3. An odd six-digit number is called "simply cool" if it consists of digits that are prime numbers, and no two identical digits stand next to each other. How many "simply cool" numbers exist?
Solution. There are four single-digit prime numbers in total - 2, 3, 5, 7. We will place the digits starting from the least significant digit. In the units place, three of them can stand $-3, 5, 7$. In the tens place, there are also three out of the four (all except the one placed in the units place). In the hundreds place, there are also three out of the four (all except the one placed in the tens place). And so on. As a result, the number of "prime cool" numbers is $3^{2}$, i.e., 729.

Answer. 729.

Recommendations for checking. Only the correct answer - 1 point. All 4 numbers are included in the units place - 3 points. The number 1 is considered a prime number, and the subsequent solution is correct based on this - 5 points.