Question 152
Ethan initially writes some numbers on a blackboard, each of which is either a $3$ or a $5$. He then repeatedly picks two numbers and replaces them with their sum, difference, product, or quotient (if the divisor is nonzero). Let $f(n)$ denote the minimum number of numbers Ethan must initially write for him to be able to eventually write the number $n$. For example, $f(2025) \leq 6$ because Ethan could start with $3$, $3$, $3$, $3$, $5$, and $5$ on the board, then repeatedly multiply two numbers at a time to eventually get $2025$.
Submit a comma-separated ordered 8-tuple of integers corresponding to the values of $f(164)$, $f(187)$, $f(191)$, $f(224)$, $f(255)$, $f(286)$, $f(374)$, and $f(479)$, in that order, or an X for any value you wish to leave blank. For instance, if you think $f(164) = 9$ and $f(224) = 8$, you should submit "9, X, X, 8, X, X, X, X". You will earn $\left|0.6^{W} \cdot \frac{(C + 1)^{2}}{4}\right|$ points, where $C$ is the number of correct answers you submit and $W$ is the number of incorrect (non-blank) answers.
Submit a comma-separated ordered 8-tuple of integers corresponding to the values of $f(164)$, $f(187)$, $f(191)$, $f(224)$, $f(255)$, $f(286)$, $f(374)$, and $f(479)$, in that order, or an X for any value you wish to leave blank. For instance, if you think $f(164) = 9$ and $f(224) = 8$, you should submit "9, X, X, 8, X, X, X, X". You will earn $\left|0.6^{W} \cdot \frac{(C + 1)^{2}}{4}\right|$ points, where $C$ is the number of correct answers you submit and $W$ is the number of incorrect (non-blank) answers.