Are You Smarter Than a 6th Grader?

Are You Smarter Than a 6th Grader?

The following is a list of challenging math questions taken from various math olympics across the country. One of these questions was given to a 6th-grade class and it was correctly answered by a student. (Can you guess which one?)

1. Augustus, Benedict, Claudio, and Diana have been accused of stealing the golden mean. It is known that one of these four people must have done it. Augustus says, “Benedict did it.” Benedict says, “Diana did it.” Claudio says, “I didn’t do it.” Diana says, “Benedict is lying when he says I did it.” If it is known that exactly one of them is lying, which one did it?
2. Many people recognize and celebrate Columbus Day, but fewer people realize that Oct 9th is Leif Eriksson Day, which is the celebration of the first European known to set foot on North American soil. Unfortunately, his expeditions did not lead to a lasting settlement in the new land. About a year after creating settlements in mainland North America his father, Erik the Red (who discovered Greenland) passed away, and Leif was called back to Greenland to rule the colonies there. First, write the names Leif Eriksson and Erik the Red on a piece of paper. Then cut up the paper such that each of the 22 letters is on its own piece. Finally, put all 22 pieces of paper into a bag. Assuming that each piece of paper is equally likely to be drawn, what is the probability that the first two pieces of paper drawn from the bag, without replacement, each have a letter that is in “Leif Eriksson” and “Erik the Red”?
3. How many whole numbers less than 1,000 contain no 3's but at least one 2?
4. A middle school has 100 lockers numbered 1 to 100, and 100 students. The first student goes down the row of lockers and opens every locker. Then the second student goes down the row of lockers and closes every locker that is numbered with a multiple of two. Then the third student goes down the row of lockers, and for every locker that is numbered with a multiple of 3, if it is open, she closes it, but if it is already closed, she opens it again. The fourth student then does the same thing for the lockers numbered with multiple of 4, and so on, down to the hundredth student. In the end, how many lockers are still open?

Courtesy of the Saginaw Valley State University Math Olympics and the MATHCOUNTS National Math Contest for middle school. (These guys are supposedly broadcast on ESPN ... pretty cool.) Hint: Some of these questions do not necessarily have to be answered by a formula. Think manual calculation, chart, etc.