Q1. Suppose a jogger wants to use her ten fingers to count laps as she circles a track. Each finger can be in two different states to represent a binary digital. How many laps can she conveniently count? Briefly justify your answer.

Q2. How many trinary (base 3) digits are required to represent numbers in the following ranges?

(a) The integers from 0 to 255 inclusively.

(b) The integers from 0 to 4,095 inclusively.

(c) The integers from 0 to 1,234,567 inclusively.

Q3. An expedition to Mars found the ruins of a civilization. The explorers were able to translate the mathematical equations: 5x² - 50x + 125 = 0 with the solutions: x = 5 and x = 8. The x = 5 solution seemed okay, but x = 8 was puzzling. The problem should be because Martians were using a non-decimal number system. Therefore, "50" is not fifty, but "50" in base b (50b=5×b+0×1=5b). The explorers reflected on the way in which Earth's number system developed. How many fingers would you say the Martians had? Hint: What should be the value of the base b such that both 5 and 8 are solutions of the equation?