It's much more wonderful to know what something is really like than to sit there and simply, in ignorance, say, "Oooh, isn't it wonderful?"

STRATEGY 4: DIVIDE AND CONQUER

There is a folk tale about a rich farmer who had seven sons. He was afraid that when he died, his land and his animals and all his possessions would be divided among his seven sons, and that they would quarrel with one another, and that their inheritance would be splintered and lost. So he gathered them together and showed them seven sticks that he had tied together and told them that any one who could break the bundle would inherit everything. They all tried, but no one could break the bundle. Then the old man untied the bundle and broke the sticks one by one. The brothers learned that they should stay together and work together and succeed together.

The moral for problem solvers is different. If you can't solve the problem, divide it into parts, and solve one part at a time.

An excellent application of this strategy is the magic squares problem. It is well known. You have a square formed from three columns and three rows of smaller squares.

Part 1: Draw a Diagram

Into these squares you will enter the numbers from 1-9 in such a way that the sum of each column, each row, and each diagonal is 15. Since we know the way to solve problems is to start, let's start by guessing and checking.

Part 2: Guess and Check

Begin by entering the numbers in order, just to see what happens:

The middle column and the middle row add up to 15 and the two diagonals add up to 15. It has become obvious that 5 is a good choice for the middle, but we have to adjust the other squares.

Part 3: Make a List

It would be helpful now to identify all the combinations of three digits that add up to 15.

Part 4: Guess and Check Again

Now we can quickly see which combinations of numbers will solve the puzzle. There are four combinations that have the number 5, and all four combinations are needed—with number 5 in the middle.

When these four combinations are placed correctly, the other three combinations needed to finish the puzzle are easy to find.

Thus, by dividing the problem into 4 parts, it can be solved systematically.

Mixture Problems

Mixture problems often appear in mathematics text books. Here is an example of a mixture problem.

A mixture is 25% red paint, 30% yellow paint, and 45% water. If 4 quarts of red paint are added to 20 quarts of the mixture, what is the percentage of red paint in the new mixture?

This problem is taken from the book Crossing the River with Dogs and Other Mathematical Adventures , by Ken Johnson and Ted Herr, a book about problem solving strategies: http://www.keypress.com .

A student solving this problem has divided it into four parts:

Part 1: Find the amount of red paint in the original mixture:

Part 2: Find the total amount of red paint:

Part 3: Find the total amount of the whole mixture:

Part 4: Calculate the new percentage:

A problem which at first seems difficult becomes easier if you divide it into parts and solve one part at a time.

Problems:

Problem 1: Another Magic Square

Problem 2: Making a Profit(A)

Problem 3: Making a Profit(B)

Problem 4: Jogging

Problem 5: The Car Bargain

Problem 6: Arranging the Classroom

Problem 7: Parquet Floor Problem

Problem 8: Make the Table Bigger

Problem 1: Another Magic Square

Can you solve a magic square problem in which there are five columns and five rows? Use all the numbers from 1 to 25, and the sum of each row, each column, and each diagonal should be 65.

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Problem 2: Making a Profit(A)

Joe Curry owns a furniture shop. He sets his prices at 20% above wholesale. When he reduces his prices for a sale, he still wants to make at least 10% profit on each item. The regular price for a couch was $240. During the sale, he reduces this price by 10%. Will Joe make his 10% profit?

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Problem 3: Making a Profit(B)

A more challenging version of Problem 2 omits the price of any one item. Everything is priced at 20% over wholesale. During the sale, Joe reduces everything by 10%. Will he make at least a 10% profit on each item?

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Problem 4:Jogging

When Laura goes to the gym, she jogs for 20 minutes on the treadmill, equivalent to a distance of 2.5 kms. In good weather, instead of going to the gym she jogs from Arnavutkoy to Rumeli Hisar, a distance of 3 kms. If she wants to jog at the same speed as in the gym, how long should it take her to go from Arnavutkoy to Rumeli Hisar and back again?

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Problem 5: The Car Bargain

Paul went to the car dealer to buy a car. He wanted the same car that his friend Barbara had bought the day before, which had a sticker price of $15,000. The salesman said he could give a discount and offered Paul a significantly reduced price. But Paul knew that Barbara had received a 30% discount, and the salesman was offering him only a 20% discount. When he pointed out that his friend had received a 30% discount the day before, the salesman took another 10% off the 20% discounted price. Paul was satisfied with the new price and bought the car, thinking he had paid the same price as Barbara. Was he right? Did they both pay the same price?

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Problem 6: Arranging the Classroom

The furniture in a classroom consists of tables and chairs. The homeroom teacher is making a seating plan. If 2 students sit at each table, 8 students will be left without a place. If 3 students sit at each table, 4 tables will be left empty. How many studens are there in the homeroom.

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Problem 7: Parquet Floor Problem

After making a parquet floor in an office building, the carpenters had left-over pieces of wood in the shape of right triangles with sides of 1, 2, and Ö5. The architect would like to use these pieces for a parquet floor in his own house. He wants to know: can he make a perfect square from 20 of these triangles? If so, what would it look like?

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Problem 8: Make the Table Bigger

Mrs. Summersby has an antique circular table with a diameter of 1.5 meters.  It is big enough to accommodate six people for dinner.  The table is divided in the middle so that leaves can be added to make the table bigger, thus creating a rectangle with two semi-circular ends. Unfortunately the leaves for making the table bigger have been lost.  Mrs. Summersby has asked a carpenter to make her new leaves so that she can accommodate ten people for dinner.  Each leaf should be 35 centimeters wide and 1.5 meters long.  To seat ten people, the perimeter of the table should be at least 6.8 meters and the area at least 3 square meters.  How many leaves should she ask the carpenter to make?

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PROBLEM SOLVING STRATEGIES

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