How can I defeat a "Take No Damage" fairy chest challenge when there are spikes in the way?

There are more ways of doing this one.

Dragon class

The easiest way would be to use a Dragon class as they can fly without any limit. That way you would just have to press A or S to fly, go down and fly to the chest.

I think it's needless to say that this approach will always have 100% success chance as it's impossible to mess it up.

1 Dash 3x Vault Runes and no platform activation

Hug the wall, drop down and dash left immediately, then use triple jump to reach the chest. This is also one of the easiest ways to do this.

Hokage + 3x Vault Runes

This is also an easy way without the need to activate the platform. Simply drop down and use your special ability to propel yourself forward the chest wall, then use triple jump to reach the chest itself.

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There are more ways of doing this, but this three i mentioned are the easiest ones and none of them require platform activation.

Also note that you might actually need only 2 vault runes for second and third solution if your timing is good. But 3x vault runes will help you be 100% successful.

You'll need a Sprint Rune (allows dashing) and a Vault Rune (allows multi-jumping) to retrieve this fairy chest. Basically, hug the wall, jump below the platform, immediately dash to the left, jump again while in mid-air, down-thrust the platform to open it up, and stand on it. Afterwards, double jump to reach the chest.

If you want to do it more safely, have two Sprint Runes equipped. I believe it can be done with one alone but, it's kind of risky.