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a Hive layout photo

Here. Is it possible to move this way?

I figure that neither the rule of hive continuity nor tile sliding rule are violated here. The rules also don't mention any direct contact requirements like with the Spider. So... is the move valid?

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No, the one hive rule is continuous, even during a move. At least in the version of the rules that I have, the second example under the One Hive rule states

Moving the black Queen Bee to a position where it re-links the Hive is also an illegal move as the Hive is left unlinked while the piece is in transit.

This would apply to your beetle move as well, since while it was in transit, it would not be connected to the hive at all.

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Well, my perception of that example was that it illustrated that even while moving you have to keep the hive's connectivity. That is, because of that Grasshopper at bottom-left you cannot take off the Queen Bee. At the same time, for most moves (especially with Grasshopper) you really must disconnect the piece you're moving. So, to me it wasn't really clear that pieces must slide along the hive's edge all the time. –  ulidtko Apr 12 '13 at 13:45
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I am pretty sure that all piece movements are considered to be of a sliding nature around the outside of the hive as the baseline movement.

The clarification on Spider isn't specific to Spider, that is just where they mention it.

A basic move (as the Bee or Beetle not going up or down does) is defined as moving around the outside of the hive, making that move illegal. Put another way, each move you make like this involves still touching at least one similar piece.

Note that the Spider and Ant both make compound moves by having repeated normal moves.

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