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But does it use an open source toolchain? Also, where does it say that it runs on ICE40?

PicoRV32 absolutely runs on iCE40 and the open toolchain. Check out scripts/icestorm and http://www.clifford.at/papers/2015/yosys-icestorm-etc/slides...

The target is the iCE40 HX8K according to the slides (p17) not the icestick unfortunately.

It uses 1521 LUTs with Yosys. The icestick is a HX1K which has 1280 LUTs I think.