Gautam Soundararajan

Math 7

Professor Winkler

4/17/13

What Price Pets?

Martin Gardner was a renowned writer who captured the hearts of readers with his enticing, yet incredibly difficult puzzles. He has earned abundant praise. Douglas Hofstadter, a cognitive scientist, said “Martin Gardner is one of the great intellects produced in this country in the 20th century.”1Gardner was an incredible mathematician. He says “I was very good at math in high school. In fact, it and physics were the only subjects in which I got good grades. I was bored to death by the other classes. I flunked a class in Latin and had to take it over. I just don’t have a good ear for languages.”2Gardner may have disliked subjects aside from mathematics, but he is still able to attract a wide range of readers. Math-lovers, math-haters, and people in between have been fascinated by his collection of puzzles.

One puzzle that caught my attention was in the Algebra section of his book “The Colossal Book of Short Puzzles and Problems.” Looking at the table of contents of the book, Algebra seems like one of the easier topics that Gardner includes. Students had to take the course in high school. What could possibly be hard about that?But Gardner’s puzzles are not simply about canceling Xs and dividing by Ys. These mathematic operations are used to solve some of the problems in the section, but they are not the difficult part. Setting up the equations and knowing where to go with them is the true test of the puzzle. If the former were true, the puzzle would not even deserve to be called a puzzle because it would be far too simple.

The puzzle I enjoyed tackling is called “What Price Pets?” As an animal enthusiast, I could not resist the problem. It discusses hamsters and parakeets. In the past, I have had both as pets and Gardner’s puzzle brings back childhood memories of playing with them. The hamsters and parakeets could easily be replaced by some other set of animals, or even entirely different objects, such as apples and bananas. The particular way that Gardner phrased the problem makes it resonate with me. This brings up an interesting point: there are many different ways to phrase a puzzle. Each makes the puzzle slightly different, even if the same mathematical concept is used to solve it. The story behind each puzzle gives it a new meaning and affects readers differently. Another reason why I enjoy the puzzle is because of the false sense of simplicity the puzzle gives. As I first attempted to solve the puzzle, the path to the solution seemed straightforward. But then I hit a wall. I knew that I was on the right track to solving the problem, but did not know where to go next. To me, this feeling is much worse than not knowing how to even begin a problem. In the latter case, I don’t have any attachment to the problem since it is too difficult for me to handle. But if I can make some progress, I start to think that I can actually find the solution. This feeling makes me strive to find the answer. The following is the puzzle.

Raymond works at PetSmart. He purchases a number of hamsters and half that many pairs of parakeets. Each hamster cost $2 and each parakeet was $1. In order to make a profit, Raymond had to sell the animals for more than he bought them for. He feels that a 10 percent increase in the price is fair to him and his customers. At the end of the week, he sees that there are seven animals left. He also notices that he has exactly covered his costs. In other words, the money he spent to purchase the hamsters and parakeets has been recouped by the sale of animals after one week. What is the dollar value of the remaining animals?

This puzzle seems easy at first glance. How hard could it be to throw in some Xs and Ys, find a relationship, and find the dollar value of the seven hamsters and parakeets? In order to solve the puzzle, we must first introduce some variables. Call X the number of hamsters Raymond purchased. The number of parakeets purchased is also X. It is easy to overlook this since the word “half” stands out. But we must also notice the word “pairs.” Half the number of pairs means the same total number of parakeets, or X. Let us now call Y the number of hamsters left at the end of the week. We could label the number of parakeets remaining at the end of the week as Z, but that is unnecessary since we have enough information to get the number of parakeets in terms of other variables. The number is 7 – Y. The intuition is straightforward. There are seven animals and Y of them are hamsters. The difference is the number of parakeets. Some other relationships that can be gleaned from this information are:

X – Y the number of hamsters sold

X – (7 – Y) the number of parakeet sold

This is deduced by simple subtraction. The next step is to introduce the dollar values to the animals. Since hamster were purchased for $2 and parakeets for $1, we have

2X + 1X = 3X

This is the total dollar amount spent to buy the animals. The price that Raymond is selling the animals for is:

2 * 1.1 = 2.2 for hamsters

1 * 1.1 = 1.1 for parakeets

Now we connect these relationships.

2.2 * (X – Y) revenue from hamsters sold

1.1 * [X – (7 – Y)] revenue from parakeets sold

The revenue from selling hamsters and parakeets is equal to the amount Raymond paid to purchase the animals.

3X = 2.2 * (X – Y) + 1.1 * [X – (7 – Y)]

3X = 3.3X – 1.1Y – 7.7

0.3X = 1.1Y + 7.7

3X = 11Y + 77

Now comes the tricky part. Many people would assume that they did something wrong since they expected an answer to emerge. However, the result above is to be expected. There was only one equation with two variables. There is no way to solve the puzzle using just this. We know that Y can take on any integer value from 0 to 7. After substituting each of the eight numbers into the equation, only values of 2 and 5 yield integer values of X. If there were 2 hamsters at the end of the week, X is 33. This says that 33 parakeets were bought. But the parakeets were bought in pairs. This slight detail in the question is hard to pick up, but plays a vital role in solving the puzzle. With this in mind, we know that 5 hamsters were left and X is 44. There were 5 hamsters, priced at $2.2, and 2 parakeets, priced at $1.1. Using multiplication and addition, we get the value of the remaining animals to be $13.20.

The puzzle can be made even more complex by simply adding another pet. Doing so introduces another variable into the equation. In order to solve the new puzzle, additional information must be given.

It is interesting to note that in solving the puzzle, only basic mathematical operations were used. There was no need for derivatives or integrals. The puzzle was challenging already. This is a common trait of mathematical puzzles. They do not require deep and complex knowledge of mathematics. Logic and persistence are the only requirements.

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