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• Ragnarok. The fabled end of the world,

• when giants, monsters, and Norse gods battle for the future.

• The gods were winning handily until the great serpentrmungandr emerged.

• It swallowed Valhalla, contorted itself across the land,

• and then merged into one continuous body with no head and no tail.

• As it begins to digest Valhalla,

• an exhausted Odin explains that he has just enough power to strike the creature

• with one final bolt of lightning.

• If you magnify his blast with your fabled hammer, Mjölnir,

• it should pierce the massive serpent.

• You'll run with super-speed along the serpent's body.

• When you hold your hammer high, Odin will strike it with lightning

• and splitrmungandr open at that point.

• Then, you'll need to continue running along its body

• until every part of it is destroyed.

• You can't run over the same section twice

• or you'll fall into the already blasted part of the snake.

• But you can make multiple passes through points where the creature

• intersects its own body.

• If you leave any portion un-zapped, Jörmungandr will magically regenerate,

• Odin's last power will be spent, and Valhalla will fall forever.

• What path can you take to destroy the serpent?

• Pause now to figure it out yourself!

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• One powerful way to solve problems is to simplify.

• And in this case, we can focus our attention on the two things

• that are important for our path:

• intersections and the stretches of snake between them.

• Or, as they're referred to in graph theory, nodes and edges.

• The edges are important because they're what we need to travel.

• And the nodes matter because they connect the edges,

• and are where we may need to make choices as we run from edge to edge.

• This simplification into nodes and edges leaves us

• with a ubiquitous and important mathematical object known as a graph,

• or network.

• We just need to figure out how to travel what mathematicians call an Eulerian path,

• which traces every edge exactly once.

• Instead of looking at the path as a whole, let's zoom in on a single node.

• During some moment in your run, you'll enter that node, and then exit it.

• That takes care of two edges.

• If you enter again, you'll need to exit again too,

• which requires another pair of edges.

• So every point along your path will have edges that come in pairs.

• One edge in each pair will function as entrance; the other as exit.

• And that means that the number of edges coming out of every node must be even.

• There are just two exceptions: the start and end points,

• where you can exit without entering, or vice versa.

• If we look at the network formed by the serpent again,

• and number how many edges emerge from each node,

• a pattern jumps out that fits what we just saw.

• Every node has an even number of edges emerging from it, except two.

• So one of these must be the start of your route, and the other the end.

• Interestingly enough, any connected network that has exactly 2 nodes

• with an odd number of edges will also contain an Eulerian path.

• The same is true if there are no nodes with an odd number of edges

• in that case the path starts and ends in the same spot.

• We can begin by taking care of this edge here.

• Now we can zig-zag back and forth across the whole snake

• until we reach the end.

• And that's just one solutionit helps to be systematic,

• but you're likely to happen upon many others

• once you know where to begin and end your run.

• You hold your hammer high at the opportune moment,

• and Odin sends the world-saving surge of lightning at you.

• Then you run like you've never run before.

• If you can pull this off, surely nothing could stop the might of the Norse Gods.

• And if something like that were out there, slouching its way towards you

• well, that would be a story for another day.

Ragnarok. The fabled end of the world,

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# Can you solve the Ragnarok riddle? - Dan Finkel

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林宜悉 posted on 2020/07/03
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