The Chessboard - Mr. Seccareccia`s Math

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.
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
cup of uncooked rice per student
24 teaspoons in
cup of uncooked rice
** You must show all of your calculations **
** You must also type out each question **
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?
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?
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!