by Bethany Therriault

Encouraging kids to play games that involve puzzle-solving is something that many experts have been telling parents to do for years. However, as kids grow up and trade games for homework, many parents allow puzzle-solving to take a back seat to grade-boosting. However, this may in fact have a negative impact on their children.

The issue is this: while homework and school lessons are certainly important for kids, learning how to think creatively is equally important. In many cases, the value of puzzles is in their ability to teach children how to approach a problem in a new and different way - something that no amount of homework can do. While learning facts and figures is definitely important, learning how to solve problems in creative ways is something that will benefit kids throughout their lives.

One of the specters of preparing for college, at least in the US, is taking the SAT. Each year, hundreds of thousands of high school kids across the country sit down to take the test that will determine whether or not they can attend the college of their dreams. As an SAT prep tutor, I see many, many kids spend frustrating hours studying and studying for the SAT. For many, the math sections are a particular stumbling block. Why? Because the SAT doesn't always display problems in the same way as their math text book. Many students, although adept at solving the math problems presented in their classes, have great difficulty when it comes to the problems on the SAT, particularly ones like this:

Matt and Carrie are siblings. Two years ago, Carrie's age was three times Matt's age. In two years, Carrie's age will be only twice Matt's age. How old will Carrie be in two years?

A) 6
B) 8
C) 12
D) 14
E) 16

Many students are unable to decide how to proceed, but more advanced students can often figure out the following two equations:

(C-2) = 3(M-2), where Carrie's age is represented by C and Matt's age is represented by M, and the "-2" means that this equation represents their ages two years ago.

(C+2) = 2(M+2), where Carrie's age is represented by C and Matt's age is represented by M, and the "+2" means that this equation represents their ages in two years.

At this point, many students get stuck. They find themselves with two different equations, each having two variables, and are unable to solve either one. However, there are in fact three ways to approach this problem. The first involves using equation subtraction, a method most students remember from Algebra II. The process looks like this:

(C-2) = 3(M-2)
C - 2 = 3M - 6
C - 2 + 2 = 3M - 6 + 2
C= 3M - 4

(C+2) = 2(M+2)
C + 2 = 2M + 4
C + 2 - 2 = 2M + 4 - 2
C = 2M + 2

  C = 3M - 4
- C = 2M + 2
  0 = M - 6

0 + 6 = M - 6 + 6
6 = M

The second method involves substitution, another method from Algebra II. The process looks like this:

(C-2) = 3(M-2)
C - 2 = 3M - 6
C - 2 + 2 = 3M - 6 + 2
C = 3M - 4

(C+2) = 2(M+2)
C + 2 = 2M + 4
C + 2 - 2 = 2M + 4 - 2
C = 2M + 2

Since, C = 3M - 4 and C = 2M + 2, then:
3M - 4 = 2M + 2
3M - 4 + 4 = 2M + 2 + 4
3M = 2M + 6
3M - 2M = 2M + 6 - 2M
M = 6

The third method is one which will appeal to those students who are unfamiliar with Algebra II. This method applies only skills learned in Algebra I, but requires the discovery of this equation:

3(M-2) + 4 = 2(M-2)

This equation shows, on the left, Carrie's age two years ago, and adds four to that number to get the number on the right, which is Carrie's age two years from now. This problem can be solved thus:

3(M-2) + 4 = 2(M+2)
3M - 6 + 4 = 2M + 4
3M - 2 + 2 = 2M + 4 + 2
3M = 2M + 6
3M - 2M = 2M + 6 - 2M
M = 6

So, the answer is A, right? Wrong. The question actually asked, "How old will Carrie be in two years?" So, to solve THAT problem, we can plug Matt's age into the expression which shows Carrie's age in two years:

2(M+2)
2(6+2) = 16

The answer is, in fact, E, 16.

This problem shows three key areas in which puzzle solving skills are easily applied to SAT math. First, puzzle solvers are often quick to identify the way in which this problem should be solved. They find it easy to translate words into equations, to see the relationships between variables. Second, problem solvers are able to decide which methods will be effective in solving their problem, and if, for example, they are unfamiliar with Algebra II methods, they are often able to find methods with which they are familiar which will allow them to solve the problem. Third, puzzle solvers learn quickly that they need to read problems carefully to find out what information they need. Puzzle solvers are less likely to miss the end of the problem, which asked them to find Carrie's age in two years.

While this is not to say that quick-witted students won't be able to solve this problem without a background in puzzle-solving, it is clear that a developed ability to approach problems creatively is important when dealing with a test like the SAT. Puzzles are a great way to develop this ability in children, even at an early age, and it is a skill that they will be able to apply in many areas of life. Many of the puzzles posted here on the GM forums are strikingly similar to problems found on the SAT, and provide a fun and stress-free way to expose kids to these problems well before they have to see them on the SAT.