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This stone was found at the foot of an ancient door.

1

This old note with some mathematical relationships was discovered nearby.

2

Can you use this information to figure out what number each of the 9 symbols on the stone represent?

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Overall solution

The nine symbols are, in the arrangement given:

9 0 3
5 1 2
4 7 8

Step-by-step deductions

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 $3^2=9$.

So

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 $6$.

Finally then

the double-triangle symbol (the only one not appearing in the given relations) must be $7$.

Note that

the jagged symbol for 6 does not appear on the stone!

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