 Home > Problem of the Week > Archive List > Detail << Prev 1/11/2009 Next >>

## A Difficult Problem? The American Mathematics Contest 8 exam is designed for students in grades 6-8. Based on the scores for the 2006 exam, this problem was considered to be the "hardest" problem:

In the multiplication problem ABA x CD = CDCD,
where A, B, C, and D are different digits,
what is A+B?

Note: Another difficult problem on the exam was: "Circle X has a radius of π. Circle Y has a circumference of 8π. Circle Z has an area of 9π. List the circles in order from smallest to largest radius." Do you see why almost 40% of the students selected the wrong answer of X, Y, Z?

Source: Adapted from MAA FOCUS, February 2007

Hint: The problem actually was a multiple choice problem, where 15.39% of the students chose (A) 1, 16.15% chose (B) 2, 26.01% chose (C) 3, 17.68% chose (D) 4, 12.09% chose (E) 9, and 12.55% of the students did not answer the question.

Does this information help?

Solution Commentary: It is perhaps easier to turn the problem into a division problem: CDCD/CD = ABA. Now CD "gozinta" CD** how many times...etc.

A Sidenote: The perplexity of the problem perhaps is increased by dual roles of letters. For example, B is a digit (actual value is 0)...but also is a choice for an answer (B) 2, etc.