"Trumpet connection" is an expression from the game of Go (Wei Qi/Badouk). It refers to the shape formed by the white (o) stones in the diagram shown here:

.......
..xxo..
.xxoab.
.xoao..
...b...

(No stones at the points marked a and b, but see below)

Connection is very important in Go, because stones that are connected to one another live or die as a group, and are generally stronger in groups. Consider the black (x) stones. They are all connected. At first glance, the white (o) stones don't look connected, and they aren't, by the rules of the game. Theoretically, one could still get captured individually, since it isn't orthogonally adjacent to one of the others. However, imagine what would happen if the opponent tried to cut at one of the two possible places marked "a". The stone placed there would be in atari, and white could immediately play at the corresponding "b" and capture the stone before it could do any damage. Therefore, the stones are effectively connected, just as if there were white stones at both points marked "a".

Now, for the advantages and disadvantages of this form of connection over a "solid" connection. The advantage is that, given a line of three diagonal stones, you can protect both the cutting points (a) with a single additional stone, whereas solidly connecting at both cutting points would require two moves. Since Go is a game of efficiency, this is critical.

There is a big "but" here. A solid connection leaves no kikashi (forcing moves), while a trumpet connection allows the opponent two "peeps" at the points marked "b." What this means is, if black plays at either "b," white must immediately connect solidly at the corresponding "a" or else be cut. Depending on the circumstances, this may not matter, but allowing black two free stones behind white's lines may pave the way for an invasion.

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