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I am thinking of making a quilt from an illustration that appeared on the
cover of Scientific American magazine a while ago. I need some help
figuring a few things out, though.
The picture below illustrates the problem. All I know is that all of the pieces are squares, and that the area of C is 64 square inches and the area of D is 81 square inches. Here's what I need to know: Will the finished shape be a square? If I want to quilt along all of the lines (including the outside edge), how much thread will I need? (Assume that it takes 1.5 inches of thread to quilt 1 inch of the line.) ______________________ ________________ | | | | | | | | | | | | | | | | A | B | | | | | | | | | | | | | | |_______ ________| |________________ _____| | | | | | H | | | | G | | C | | |_____|_______| | | | I->|_|________| | F | | | | | E | | | | | D | | | | | |________________|___________|__________| SolutionsThis problem proved to be a little more difficult than a lot of people thought it would be. 115 people got it right, but 71 people got it wrong. Most of the errors were because the length of the thread was tough to figure out. It's sort of like Stonehenge - every time you count it, you could come up with a different answer. The way to combat this is to come up with a good strategy to make sure you count all of the pieces only once. One person used a highlighter to cross them off. Here are two more strategies. Edward Yoo of O'Neill Collegiate and Vocational Institute counted all of the vertical lines, then all of the horizontal lines to help him keep them clear. Ha Trung Ho of Sefton High School added up all of the edges and divides by two - since they are all shared by other pieces (except for the outside, and he deals with that). Their answers are included below. Figuring out the sizes of the squares was not as difficult, but not everyone got that right, either. I have included the explanations of Jonathan Michaels of Pleasanton Middle School and Chaim Bloom of Akiba Hebrew Academy. They did a very thorough job with both part of the solution. Some people came up with answers that weren't even rectangles - that all four edges were different lengths. Since we are starting with squares, the finished piece has to be at least a rectangle - everything is 90 degrees. One thing that caught a few people is that they tried to add the perimeters somehow. You'll find that adding the perimeters in something like this doesn't work the same way that areas does, because much of the perimeter is shared by two pieces, so be careful! A list of all the people who got this problem right follows the highlighted solutions below, and most of the solutions are also available.
From: Jonathan Michaels
Grade: 8
School: Pleasanton Middle School, Pleasanton, California
C is a square and it has an area of 64 square inches. The
formula for the area of a square is A = s^2 where s is the
length of a side of the square. I can replace A with 64 and
solve (by taking the square root of both sides of the equation)
to find that the length of a side of C is 8. Using the same
method for D, I find out that a length of the side of D is 9.
Since, by looking at the illustration (assuming that all lines
that appear to be straight are straight and all angles that
appear to be right angles are right angles), I can tell that the
length of the side of I plus the length of the side of C equals
the length of the side of D. In an equation, I + C = D. I know
C and D, and so I can replace them with their values (8 and 9
respectively). Then I can solve (by subtracting 8 from both
sides of the equation) to find that the length of a side of I is
1. Henceforth "the length of a side of x" is "x". I can see
that I[1] + D[9] = E. I simplify to see that E = 10. I also see
that H + I[1] = C[8]. I solve by subtracting 1 from each side to
see that H = 7. I can also see that E[10] + I[1] = H[7] + G.
Simplifying and solving that, I find that G = 4. Since G[4] +
E[10] = F, F = 14. Because H[7] + C[8] = B, B = 15. Finally,
since F[14] + G[4] = A, A = 18.
One side of the large quadrilateral is F[14] + E[10] + D[9] =
A[18] + B[15] = 33. The other side is A[18] + F[14] = B[15] +
C[8] + D[9] = 32. Since the two perpendicular sides are not
equal in measure, the large figure is not a square. To find out
the total length of the lines in and making up the large figure,
I add 4A + 3B + 3F + 3E + 3D + 2C + G + I. This comes out to be
237. There are 237 inches of line to stitch on. Since it takes
1.5 inches of thread to stitch 1 inch of line, the amount of
thread needed is 1.5 * 237 = 355.5 inches.
From: Edward Yoo
Grade: 13
School: O'Neill Collegiate and Vocational Institute, Oshawa, Ontario, Canada
Let's assume that each capital letter
represent the length of that
sqaure.
c = square root of 64 = 8
D = square root of 81 = 9
I = D - C = 1
H = C - I = 7
E = D + I = 10
G = E + I - H = 4
F = E + G = 14
A = F + G = 18
B = A + G - H = 15
Therefore, A = 18, B = 15, C = 8, D = 9,
E = 10, F = 14, G = 4, H = 7,
I = 1
Since A + B = 33 and A + F = 32, the
finished shape is not a square.
The total length of every hor. lines
= A+B+A+B+G+H+I+C+F+E+D
= 119 inches
The total length of thread needed
for hor. lines
= 119 x 1.5
= 178.5 inches
The total length of every ver. lines
= A+F+A+F+G+E+H+I+B+C+D
= 118 inches
The total length of thread needed
for ver. lines
= 118 x 1.5
= 177 inches
Therefore, the total length of thread
needed for the quilt is
355.5 inches
From: Hai Trung Ho
Grade: 11
School: Sefton High School, Sefton, New South Wales, Australian
From now, just assume that all the units are in the appropriate
units (eg. inches, square inches...).
Let X be the assigned area, and x be the side length of the square.
(the difference is that the area is in uppercase, the side is in
lowercase)
Since C = 64 then c =8
D = 81 then d =9
from the diagram, i+c=d, so i=1
similarly, e=10 (e=d+i)
h=7 (h+i=c)
g=4 (h-i+g=e)
b=15 (b=h+c)
a=18 (a=b+h-g)
f=14 (a+b=f+e+d)
From this, the 'horizontal' side is a+b = 33
the 'vertical' side is a+f = 32
Thus, the resulting figure is not a square.
Now, we are required to find the length of thread. We shall find
the length of the way round and then multiply by 1.5 .
Note that every side belongs to exactly two squares. for example, the
side common between A and B is shared between A, B, and H
note that this is counted twice if we add the perimeter of A, B and H.
Thus, twice the 'length' is the perimeter of all the squares plus
the outer perimeter.
This results in 4(a+b+c+d+e+f+g+h+i)+a+a+b+b+c+d+d+e+f+f
= 4(86) + 130
= 474 inches
Note that this is twice the required length
Thus, the required length is 287 inches
To quilt along the lines, it would require
287 * 1.5 = 355.5 inches of thread.
From: Chaim Bloom
Grade: 10
School: Akiba Hebrew Academy, Merion, Pennsylvania
Subject: Making A Quilt Of Squares
I liked this one. :-)
Since the area of C is 64 sq. in. and D is 81 sq. in., you know that the
sides of C and D are 8 and 9 respectively. From there you can deduce
that each side of I is 1. (9-8). Then, you know that H is 7 because
8-1=7. You also kow that E is 10 because 9+1=10. G + H = E + I, so G
must be 4. G + E = F, so F is 14. G + F = A, so A is 18. And C + H =
B, so B is 15.
Now you know that the figure is 33 x 32 x 33 x 32. Hence, it is not a
square.
On to part two.
Take the perimeter (130).
Add 18 twice. (Where A meets B and the top of H, and where A meets F and
G.)
Add 15. (Where B meets H and C.)
Add 14. (Where F meets G and E.)
Add 10 twice. (Where E meets G and part of H, and where E meets D and
I.)
Add 9. (Where D meets I and C.)
Add 8. (Where C meets H and I.)
Add 4. (Where G meets part of H.)
Add 1. (Where I meets part of H.)
This adds up to 237 inches to quilt, resulting in a need for 355.5 inches
of thread.
Thanks,
Chaim Bloom
10th grade, Akiba Hebrew Academy
The following students submitted correct solutions this week:Robert Huang, Grade 9, Taipei American School, Taipei, TaiwanTiffanie Lam, Grade 8, Sequoia Middle School, Pleasant Hill, California Alex Morgovsky, Grade 11, Akiba Hebrew Academy, Fort Washington, Pennsylvania Zach Rentz, Grade 10, Akiba Hebrew Academy, Merion, Pennsylvania Jonathan Michaels, Grade 8, Pleasanton Middle School, Pleasanton, California Edward Yoo, Grade 13, O'Neill Collegiate and Vocational Institute, Oshawa, Ontario, Canada Will Marrs, Grade 9, Granada High School, Livermore, California Brandon Levtzow, Grade 9, Southeast Raleigh High School, Raleigh, North Carolina Tom Atkins and Kevin Kelly, Grade 7 & 8, St. Gregory School, Plantation, Florida Kale Stedman, Grade 10, Carlisle High School, Carlisle, Pennsylvania Libbie Gies, Grade 9, Highland Park Senior High School, St. Paul, Minnesota Alison Falkenhagen, Grade 9, Highland Park Senior High School, St. Paul, Minnesota Michael Buckler, Grade 10, Germantown Academy, Fort Washington, Pennsylvania Jenny Lurie, Grade 10, Akiba Hebrew Academy, Merion, Pennsylvania Alessandro Binotti, Grade 9, American Embassy School, New Delhi, India Susannah Goodwin, Grade 10, Carlisle High School, Carlisle, Pennsylvania Saurabh Sarkar, Grade 8, Albright Middle School, Houston, Texas Noah Ribeck, Grade 9, Camden-Rockport High School, Camden, Maine Perry Scofield, Grade 9, Camden-Rockport High School, Camden, Maine Wassia Khaja, Grade 8, Albright Middle School, Houston, Texas Christie Heyer, Grade 8, Odle Middle School, Bellevue, Washington Brian Dorning, Grade , Redmond High School, Redmond, Oregon Tracy Steed, Grade 12, Wilburton High School, Wilburton, OK Justin McCoy, Grade 8, Albright Middle School, Houston, Texas Nagib Haque, Grade 10, Druid Hills High School, Atlanta, Georgia Avrum Tilman, Grade 10, Akiba Hebrew Academy, Merion, Pennsylvania Gordon Bockus Jr., Grade Freshman, Eastern Oklahoma State College, Wilburton, Oklahoma Melanie Seng, Grade 10, Germantown Acadmey, Fort Washington, Pennsylvania Erin Gates, Grade 9, Skyview High School, Vancouver, Washington Devyn Killingbeck, Grade 11, South Windsor High School, South Windsor, Connecticut Barbara van Santen, Grade 9, American Embassy School, New Delhi, India Aya Koshinaka, Grade 9, American Embassy School, New Delhi, India Jia-Xin Wang, Grade 9, American Embassy School, New Delhi, India Hai Trung Ho, Grade 11, Sefton High School, Sefton, New South Wales, Australian Chris Lauber, Grade 9, Smoky Hill High School, Aurora, Colorado Catherine Mangasi, Grade 12, Wilburton High School, Wilburton, Oklahoma Ann Metiva and Nicole Williams, Grade 8, Center for the Arts and Sciences, Saginaw, Michigan Jessica Barclay-Strobel, Grade 10, Oak Park and River Forest High School, Oak Park, Illinois Rob Eagle and Darren Kerstien and Jessica Black and Rachel Meyer, Grade 7 & 8, The Challenge School, Denver, Colorado Travis Rowland, Grade 9, Smoky Hill High School, Aurora, Colorado LeMecha' Bush, Grade 12, Shelby County High School, Columbiana, Alabama Noam Abrams, Grade 10, Akiba Hebrew Academy, Merion, Pennsylvania Lisa Schwartz, Grade 11, South Windsor High School, South Windsor, Connecticut Dave Schilberg, Grade 11, South Windsor High School, South Windsor, Connecticut Jordan Dubay, Grade 11, South Windsor High School, South Windsor, Connecticut Julia Fischer, Grade 10, Granada High School, Livermore, CA Candace Murray, Grade , Redmond High School, Redmond, Oregon Bridget Timony, Grade , Redmond High School, Redmond, Oregon Laura Roos, Grade 10, Shaler Area High School, Pittsburgh, Pennsylvania Joanna Frankel and Joanna Mack, Grade 9, Germantown Academy, Fort Washington, Pennsylvania Kelsey Long, Grade 9, Highland Park Senior High School, St. Paul, Minnesota Nicholas Kim, Grade 10, American Embassy School, New Delhi, India Sarah Lawrence, Grade , Redmond High School, Redmond, Oregon Hyun-Il Koh, Grade 10, American Embassy School, New Dehli, India Chaim Bloom, Grade 10, Akiba Hebrew Academy, Merion, Pennsylvania Kerianne Schubert, Grade , Redmond High School, Redmond, Oregon Camille and Ashley, Grade , Redmond High School, Redmond, Oregon Megan Morgan, Grade , Redmond High School, Redmond, Oregon Eve L. and Katy F-M., Grade 9, Germantown Academy, Fort Washington, Pennsylvania Adam Skoglund, Grade , Wgy, Enkoping, Uppland, Sweden Aaron Mertz, Grade 8, Plum Grove Junior High School, Rolling Meadows, Illinois Steve King, Grade 9, Carlisle High School, Carlisle, Pennsylvania Frannie Laks, Grade 10, Akiba Hebrew Academy, Merion, Pennsylvania Mathew Steadman, Grade 8, Odle Middle School, Bellevue, Washington Blake Hall, Grade 10, Carlisle High School, Carlisle, Pennsylvania Alex Chen, Grade 7, Odle Middle School, Bellevue, Washington C. J. Walthall, Grade 8, Albright Middle School, Houston, Texas Josh Sarfaty, Grade 8, Valley View Middle School, Simi Valley, California Eric Hanau, Grade 10, Amador Valley High School, Pleasanton, California Alicea Klemas, Grade 10, South Windsor High School, South Windsor, Connecticut David Grant, Grade , Livermore High School, Livermore, California Jie Tang, Grade 6, Odle Middle School, Bellevue, Washington Rachelle Bermingham and Jennifer Mutch, Grade 9, Granada High School, Livermore, California Goodwin, Grade 10, MD Kamila Sikora, Grade 9, Smoky Hill High School, Aurora, Colorado Xiaochang, Grade 8, Odle Middle School, Bellevue, Washington Alexandra Niculce and Teodora Niculce, Grade 10, Reading High School, Reading, Pennsylvania Janas, Grade , Oswego, New York Cristin Kenney, Grade 8, homeschooled, Turbot, Pennsylvania Michael Knickerbocker, Grade 6, Lowther North Intermediate School, Emporia, Kansas Jen Ramirez, Grade 10, Germantown Academy, Fort Washington, Pennsylvania Emily Buzicky, Grade 9, Highland Park Senior High School, St. Paul, Minnesota Kendra Cutter, Grade , Redmond High School, Redmond, Oregon Jody Rogers, Grade , Redmond High School, Redmond, Oregon Mike Montreuil, Grade , Redmond High School, Redmond, Oregon Kari Kristensen, Grade , Remond High School, Redmond, Oregon Jordie Kvidera, Grade 7, Odle Middle School, Bellevue, Washington Joe Thomer, Grade 9, Germantown Academy, Fort Washington, Pennsylvania Alex Chernyavsky, Grade , Akiba Hebrew Academy, Fort Washington, Pennsylvania Bassem Amin, Grade 10, American Embassy School, New Delhi, India Matthew Harrison, Grade 9, Germantown Academy, Fort Washington, Pennsylvania Justin Chaudrue, Grade , Carlisle High School, Carlisle, Pennsylania Alison Miller, Grade 6, homeschooled, Niskayuna, New York Erin Gehrke, Grade , Redmond High School, Redmond, Oregon Jason Chiu, Grade 9, Laramie Junior High School, Laramie, Wyoming Jennifer Liang, Grade 8, Odle Middle School, Bellevue, Washington Meredith Blackman and Renee Bacino, Grade 9 & 10, Germantown Academy, Fort Washington, Pennsylvania Anne Hines and Lauren Rossi, Grade 10, Germantown Academy, Fort Washington, Pennsylvania Tim Peterson, Grade 7, homeschooled, Rochester, New York Emily Rozak, Grade , Germantown Academy, Fort Washington, Pennsylvania Zach Dillon, Grade , Odle Middle School, Bellevue, Washington Scott Brown, Grade 9, Germantown Academy, Fort Washington, Pennsylvania Sonya Garg, Grade 10, American Embassy School, New Dehli, India Marc Horton, Grade 8, Odle Middle School, Bellevue, Washington Charlie Wenger and Kate Boyd, Grade 8, Odle Middle School, Bellevue, Washington Rob Johnson, Grade , Redmond High School, Redmond, Oregon Mark Goodman, Grade , Redmond High School, Redmond, Oregon Marisa Gillespie, Grade , Redmond High School, Redmond, Oregon Peter Hoover, Grade , Redmond High School, Redmond, Oregon Lisa Simmons, Grade , Redmond High School, Redmond, Oregon Ryan Holcomb, Grade , Redmond High School, Redmond, Oregon Ben Wagenblast, Grade , Redmond High School, Redmond, Oregon Lindsey Fangman, Grade , Redmond High School, Redmond, Oregon Kristen Allegue, Grade 9, Ethel Walker School, Simsbury, Connecticut Leia Nitschelm, Grade , Redmond High School, Redmond, Oregon Carrie Tsui, Grade , Smoky Hill High School, Aurora, Colorado |
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