Hardest Question at MathCounts 2012?

Rumor has that some big professional football competition will occur on Sunday, but I’ll be setting my sights on a more intellectual competition that takes place on Saturday. Middle school students across the country will be participating in the MathCounts Chapter Competition. Good luck to all mathletes!

As a member of the MathCounts Question Writing Committee, one of my responsibilities is to ensure that the questions on the competition are not too hard, not too easy, but just right. Consequently, the committee reviews a lot of data regarding the percentage of students that answer each question correctly.

Rather accidentally, I wrote the most difficult question that appeared on the 2012 Chapter Competition.

The chart below shows the number of students from a random sample (n = 1200) who answered each of the 30 questions correctly on the Sprint Round:

Although Problem 29 was answered correctly by just 42 students, it was not the most difficult. Students are given 40 minutes to complete all 30 problems in the Sprint Round. Consequently, many students never get to the problems at the end of the round, and the last several questions are typically answered correctly by only a handful of students.

On the other hand, Problem 14 was answered correctly by only 50 students, and it occurs in the middle of the round. By design, the questions in the round proceed from least difficult to most difficult, so problem 14 was intended to be a medium question. Questions on either side of Problem 14 were answered correctly by 500 to 700 students, and the members of the question writing committee suspected that Problem 14 would be answered correctly by a similar number. Boy, did we miss the mark!

(You’ll notice that we also incorrectly guaged the difficulty of Problem 9, though not quite as dramatically.)

So, what was this doozie of a question? Try it for yourself:

If Friday the 13th occurs in a month, then the sum of the calendar dates for the Fridays in that month is 6 + 13 + 20 + 27 = 66. What is the largest sum of calendar dates for seven consecutive Fridays in any given year?

What do you think? Are you as smart as the 4.167% of mathletes who got this problem correct? To find out, take a look at the calendar for May and June 2013.