white to move and mate in 5

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wouterkabouter wrote:
f you let them play g2 it will be a minimum of six;

So don't let black play g2. When you got that far in your logic, the rest is entirely obvious.

1. Be5 Bh1 2. Bxg3 and then 3. Bd6, 4.Bf8, 5. Bg7 mate.

And when you write "it can be done faster (4)" it is just plain wrong. I don't even know what you meant by that, it is just weird.