Question 1: 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.

Question 2: How many binary (base 3) digits are required to represent numbers in the given ranges?

Part (a): The integers from 0 to 255 inclusively.

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

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

Question 3: 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.

Q: How many fingers would you say the Martians had?

Note: What should be the value of the base b such that both 5 and 8 are solutions of the equation?