This puzzle replaces all numbers with other symbols.
Your job, as the title suggests, is to find what value fits in the place of $\bigstar$. To get the basic idea, I recommend you solve Puzzle 1 first.
All symbols follow these rules:
- Each numerical symbol represents integers and only integers. This means fractions and irrational numbers like $\sqrt2$ are not allowed. However, negative numbers and zero are allowed.
- Each symbol represents a unique number. This means that for any two symbols $\alpha$ and $\beta$ in the puzzle, $\alpha\neq\beta$.
- The following
equationsinequalities are satisfied (this is the heart of the puzzle): $$ \text{I. }a^{a}-a<a\times a \\ \space \\ \text{II. }b\times b-a^{a}\times b<a \\ \space \\ \text{III. }b^{c}<a\times a\times c \\ \space \\ \text{IV. }d\times d-c\times d+c\times c<c\times d+a \\ \space \\ \text{V. }a\times e^a\times c>d^{e} \\ \space \\ \text{VI. }e\times c<a\times b-a\times d \\ \space \\ \text{VII. }e^{a}<\bigstar<a\times b^{a} $$
What is a Solution?
A solution is a value for $\bigstar$, such that, for the set of symbols in the puzzle $S_1$ there is a subtitution $f:S_1\to\Bbb Z$ that satisfies all given equations.
Can you prove that there is only one possible value for $\bigstar$, and find that value?
Good luck!
Side Note: to get $\bigstar$ use $\bigstar$
, and to get $\text^$ use $\text^$
Previous puzzles: