Math 186 IB The Chessboard Evaluated using Criterion B and C Due date: 15 December 2014 There is an old tale of a boy who did a great service for a King. The King offered the boy any prize that he wanted, so the boy asked for a quantity of rice. “Put one grain of rice on the first square of a chessboard,” said the boy, “and put two on the second square, then double to 4 for the third square and keep doubling until you reach the last square of the board”.1 The squares were filled for 64 days in a row. The king felt relieved that the boy had asked for such a modest prize. What the king did not realize was that on the last square alone there would be 9 223 372 036 854 775 808 grains of rice! This number is 9 quintillion 223 quadrillion 372 trillion 36 billion 854 million 775 thousand 8 hundred and 8. You are now going to answer the following questions. Remember to explain what you are doing – communicate in mathematical language. Always show your steps. You may use any resources that you want but you must always answer in your own words. I will be also checking how you investigate patterns, so be precise. The Project Draw two chessboards On a piece of graph paper, draw a chessboard. You may use any size to represent a square on the chessboard. On each of the squares in the first three rows, enter the number of grains of rice that the boy would have received. Draw another chessboard. This time enter the number of grains of rice on each square in the first three rows, using EXPONENTIAL FORM. The numbers must all have the same base, raised to different powers. 1 David Birch, The King's Chessboard, Puffin Pied Piper Books: New York, 1988. The Mathematics behind the story 1. Describe at least 5 patterns that you see in the numbers on the first chessboard. Describe at least 3 patterns that you see in the numbers on the second chessboard. Look at the diagonals, rows and columns. 2. How many grains of rice would you need all together in order to fill up from the 1st to the 15th square, including the 15th square? 3. How many days did it take for the boy to receive a total of at least one million grains of rice? Remember to start counting at Day 1. 4. On which day did the boy first receive more than one million grains of rice? 5. If 50 grains of rice weigh about one gram, estimate the mass of the rice that the boy received on the tenth day. 6. On which day would enough rice arrive to feed everyone in your class? The information that you need is: __________ students in the class 1 cup of uncooked rice per student 2 24 teaspoons in 1 cup of uncooked rice 2 ** You must show all of your calculations ** ** You must also type out each question ** Reflecting 1. Suppose the boy had tried to stack all the rice on the last square of the chessboard in a tall tower, each grain lying on top of the one below it. A grain of rice is about 1 mm thick. 1 km = 1000 m, 1 m = 1000 mm. Would the column be higher than Mount Everest? Would it be higher than the distance from earth to the moon? Would it be higher than the distance from earth to the sun? 2. If you had to count the number of grains of rice in a plastic bag, how would you do it? (Please do not answer by saying – “Just count them all.” It would take far too long!). Would your answer be exact or “as close as possible”? Can you think of more than one way to do this? Extension Write your own “doubling story or folktale”. It only needs to be a paragraph in length. An example might be the following. You must also provide a solution to your story. Use your imagination to create an original story. When a boy was asked by his parents what he wanted as a reward for all his good deeds, he asked for money, starting on his fifth birthday. On his fifth birthday he received one penny to put in the jar. On his sixth birthday he got two pennies to put in the jar. Now he has three pennies in the jar. On his seventh birthday he put four pennies in the jar. On which birthday would he have more than $1000.00 in the jar? I think the boy might have a problem. Do you know what it is? (Answer: on his 21st birthday. The problem might be: “Can he find a big enough jar?”) Good Luck!

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