Final answer:
The question involves maximizing the use of a multiplication operator and minimizing the use of an addition operator to reach a target number from the starting number 1. The strategy is to multiply as much as possible without exceeding the target number, then add to reach the exact target number, necessitating careful consideration of the operations at each step.
Step-by-step explanation:
The student's question involves using two mathematical operations—with constant values x and y—to transform the starting number 1 into the target number t. The objective is to maximize the usage of the multiplication operator (operator 2) and minimize the usage of the addition operator (operator 1). Since multiplication by a nonzero number has a more significant impact on increasing the value of a number than addition, in order to maximize operator 2's use, we need to apply the multiplication operator as frequently as possible before the resulting product exceeds the target number t. Only then should we use the addition operator to adjust the number to exactly t.
When the target number t is approached, we may need to use the addition operation if the next multiplication would result in a number greater than t. The optimal strategy focuses on the trade-off between the two operations to reach t with the minimum number of additions. The process requires iterative application of these operations, considering each step carefully based on the current number and the fixed values of x and y. In a step-by-step approach, applying operator 2 (multiplication) would repeatedly increase the number exponentially, which aligns with the concept that exponentiation represents a series of multiplications (as mentioned in hint 22). Once the multiplicative increases get near to t, operator 1 (addition) can make fine adjustments.