnie is less like a chipmunk. Not pictured here is a heart balloon that I attached to Minnie’s hand with “Happy Birthday!” written on it.
This bird design by Brian Walendziak is super cute! I’ve been trying to figure out other ways to use geo blossom balloons. This design uses a geo blossom balloon turned inside out.
I made a couple small changes – the back of the balloon is all purple. Instead of twisting a cluster of four white bubbles, I used a purple 260 for two bubbles and a white 260 for two bubbles (Brian suggests using four white bubbles to save time.) You also don’t need to twist a bubble to attach the beak. You can just wrap the knot around the balloon where you want to attach it. If you do twist a cluster of four white bubbles, you could make the bird have a face on both sides.