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## When My Dear Aunt Sally Did Not Help!

When the twins Stu Dent and Polly Dent were caught using their cell phones in class, their mathematics teacher gave them an extra assignment to do: Evaluate 999.

That night, when facing the assignment, they could not determine whether this meant that 999 = 9(99) or 999 = (99)9.

So, the Dent twins decided to flip coins as to who would do which version, thereby protecting themselves when submitting their solutions the next day. By the flip, Stu "won" the first interpretation and Polly the second.

Which of the two twins did not get much sleep that night? Explain.

Source: Adapted from Oregon Mathematics Teacher, September 2000, p. 44.

Hint: Start calculating...or stop and do some thinking. How many digits are involved?

Solution Commentary: For Polly, 999 = (99)9 = (387420489)9, while for her brother Stu, 999 = 9(99) = 9387420489.

Rather than evaluating these expressions further, set them both equal to 10x, so the number of digits involved can be computed (and the size of the number known). Now, take the ln of both sides of each expression. That is, for Polly, x = [9 ln(387420489)]/ln(10), while for Stu, x = [387420489 ln(9)]/ln(10). Solving for x, Polly is confronting a number with about 78 digits and Stu is confronting a number with about 369,693,100 digits.

To compare their two tasks, suppose they each could write 5 digits per inch on a roll of old adding machine tape. Polly would need about 15 1/2 inches to express her result for 999 = (99)9, while poor Stu would need about 73,938,620 inches or 1,167 miles of tape to express his result for 999 = 9(99).