A word of 6 Letters. I always play fair. What am I ?

What am I?

A word of 6 letters,
I always play fair, Minus my first I am cold
Minus my last and jumbled, I'll give you a massage
Jumble my whole, You'll learn nothing from me
I always play fair

What am I ?

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The answer is: CHEAT
A word of 6 letters, I always play fair :
A CHEAT doesn't play fair, and it's not 6 letters.
Minus my first I am cold:
HEAT isn't cold.
Minus my last and jumbled, I'll give you a massage :
You might need a massage if you have an ACHE.
Jumble my whole, You'll learn nothing from me:
TEACHing makes you learn things.
I always play fair
Every clue is wrong because the narrator is cheating.