Answers

Challenge 7-8/Ages 12-13

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Presented to MathWorld Interactive by Lacey, Katie, Sarah, Gabrielle, and Ashley of Bloomington, Illinois

Write what you know..

We know checkers is played on a board that has squares on it that alternate between two colors, usually black and red.  We know the squares are arranged into 8 rows and 8 columns.   We know it takes 2 players to play.  We know that when played, each player fills one color of 3 rows with checkers (one player has black, the other has red.)  We know checkers has empty squares.  We know a domino covers exactly 2 squares.

Write what you are looking for...

1.  We are looking for how many squares are on a checkerboard.  2.  We are looking for how many checkers there are.  3.  We are looking for how many empty squares are there when the checkers are placed in starting position on the checkerboard. 4.  We are looking for what fraction of the squares are filled with red squares.  5.  We are looking for how many grains of rice would be on the last square in the 3rd row if 1 was placed on the 2st square, 2 on the second, 4 on the third and so on..  6.  We are looking for how many grains would be on the last square.  7.  We are looking for how many miles long would the 1/4 inches of rice be if they were placed end-to-end.  8.  We are looking for if each grain was 1/4 inches long, how many miles long would the grains be if placed end to end.  9.  We are looking for how many dominoes will be needed to cover all the squares.  b)  We are looking for how many dominoes are needed to cover all squares except 1 black and 1 red.  c)  We are looking for how many dominoes are needed to cover all the squares except for 2 red squares.

Strategies....

1.   Since there are 8 squares vertically and 8 squares horizontally, we multiplied the 8 x 8 to find the amount of squares.  We did this in our heads and figured that 8 x 8 = 64. 2.  We are finding the amount of checkers there by drawing a checkerboard on the markerboard in the classrsoom and then counting how many checkers there would be.  We know that hwen you set up the board, a checker goes on every othe square.  We figured out that there would be 24 checkers.  3.  We found the amount of empty squares there are by taking the amount of checkersminus the number os squares on teh checkerboard.  WE did this with a calculator.  This means there are 40 empty squares.  4.  We found the fraction of red squares because we know that there are an equal amount of red squares and black squares.  WE found the amount of red squares by dividing 64 by 2 with a calculator.  The answer we got was 32.  So, the fraction of red squares would be 32/64.  5.  We found this answer by drawing the first three rows and then writing the amount that would be in each square.  We used a calculator to find the amount.

65,536

131,072

262,144

524,288

1,048,576

2,097,152

4,194,304

8,388,608

256

512

1,024

2,048

4,096

8,192

16,384

32,768

1

2

4

8

16

32

64

128

We found out that there would be 8,388, 608 grains of rice in the third row.  6.  We are going to figure out how many grains of rice would be on the last square by continuiing our drawing and after each number multiplying it by 2 with a calculator. 20,000,000,000,000,000,003,058.  Pretty soon we gave up on this way to do it because we realized that our calculator wouldn't hold all of  the numbers.  We asked our teacher for assistance and our teacher showed us a pattern by going vertically down the last column multiplying each number by 256.  We did this by using a pencil and paper.  I finally found the amount of rice that would be on the last square.  The number of rice grains on the last square would be 2, 361,183,241,434,822,606,848.  7.  I takled this problem by taking the amount of rice grains and dividing the number by 4 to find out how many in. of rice there would be.  I again did this by hand.  I found that there would be 590,295,810,358,705,651,712 inches of rice.  We found out that there were 63,360 inches in a mile.  I divided 590,295, 810, 358, 705, 651, 712 by 63, 360 by hand.   The answer I got was 9, 316,537, 410,964,420 R 512.  So, there would be 9,316,537,410,964,420 miles and 512 inches if each grain of rices were lined up.  4.  a)  We can use divisioin and pencil/paper to solve this problem.  We know that there are 64 squares on a checkerboard.  So, if each domino covers 2 squares, you'd take 64/2=32 dominoes to cover all the squares.  Answer:  32 dominoes.  b)  We figured out on A that it takes 32 dominoes to cover the whole board.  Assuming underneath one domino are one black and one red square, to one black and one red square exposed, you'd have to subtract one domino from 32 which equals 31.  Answer:  31 dominoes.  c)  Assuming that more than one domino can cover a square in this instance, the number of dominoes you could use would be 34.   Really you could use any number arrangement.

Solutions....

The solution to question #1 is 64 squares.  The solution to question #2 is 24 checkers.  The solution to question #2B is 40 empty squares.   The solution to question #2C is 1/2 of the squares are red.  The solution to 3A is 8,388,608 grains of rice.  The solution to #3B is 2,361,183, 241, 434, 822, 606, 848.   The solution to question #3C is 9,316, 537,410,964, 420 miles and 512 inches.  The solution to #4A is 32 dominoes.  The solution to #4B is 31 dominoes.  The solution to #4C is 34 dominoes.

Check your work...

We checked our work by reading over the problems and checking our multiplication to see if it is correct.  All our work is correct.

Extend....

How many squares total are there on a checkerboard if you can combine squares to form a new square?  Meaning...a square can be 2x2 or 3x3?

AWARD:  OUTSTANDING SOLUTION FORMAT

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AwardPresented to MathWorld Interactive by Tony Boy of Staten Island, New York

HI MY NAME IS TONY AND I AM FROM ST. MARY'S SCHOOL IN STATEN ISLAND. HERE ARE MY ANSWERS TO THE CHECKERBOARD MATH CHALLENGE. TO SOLVE PROBLEM 1, WHICH WAS TO SEE HOW MANY SQUARES ARE ON A CHECKERBOARD, I MULTIPLIED THE NUMBER OF ROWS WHICH WAS 8 TIMES THE NUMBER OF COLUMNS WHICH WAS 8. THE ANSWER I CAME UP WITH IS 64 SQUARES ON THE CHECKERBOARD. 8 X 8 = 64

TO SOLVE PROBLEM 2A, I REACHED MY ANSWER OF 24 BY THE FOLLOWING METHOD. I KNEW THAT EACH PLAYER FILLS ONE COLOR OF THREE ROWS WITH CHECKERS. FIRST I HAD TO FIGURE OUT HOW MANY OF EACH COLOR WAS ON A ROW. I KNOW THERE ARE EIGHT SQUARES IN EACH ROW AND THAT THEIR ARE EQUAL AMOUNTS OF RED SQUARES AND BLACK SQUARES IN EACH ROW. I THEREFORE KNOW THAT THERE ARE 4 OF EACH COLOR IN EACH ROW.

I MULTIPLIED 4 (WHICH IS THE AMOUNT OF SQUARES) BY 3 (WHICH IS THE AMOUNT OF ROWS) AND CAME UP WITH 12. I THEN MULTIPLIED THAT NUMBER BY 2 BECAUSE THERE WERE TWO PLAYERS. TO FIGURE OUT PROBLEM 2B, WHICH WAS HOW MANY SQUARES WERE EMPTY, I SUBTRACTED 24 WHICH IS THE NUMBER OF SQUARES FILLED IN FROM 64 WHICH WAS THE TOTAL AMOUNT OF SQUARES AND CAME UP WITH MY ANSWER OF 40. TO FIGURE OUT

PROBLEM 2C, WHICH IS WHAT FRACTION OF THE SQUARES IS FILLED WITH RED SQUARES, I DIVIDED 64 BY 2 BECAUSE THERE WAS AN EVEN AMOUNT OF RED AND BLACK SQUARES. I CAME UP WITH MY ANSWER OF 32/64 WHICH I REDUCED TO 1/2. TO SOLVE PROBLEM 3A, WHICH WAS TO TRY TO FIGURE OUT HOW MANY GRAINS OF RICE WOULD BE ON THE LAST BOX IN THE THIRD ROW, THESE ARE THE STEPS I DID. THEY GAVE US INFORMATION THAT

THERE WAS ONE GRAIN OF RICE ON THE FIRST SQUARE, 4 ON THE THIRD AND 8 ON THE FOURTH. THIS TELLS ME THAT IN ORDER TO GET MY ANSWER, I HAVE TO DOUBLE THE AMOUNT OF GRAINS EACH TIME I MOVE OVER A SUCESSIVE SQUARE. I WILL HAVE TO USE MY MULTIPLICATION SKILLS ON THIS PROBLEM. I WILL HAVE TO USE THIS PROCEDURE FOR EACH OF THE 24 SQUARES IT TAKES ME TO GET TO THE LAST BOX ON THE THIRD ROW.

SO IF THERE IS 8 ON THE FOURTH SQUARE, I KNOW THERE IS GOING TO BE 16 ON THE FIFTH, 32 ON THE SIXTH, AND SO ON. I WILL CONTINUE TO DO THIS UNTIL I REACH THE 24TH SQUARE. ONCE I DO THIS METHOD 23 TIMES, I WILL GET MY ANSWER WHICH WAS 16,777,216. TO SOLVE PROBLEM 3B, I DID THE SAME STEPS AS PROBLEM 3A, BUT KEPT ON GOING TILL I HIT THE 64TH SQUARE. MY FINAL ANSWER TO THAT PROBLEM WAS

25,801,311,048,658,816. TO SOLVE PROBLEM 3C, I MULTIPLED MY FINAL ANSWER FROM 3B WHICH WAS 25,801,311,048,758,816 BY .25 I DID THIS BECAUSE I KNEW THAT EACH GRAIN IS 1/4 SO I CONVERTED IT TO A DECIMAL. THAT GAVE ME A FINAL ANSWER OF 6,450,327,772,164,704.00 WHEN SOLVING PROBLEM 4A, I TOOK MY MODEL AND FOUND OUT THAT THE AMOUNT OF SQUARES WAS 64. I THEN DIVIDED THT NUMBER BY 2 BECAUSE

EACH DOMINOE COVERED EXACTLY TWO SQUARES. MY ANSWER CAME OUT TO 32. AFTER I GOT MY ANSWER, I PHYSICALLY CHECKED BY USING MY MODEL THAT I MADE AND USED TWO FINGERS TO CHECK TWO SQUARES AT A TIME. WHEN SOLVING PROBLEM 4B, I FOLLOWED THE SAME STEPS FROM 4A. I THEN SUBTRACTED TWO SQUARES BECAUSE THE PROBLEM SAID THAT ONE BLACK AND ONE RED SQUARE WERE NOT COVERED. MY FINAL ANSWER IS 31.

WHEN SOLVING PROBLEM 4C, I USED THE SAME METHODS THAT I USED IN 4A AND 4B.

MY EXTENSION PROBLEM FOR THIS PARTICULAR MATH PROBLEM WOULD BE HOW MANY DOMINOES WOULD YOU USE, IF YOU ONLY WANTED TO COVER 48 SQUARES AND HOW MANY GRAINS OF RICE WOULD YOU NEED TO COVER THE REMAINING SQUARES IF YOU FOLLOWED THE RULES IN QUESTION 3 FROM THE ORIGINAL PROBLEM. YOU WOULD GET THE FIRST PART OF YOUR ANSWER BY DIVIDING 48 BY 2 WHICH GIVES YOU AN ANSWER OF 24. YOU WOULD GET

THE SECOND PART OF YOUR ANSWER BY SUBTRACTING 48 FROM 64 WHICH COMES OUT TO 16. THEN YOU WOULD HAVE TO START WITH 1 AND TIMES THAT BY 2 WHICH EQUALS 2, TIMES THAT BY 2 WHICH EQUALS 4, ETC., ETC. YOUR REMAINING BOXES WOULD CONTAIN 65,536 GRAINS OF RICE

AWARD:  OUTSTANDING STRATEGIES

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AwardPresented to MathWorld Interactive by M.A.D. Math of Staten Island,New York

Our Problems are to find out if we ever played checkers,a checker has how many squares,how many checkers there are,how many empty squares there are,what fraction of the squares is filled with red squares,on the third row how many grains of rice would be on the last square,on the last square how many grains of rice would there be,if each grain is 1/4 inches long. How many miles long would the grains be if placed end-to-end,to cover all the squares how many dominos will be needed,to cover all squares except for 1 black and1 red square how many dominos will be needed& how many dominos are needed to cover all the squares except for 2 red squares.

Knowns: We know that the game is played on a board. We know that the board is divided into squares. We know that the squares alternate between two colors, usually black and red. We know the squares are arranged into 8 rows and 8 columns. We know that the game is played with two players. We know each player fills 3 rows with 1 color checker. We know that 1 domino is the same size as two checker squares. We know each grain of rice is 1/4 of an inch. WE Know we've played checkers before but since it does not say that on the sheet we will put it under assumptions.

Unknowns: We do not know how many checkers there are. We do not know how big a checker is. We do not know how many squares there are. We do not know how many empty squares there. We do not know how many grains of rice there are. We do not know how many grains of rice fit in one square. We do not know how many dominos there are.

Assumptions: We assume we played checkers before. We assume there is a checker for each square. We assume there is exactly the same number of black squares as there are red.

Strategies:Write an equation.1)8V8=64{To find out how many squares are on the checkerboard we multiplied the 8 columns and the 8 rows together.}2)A)8@3=24 +24=48 this answer shows both the red and black checkers added up.For this answer we multiplyed 3 rows by 8 squares.2)B)64-48=16empty squares.I subtracted 64, all the total of the squares by 48 the total of checkers on the board. 2)C)64-32=32.Half of the squares are filled with red squares.We found this out by counting the squares on a checker board.3)A ) We found this out by using a calculator.We wrote this on the calculator on the following:2 to Y\X 23=8388608 because there are 24 squares made up of 2s , but one of them is 1.3)B)2 to Y\X 63=9.22337203718 We found this out by using a calculatorwe used the same process as in the problem before but used 64.3)C)253440 are 1\4 of an inch in one mile.We found this out by finding how many inches in a mile and finding how many 1\4 are in an inch. 4)A) 64)2= 32 you divide the number of checker squares by 2 because each domino takes two squares.4}B)64)2-1=31you divide the number of squares by two and then minus it by 1 because the colors alternate.4)C)64)2-2=30 you divide the number of squares by two and then minus it by 1 because the colors alternate.

Solution:We have played checkers before.1.There are 64 squares on a checkerboard .Part2 There are 48 checkers all together. There are 16 empty squares.There are 32 Red squares. Part3 A)8388608 we wrote how we got our anwer on the strategies. B)9.22337203718 we wrtote how we got this on the strategies. C)253,440 miles.we wrote how we got this on the strategies. Part 4 A)32 dominos cover all the squares B)31dominos cover everything but1 black and 1 red square C)30 dominos cover every square but 2 red.

Etend Problem: How many dominos are needed to cover half of the checkerboard Answer:16dominos

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