Math for Knitters - Changing Gauge in a Pattern

There is a pattern you'd like to knit, but you'd like to substitute a thinner or thicker yarn, and/or a different needle size.  Can you do it?  Here's how to determine if it's possible, and how to do the math when it is.

Information on things you'll see in this video:

The pattern for the sweater I’m wearing (not my design, in a discontinued cotton yarn) can be found here:
http://knitty.com/ISSUEspring03/PATTs...

The poncho on the mannequin behind me is my Rodeo Drive Poncho, pattern + video tutorial:
http://verypink.com/2013/12/20/rodeo-...

My Barber Pole Cowl, free tutorial and pattern, can be found here:
http://verypink.com/2014/04/02/barber...

The blue hat pattern (not my design) can be found here, knit with Shalimar Breathless in color Neptune:
http://www.ravelry.com/patterns/libra...

The bamboo pins I use to check gauge can be found here:
http://astore.amazon.com/verypinkcom-...

The pink yarn I used in my bulky swatch is Cascade 128 Wool.

The double-pointed bamboo needle I use as a pointer is Knitter's Pride Bamboo:
http://astore.amazon.com/verypinkcom-...

The nail polish I’m wearing is Julep, color “Maddy”.
http://www.julep.com/rewardsref/index...

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• #using_thinner_yarn