This stone was found at the foot of an ancient door.
This old note with some mathematical relationships was discovered nearby.
Can you use this information to figure out what number each of the 9 symbols on the stone represent?
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The nine symbols are, in the arrangement given:
9 0 3
5 1 2
4 7 8
From the factorial relationship,
two different digits with one being the factorial of the other must be either $0!=1$ or $3!=6$.
But from the perfect-power relationship,
two different digits, one raised to the power of the other giving a third single digit, must be $2^3=8$ or $3^2=9$.
2 and 3 both appear in that perfect-power relationship, which means the factorial relationship is not $3!=6$. Therefore the hooked symbol is $0$ and the P-shaped symbol is $1$.
The top left (division) relationship
is a red herring, making you think the jagged symbol must be 1, when in fact it's the hooked symbol being $0$.
From the square root relationship,
the X-shaped one is either 4 or 9 (we've already found 1), and the h-shaped one is correspondingly either 3 or 4. Since we already know 3 is in the perfect-power relationship, that means the h-shaped one is $4$ and the X-shaped one is $9$.
Going back to the perfect-power relationship,
since we now have 9, that weird symbol on the right is $8$, the moose symbol is $2$, and the mushroom symbol is $3$.
Now for the big product relationship:
the right-hand side is $3(9+1)=30=5\times6$, so the zigzag and the jagged symbol are $5$ and $6$ in some order.
From the division relationship at the bottom,
the zigzag symbol is $5$ (the smaller one) and the jagged symbol is $6$.
the double-triangle symbol (the only one not appearing in the given relations) must be $7$.
the jagged symbol for 6 does not appear on the stone!