Proof of work - 51% attack
Since they're all competing, the hash-power used to create a block in 10 minutes, is 1/4 of the total hashing power. 


That's not correct.

If the work effort is set to $n$, that is, it takes an expected $2^n$ hashes before someone finds a valid preimage.  If Alice, Bob, Charlie and Eve are all working equally hard, then we would expect that each one would end up computing an average of $2^n / 4$ hashes before someone finds a preimage.  It might be Alice, who has computed only $2^n / 4$ hashes, but still a total of $2^n$ hashes were computed.

So, if Eve goes off on her own, she'll need to compute her own $2^n$ hashes herself before she finds a preimage; in that time, Alice, Bob and Charlie will end up computing $3 \cdot 2^n$ hashes, enough to find an expected 3 links in the hash chain, before Eve can compute one - Eve cannot win this race in the long term.

(Note: this analysis rather assumes that when Alice finds the preimage, she instantly announces it, and the others immediately start searching based for the next preimage.  This isn't precisely true; however that isn't nearly a large enough effect to give Eve an advantage when she has only 1/4 of the computing power)