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The Dragon & Adhelia?
A Dragon consists of several heads and several necks, where each neck joins two heads. When a Dragons’s head A is hit by a sword, all the necks from head A disappear, but new necks grow up to connect head A to all the heads which weren’t connected to A. Adhelia defeats a Dragon by cutting
it into two parts which are not joined. Find the minimum N for which Adhelia can defeat any Dragon with 100 necks by no more than N hits ?
No serious answers yet!?! So far I have only a guess at the answer, but here are some ideas. Perhaps someone can suggest a nice concise solution method:
First kill a few small dragons:
Necks vs Hits
1 - 1
2 - 1
3 - 1
4 - 2
5 - 2
6 - 2
7 - 2
8 - 2
9 - 3
I think I see a pattern...
Another (parallel) idea. Look for general dragon-slaying methods:
1) Pick a head and hit, not that head, but every head connected to it. Those heads will reconnect elsewhere, leaving the chosen head to fall off! Look for a head with the fewest necks.
2) Alternatively, if a head is connected to almost every other head, first hit those heads NOT connected to the chosen head, so that they connect up, then hit the head connected to every other head and it falls off. Look for a head connected to almost every head.
3) ?? Any other strategies ?? I needed only (1) and (2) for the table above, from which I have a guess for a 100 necked dragon, but no proof yet.
Some other miscellaneous observations:
For a 100-necked dragon, a 100-headed dragon with the heads and necks arranged in a ring would require 2 cuts to kill (by method 1). Hence N≥2.
A 100-necked dragon, must have at least 15 heads. Since (14)CHOOSE(2)=91, a 14 headed dragon can have at most 91 necks. Now each neck meets 2 heads, so there is an average of at most 200/15=13+1/3 necks per head. Hence some head must have at most 13 necks. So any dragon can be killed with at most 13 hits (by method 1). So N≤13. But a dragon where every head has 13 or more necks has no more than 200/13=15+5/13 heads. With at most 15 heads and 13 necks on some head, 3 hits will kill this dragon by method 2. That reduces N to at most 12....
I have 2≤N≤12, and a guess for a particular value of N. A proof would consist of a dragon which cannot be killed with less than N hits along with a proof that some method can kill every dragon with at most N hits.
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This problem comes from graph theory: Heads are vertices of the graph and necks are edges. A common technique in graph theory is to replace a subgraph with another subgraph. In this case replace the graph of all edges connected to a particular point with its complement. So the question is how many such replacements are needed to create a disconnected graph. (Could that be used as a kind of measure of how "connected" a graph is?) Unfortunately, I don't know enough graph theory to make this observation useful.
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What does it mean when my neon tree dragon bobs his head?
I have two neon tree dragons (male and female) and my male keeps bobbing is head, i'm wondering if its just a neck tweak but i have no clue!
It is a gesture of dominence. It lets other dragons know to bake off and that is his territory. If a dragon does it back than ussually a fight will start over dominence. Girls will often move their arms in a circle to show submission. If he does it to you then he is just signalling for you to back off and give him his space
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