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> Given the three numbers $x,y=x^x,z=x^{x^x}$ with $.9<x<1.0$. Arranged in order of increasing magnitude, they are:
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> $\text{(A) } x,z,y\quad \text{(B) } x,y,z\quad \text{(C) } y,x,z\quad \text{(D) } y,z,x\quad \text{(E) } z,x,y$
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Hi, hope you are doing well. I was solving this 1968 AHSME Problem 29.

Unfortunately, I could not solve this problem. Any help would be appreciated. Also, I was wondering what would happen if say $0<x<0.9$ or $x<0$ and other cases. Please help.

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> **My Thought Process**
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> Since $x<1.0$, $x^x$ must be less than $x$ but I couldn't reason out the further.
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> Also, in the case $x > 1$, $x^{x^x} > x^x > x$ as it increases due to $x$ being greater than $1$.
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