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I understand it is a medal system but I'm really new to shmup styles of games. How exactly does the medal system affect the multiplier?

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Each medal is worth one more than the last one. The first one will increase your multiplier by 1, the next one by 2, then 3, 4, etc. Thus, collecting n medals will give you a multiplier of n*(n+1)/2. E.g. after 100 medals, you'll have a multiplier of 5050.

Some enemies will give you medals worth a little more (+5, +10 and even +200). For example, collecting a +10 after 2 normal medal will give you a multiplier of 1 + 2 + 3 + 10 = 16.

There are a few points to be noted : losing a life will return your multiplier to 0, collection a power up (the big "P") while at maximum power will increase the multiplier by 100.

Here is a more general guide on Giga Wing : http://shmups.system11.org/viewtopic.php?p=73570

Here are two videos of very good players that will show you the good spots to reflect : http://super-play.co.uk/index.php?content=2&gameid=99

(The players in these videos are very good, you won't be able to mimic those paths on your first few tries :) )

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