Amy has a video on intarsia knitting in the advanced techniques section, but here's the general idea:
With this many colors, I would use bobbins. And because of the way this pattern works, that it's just puzzle piece after puzzle piece, I would use a different bobbin for each of those puzzle pieces, instead of carrying/stranding anything on the back. Just be sure you do 2 things:
- When changing colors, wrap the 2 yarns around each other so that you don't have holes between the puzzle pieces.
- Let's say you're on a knitting row and you finish with red and are about to start with blue-- look at the chart at the next row to see where you come in with the red again when you come back on the purl row. If you will have to start the red, let's pretend 6 sts earlier, on that purl row, then while you're still on the knit row, carry the red behind the blue for 6 sts so that it's ready for you on that return purl row, and you don't have a big loop where you had to bring the yarn over those 6 sts. This will probably make more sense as you actually do it.
When I say "carry", it means to cross over/twist/wind together the 2 colors every 3 sts. I know that this scarf gets doubled over and so all of this will be hidden. But if you don't strand every 3 sts, you are likely to get puckers and general confusing messiness behind your work.