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This Problem
Lettuce for Sale - posted March 29, 1999
 On Monday, the produce manager stocked the display
case with eighty heads of lettuce. By the end of the day some of
the lettuce had been sold.
On Tuesday, the manager surveyed the display case and counted
the number of heads of lettuce left. He decided to add an equal
number of heads of lettuce. (He doubled the leftovers.) By the end
of the day he had sold the same number of heads of lettuce as on
Monday.
On Wednesday, the manager decided to triple the number of heads
of lettuce that had been left in the case. He sold the same number
of heads of lettuce that day too. However, at the end of the day
there were no heads of lettuce left.
How many heads of lettuce were sold each day?
If you use variables, make sure to represent them. Explain all
equations and show your work. Don't forget to show that you
checked your work.
Comments
Basically students used four ways to solve this
problem. The most common way was to use a single variable. Other
students used three variables and solved by substitution or
elimination. A few students recognized they used a composition of
functions. Finally, we had students who used a guess and check
method. Although guess and check is certainly a valid technique,
the Algebra Problem of the Week site is trying to encourage
students to use their algebra problem solving techniques to solve
most of the problems posed.
Highlighted solutions:
| From: |
Matthew Roitstein, age
15 |
| School: |
William S. Hart High School,
Newhall, CA | |
48 heads of lettuce were sold each day.
I first read the problem all the way through before I started
anything, and then I read it a second time writing every step down to
"build" an equation. I started with 80, since the produce manager
started with 80. Then I went on from there. (In the following steps,
y stands for how many were sold that day.)
80 This is what he started with.
80 - y This is what he had after the first day.
2(80 - y) This is what he had to begin the next day.
2(80 - y) -y This is what he had after the second day.
3[2(80 - y) - y] This is what he had to begin the third day.
3[2(80 - y) - y] - y = 0 This is what he had after the third day.
On the last step I finally came up with an equation to solve for how
many heads of lettuce the produce manager sold each day. Now, I
simply solved for y.
3[2(80 - y) - y] - y = 0
3(160 - 2y - y) - y = 0
3(160 - 3y) - y = 0
480 - 9y - y = 0
480 - 10y = 0
480 = 10y
y = 48
Now, I know that the number of heads of lettuce that the produce
manager sold each day was 48. Then I checked my work by plugging 48
into the original eqaution for y.
3[2(80 - 48) - 48] - 48 = 0
3(160 - 96 - 48) - 48 = 0
3(16) - 48 = 0
48 - 48 = 0
0 = 0
48 is the amount of heads of lettuce sold each day.
| From: |
Mariko
Furukawa,
age 16 |
| School: |
Shaler Area High School,
Pittsburgh, PA | |
He sold 48 heads of lettuce each day.
Follow what the produce manager did. Let the number of lettuce he
has, Y.
Monday - Got 80 heads of lettuce.
Sold some of them which we don't know how many, so make the
number of lettuce he sold variable, X.
Y=(80-X)
Tuesday - Doubled the left overs.
Sold the same number of heads of lettuce as on Monday.
Y=2(80-X)-X
Wednesday-Tripled the left overs from Tuesday.
Sold the same number of heads of lettuce as other days.
Y=3{2(80-X)-X}-X
Now the end of the Wednesday, there were no heads of lettuce left. So
we can set the variable, Y that is the number of lettuce he has
equals zero.
3{2(80-X)-X}-X=0 Check the answer.
3{160-2X-X}-X=0 Monday (80-X)=80-48=32
480-6X-3X-X=0 Tuesday 2(32)-48=16
480-10X=0 Wednesday 3(16)-48=0
-10X=-480 There are no heads of lettuce left.
X=48
| From: |
Asher Walkover, age
16 |
| School: |
Rambam Mesivta, Lawrence,
NY | |
The grocer sold 48 lettuces every day. (Who would have guessed green
veggies were so popular? Ich)
Let the number of lettuces that the man started out with equal y.
(Sure, we are told y is 80, but since formulas look fancier with more
letters, I'm not going to add that in until the end).
We are told that on Monday a number of lettuces were sold... a number
that I will say is equal to x. At the end of monday y-x lettuces
remain.
The number of lettuces is then doubled, and 2(y-x)=2y-2x, so on
Tuesday morn' there are 2y-2x lettuces in the stall. Tuesday night
rolls around and, lo and behold, x more lettuces are gone. There are
now (2y-2x)-x lettuces left, or 2y-3x.
Of course, 2y-3x lettuces is not the kind of thing a man can live off
(have you ever tried selling 2y-3x lettuces? It's not easy!), so our
industrious grocer says "POOF", and triples the amount of lettuces.
3(2y-3x) is equal to 6y-9x, so when the store opens for business on
Wednesday, 6y-9x lettuces are for sale.
Wednesday night comes and a bright moon shines over a happy
lettuce-seller and empty lettuce-market. This mathematically inclined
businessman does some quick arithmatic and finds that, believe it or not, x
lettuces were sold. Since there are now 0 lettuces, 6y-9x-x=0
6y-9x-x=0 so combining x terms...
6y-10x=0 adding 10x to both sides...
6y=10x and finally deviding by 10...
6y/10=x
Since when this entrepreneur of a grocer first opened business on
Monday he had 80 lettuces, y=80, and...
6y/10=x replacing y with 80...
6(80)/10=x and working out the equation...
48=x
Therefore our happy green-grocer sold 48 lettuces each day
To check, I will subtract 48 (x) from the 80 (y) lettuces he started out with,
and get 32 - The number remaining at the end of Monday. I will double this
number, in imitation of our produce-manager, to find that there were 64 heads
of lettuce on the stall before business opened on Tuesday. 48 of these were
sold, so on Tuesday night 16 heads remianed. Tripling this number I see that
there are 48 lettuces for sale on Wednesday, and when those 48 are sold, 0
remain.
... hmmm... 93 cents each... 8% sales tax... change of a 50...hmm...
| From: |
Josh Clayton, age
15 |
| School: |
Rockford High School,
Rockford, MI | |
Each day, 48 heads of lettuce were sold.
Okay, what I did was go through the information step-by-step to make
sure I copied the information correctly. Here is my information I
extracted from the problem
sold x on Monday
80 - x = Monday extra
2(80 - x) - x = Tuesday extra
3(2(80 - x) - x) - x = Wednesday extra
Then, all I did was solve for x to find how many were sold.
3(2(80 - x) - x) - x = 0
3(160 - 2x - x) - x = 0
480 - 9x - x = 0
480 - 10x = 0
-10x = -480
x = 48
Next, I checked my answer.
Monday extra
80 - 48 = 32
Tuesday before selling
32*2 = 64
Tuesday extra
64 - 48 = 16
Wednesday before
16*3 = 48
Wednesday extra
48 - 48 = 0
It checks.
| From: |
Steven Barnaby, age
15 |
| School: |
Scottsburg High School,
Scottsburg, IN | |
There were 48 heads of lettuce sold each day.
Let x = number of heads of lettuce sold the first day
y = the number left over the first day
z = the number left over the second day
Monday 80 - x = y
Tuesday 2y - x = z
Wednesday 3z -x = 0
By substitution 3(2y-x) - x = 0
6y - 3x -x =0
6y - 4x = 0
by substitution again
6(80 - x) - 4x = 0
480 - 6x -4x = 0
480 - 10x = 0
+10x +10x
480 = 10x
480/10 = 10x/10
48 = x
To check find all variables and check
Using the equation for Monday and more substitution
80 - 48 = y
32 = y
Then, using Tues and substitution
2(32) - 48 = z
64 - 48 = z
16 = z
Monday 80 - 48 = 32
Tuesday 2 (32) - 48 = 16
Wed. 3(16) -48 = 0
| From: |
Tien Nguyen, age
14 |
| School: |
Johnson Middle School,
Westminster, CA | |
48 lettuces were sold each day.
x=number of lettuce sold each day
Day number of lettuce number of lettuce at
the end of the day
Monday 80 80-x
Tuesday 2(80-x) 2(80-x)-x
Wednesday 3(2(80-x)-x) 3(2(80-x)-x)-x=0
At the beginning of Monday there are 80 lettuce. At the end of the
day some lettuce were sold. 80-x. x is the number of lettuce that
were sold. At the beginning of Tuesday the manager doubled the left
over from yesterday. The left over from yesterday is 80-x so At the
beginning of Tuesday there are 2(80-x) lettuce. At the end of Tuesday
the same number of lettuce were sold like yesterday which is x so
it's 2(80-x)-x. On Wednesday the manager tripled the number of
lettuce leftover from yesterday so it's 3(2(80-x)-x). He sold the
same amount of lettuce on Wednesday too and at the end of the day
there were no lettuce left so it's 3(2(80-x)-x)-x=0
After I got the equation I solved it for x.
3(2(80-x)-x)-x=0
3(160-2x-x)-x=0
3(160-3x)-x=0
480-9x-x=0
480-10x=0
480=10x
480/10=10x/10
48=x
x=48. This means that the number of lettuce sold each day is 48.
I checked my work by putting 48 in the x place.
Day number of lettuce number of lettuce
at the end of the day
Monday 80 80-48=32
Tuesday 2(32)=64 64-48=16
Wednesday 3(16)=48 48-48=0
It works! at the end of Wednesday there are no lettuce left so it's
right.
| From: |
Kim Schneider, age
16 |
| School: |
Shaler Area High School,
Pittsburgh, PA | |
There were 48 heads of lettuce sold on each day.
The first step I did was to form the equation 80 - x = y. 80 is the
original amount of lettuce, x is the number of lettuce sold which is
unknown, and y is the number left after the first day. Since the
remainder was doubled the next day and the same amount of lettuce was
sold as the first day another equation can be formed, 2y - x = z.
Multiply y times two because that would be double the remainder of
the first day, then subtract x because that is the number sold, and z
would be the amount left after the second day. A third equation
still must be made for the number sold on the last day. This
equation is 3z - x = 0. z would be multiplied by 3 because the
remainder of the previous day was tripled, and again you would
subtract x because that was the same amount sold in the previous
days. The equation equals zero because there was no lettuce left
after that day. Now using these three equations you can solve for
x. First I made the third equation equal to x, 3z = x. Then I
substituted 3z in for x in the first two equations. Next i set these
two equations equal to y, 80 - 3z = y and y = 2z. Since both these
equations are equal to y then they are equal to each other, 80 - 3z =
2z. This reduces the variables to be solved for to one. After
solving for z you find that it equals 16, but since you want to know
the number of heads of lettuce sold each day you are not done. You
need to solve for the variable x. If z = 16 this can be substituted
in the equation 3z = x, 3z(16) = x. This tells that x = 48. Now
this can be substituted in the three equations to check if it is
correct. 80 - (48) = 32 2(32) - 48 = 16 3(16)
- (48) = 0
Since all three equations came out right the answer is correct. This
also shows that y = 32 which was not yet found.
| From: |
Lindsay Wolfson, age
16 |
| School: |
Shaler Area High School,
Pittsburgh, PA | |
There are 48 heads of lettuce sold each day.
Lindsay Wolfson
Problem of the Week
Began Sold Left over
Monday 80 X (80-X)
Tuesday 2(80-X) X 2(80-X)-X
Wednesday 3(2(80-X)-X) X 3(2(80-X)-X)-X ) or 0
Since each day X amount of lettuce is sold and on Tuesday and
Wednesday the amount left over is doubled and tripled respectively.,
then since there is none left on Wednesday:
3(2(80-X)-X)-X = 0
3(160-2X)-X)-X = 0
480-6X-3X-X = 0
480-10X = 0
480=10X
48= X
So if you put 48 into the chart then
Began Sold Left over
Monday 80 48 (80-48)=32
Tuesday 2(80-48)2(32)=64 48 2(80-48)-4864-48=16
Wednesda 3(2(80-48)-48)=48 48 3(2(80-48)-48)-48 )
48-48=0
48 heads of lettuce are sold each day
| From: |
HuiJun Chew, age
16
HuiMei Chew, age 16 |
| School: |
Raffles Girls' School
(Secondary), Singapore,
Singapore | |
48 heads of lettuce were sold each day.
Let x be the number of heads of lettuce sold each day.
No. of heads left on Monday=80-x
No. of heads on display on Tuesday=2(80-x)=160-2x
No. of heads left on Tuesday=160-2x-x=160-3x
No. of heads on display on Wednesday=3(160-3x)=480-9x
No. of heads left on Wednesday=480-9x-x=480-10x=0
480-10x=0
10x=480
Therefore, x=48
CHECK:
if x=48,
No. of heads left on Monday=80-48=32
No. of heads on display on Tuesday=2(32)=64
No. of heads left on Tuesday=64-48=16
No. of heads on display on Wednesday=3(16)=48
No. of heads left on Wednesday=48-48=0 (Q.E.D)
| From: |
Martin Angert, age
14 |
| School: |
Holland Junior High School,
Holland, PA | |
48 heads of lettuce were sold each day.
For this problem all that has to be done is to make an equation that
answers the problem. This is it: ((80-x+80-x)-x)3-x = 0, where x is
the number of heads of lettuce he sold each day. This equation works
because he sold x heads on the first day and then doubled it
(80-x+80-x). Then he sold x heads of lettuce (80-+80-x)-x. Then he
tripled it ((80-x+80-x)-x)3. Then he sold x again and was left with
zero ((80-x+80-x)-x)3-x = 0. So when worked out this is what the
problem looks like.
((160-2x)-x)3-x=0
480-6x-3x-x=0
480-10x=0
480=10x
48=x
This means that each day he sold 48 heads of lettuce. Now to check
the problem.
(80-48+80-48)-48)3-48=0
(64-48)3-48=0
48-48=0
0=0 AHA it works
| From: |
Martin McNicoll, age
19 |
| School: |
Strathclyde University,
Glasgow, Scotland,
UK | |
48 heads of lettuce were sold each day.
First set the number of heads sold each day as X.
The number of heads left on Monday is thus 80-X as there were 80 to
start with.
As the mananger doubles the number of heads before opening on
Tuesday, the number of heads left on Tuesday is
2(80-X)-X
Before opening on Wednesday the manager triples this amount and when
X heads are again sold, there are none left so
3[ 2(80-X)-X ]-X=0
Now this equation is linear and only in X so it should be easy to
solve.
Getting rid of the brackets and simplifying gives
10X = 480
Solving this gives
X=48
Now let's check whether this really works.
Start off with 80 heads.
On Monday 48 are sold leaving 32.
Manager doubles this to give 64 of which 48 are sold on Tuesday,
leaving 16.
Manager now triples this amount to give 48 and 48 are sold leaving
none.
This agrees with the condition set by the problem.
| From: |
Allan Espinosa, age
14 |
| School: |
Philippine Science High
School, Quezon City,
Philippines | |
They sold 48 lettuce heads each day
let x be the no. of lettuce heads sold each day.
Number of Lettuce heads
|
Monday |
Tuesday |
Wednesday |
| Stocks |
80 |
2(80-x) |
3(160-3x) |
| Stocks Sold |
x |
x |
x |
| Stocks left |
80-x |
160-3x |
0 | Equation:
3(160-3x)-x=0
480-9x-x=0
480-10x=0
-10x=-480
x=48
They sold 48 lettuce heads each day
Checking:
Number of Lettuce heads
|
Monday |
Tuesday |
Wednesday |
| Stocks |
80 |
64 |
48 |
| Stocks Sold |
48 |
48 |
48 |
| Stocks left |
32 |
16 |
0 |
167 students received credit this week.
Del Rasouli A. Rashid, age 20 - National
University of Singapore, Singapore, Singapore
Madeel
Abdullah, age 15 - Marple Newtown Senior High School, Newtown
Square, PA
Steve Adams, age 14 - Palm Harbor University
High School, Palm Harbor, FL
Mike Altman, age 16 -
Marple Newtown Senior High School, Newtown Square, PA
Kyle
Anderson, age 12 - Churchville Chili Junior High School,
Churchville, NY
Ashley Andreone, age 15 - Shaler Area
High School, Pittsburgh, PA
Martin Angert, age 14 -
Holland Junior High School, Holland, PA
C. Anghel, age
16 - Mackenzie High School, Deep River, Canada
Jeff
Baldovich, age 14 - Pequea Valley Intermediate School,
Kinzers, PA
Steven Barnaby, age 15 - Scottsburg High
School, Scottsburg, IN
Doruk Baykal, age 16 - Robert
College, Istanbul, Turkey
Rusty Bergren, age 17 - CESA
#1 (Ozaukee Community High School), Mequon, WI
Adam
Bezilla, age 16 - Shaler Area High School, Pittsburgh,
PA
Mike Bockus, age 16 - Wilburton High School,
Wilburton, OK
Angela Bookwalter, age 15 - Shaler Area
High School, Pittsburgh, PA
Tony Borres, age 16 - Shaler
Area High School, Pittsburgh, PA
Ivana Bosnic, age 16 -
Gimnazija F. Petrica, Zadar, Croatia
Robert Brooks, age
25+ - Not in School, Aurora, IL
Mark Brown, age 25+ -
Bawating Collegiate & Vocational Institute, Sault Ste. Marie,
Ontario, Canada
Teddy Buffington, age 10 - Heritage
Heights, Amherst, NY
Jill Burkhart, age 16 - Shaler Area
High School, Pittsburgh, PA
Ryan Button, age 14 - West
Morris Mendham High School, Mendham, NJ
Amanda C, age 16
- Dalton High School, Dalton, GA
Pat Caffrey, age 14 -
West Morris Central High School, Chester, NJ
Laura
Cancienne, age 15 - Fontainebleau High School, Mandeville,
LA
Kyle Cannon, age 15 - Rockford High School, Rockford,
MI
Kim Carlos, age 13 - Brookhurst Junior High School,
Anaheim, CA
Francesco Carnevale, age 18 - Liceo
Scientifico Gandini, Lodi, Italy
Vilma Ceazar, age 15 -
Crowley High School, Crowley, TX
Matteo Cella, age 17 -
Liceo Scientifico Gandini, Lodi, Italy
Christopher
Cellini, age 15 - Notre Dame High School, West Haven,
CT
Dolie Chacko, age 15 - Marple Newtown Senior High
School, Newtown Square, PA
Rag-Tag Cheetahs, average age
14 - Our Lady of Mount Carmel, Louisville, KY
Andrew
Chen, age 16 - Marple Newtown Senior High School, Newtown
Square, PA
Eric Chen, age 13 - Brookhurst Junior High
School, Anaheim, CA
Jimmy Chen, age 11 - Oak Hill
Elementary, Overland Park, KS
HuiJun Chew, age 16 -
Raffles Girls' School (Secondary), Singapore,
Singapore
HuiMei Chew, age 16 - Raffles Girls' School
(Secondary), Singapore, Singapore
Scott Chisholm, age 13
- West Essex Jr. High, North Caldwell, NJ
Jason Chiu,
age 13 - Laramie High School, Laramie, WY
Ben Clark, age
16 - Redmond High School, Redmond, OR
Katie Clark, age
16 - Shaler Area High School, Pittsburgh, PA
Josh
Clayton, age 15 - Rockford High School, Rockford,
MI
Monica-Ramona Costache, age 14 - Mihai Viteazul High
School, Bucharest, Romania
Luis Ernesto Criales Escobar,
age 12 - British International College, Barranquilla,
Colombia
Jenni Crom, age 15 - Watertown Senior High
School, Watertown, SD
Matt Culbertson, age 14 - West
Morris Mendham High School, Mendham, NJ
B.R Cutler, age
14 - Roosevelt High School, Minneapolis, MN
Emily
DiMatteo, age 15 - Shaler Area High School, Pittsburgh,
PA
Devin Duncan, age 15 - Dayspring Academy, Whitefish,
MT
Jerrod Early, age 16 - Shaler Area High School,
Pittsburgh, PA
Jared Elizares, age 14 - Amador Valley
High School, Pleasanton, CA
Caleb Elliott, age 15 -
Redmond High School, Redmond, OR
Jenn Elliott, age 16 -
Marple Newtown Senior High School, Newtown Square, PA
Ryan
Erlenbach, age 15 - South Medford High School, Medford,
OR
Allan Espinosa, age 14 - Philippine Science High
School, Quezon City, Philippines
Andrea F, age 14 -
Sawgrass Springs, Coral Springs, FL
Diego Falavigna, age
17 - Liceo Scientifico Gandini, Lodi, Italy
Lindsay
Falkenstern, age 14 - West Morris Central High School,
Chester, NJ
Betsy Feldman, age 15 - Marple Newtown
Senior High School, Newtown Square, PA
Robert Fischer,
age 25+ - Arizona State University, Tempe, AZ
Claudia
Fragozo, age 18 - Cigarroa High School, Laredo, TX
Ariel
Franks, age 11 - Forsyth School, St. Louis, MO
Kevin
Fung, age 13 - Brookhurst Junior High School, Anaheim,
CA
Mariko
Furukawa,
age 16 - Shaler Area High School, Pittsburgh, PA
Juan
Ignacio Fuxman Bass, age 18 - Moorlands, Tortuguitas,
Argentina
Lisa Gardner, age 8 - Boyette Springs
Elementary School, Riverview, FL
Kristen Geubtner, age
15 - Shaler Area High School, Pittsburgh, PA
Josh
Godick, age 14 - Deland High School, Deland, FL
Jen
Gordon, age 15 - Marple Newtown Senior High School, Newtown
Square, PA
Emily Guh, age 11 - Wissahickon Middle
School, Ambler, PA
Bob Haas, age 16 - Shaler Area High
School, Pittsburgh, PA
Cristina Haro, age 14 - Culver
High School, Culver, OR
Christine Hu, age 11 - Kennedy
Junior High, Cupertino, CA
Carey Hull, age 16 - Shaler
Area High School, Pittsburgh, PA
Lauren Hunter, age 15 -
Marple Newtown Senior High School, Newtown Square,
PA
Sravisht Iyer, age 13 - Kennedy Junior High School,
Naperville, IL
Briana Johnson, age 17 - Highwood High
School, Highwood, MT
Ann Jones, age 16 - Redmond High
School, Redmond, OR
Junior High Project Challenge Class,
average age 13 - St. Angelas Church School, Cleveland,
OH
Samantha Kaplan, age 13 - West Essex Jr. High, North
Caldwell, NJ
Konrad Karczewski, age 12 - Wissahickon
Middle School, Ambler, PA
Varun Karim, age 15 - Culver
City High School, Culver City, CA
Jeff Keenan, age 16 -
Shaler Area High School, Pittsburgh, PA
Dave Kennedy,
age 25+ - none, Lethbridge, Canada
Deborah Kim, age 18 -
Villa Devoto School, Buenos Aires, Argentina
Joyce Krow,
age 25+ - Middleburg High School, Middleburg, PA
Chrissy
Laverdier, age 19 - Arizona State University, Tempe,
AZ
Stephen Legg, age 11 - Robert Miles Junior School,
Bingham/Nottingham, England
Steve Li, age 14 - Sentinel
Secondary School, West Vanocuver, Canada
Karlene Lihota,
age 16 - Marple Newtown Senior High School, Newtown Square,
PA
Jordan Lillian, age 13 - Sawgrass Springs Middle,
Coral Springs, FL
Sloan Lindsey, age 15 - West Florence
High School, Florence, SC
Russ Lutz, age 16 - Shaler
Area High School, Pittsburgh, PA
Thomas Maffai, age 14 -
Redmond High School, Redmond, OR
Mathamagicians of
Mathachusetts, average age 13 - Dedham Country Day, Dedham,
MA
Seth Maxwell, age 15 - Deland High School, Deland,
FL
Gary McBurney, age 16 - Shaler Area High School,
Pittsburgh, PA
Ryan McHugh, age 15 - Shaler Area High
School, Pittsburgh, PA
Martin McNicoll, age 19 -
Strathclyde University, Glasgow, Scotland, UK
Debbie
Meiners, age 15 - Fontainebleau High School, Mandeville,
LA
Jeff Meyer, age 16 - Redmond High School, Redmond,
OR
Roberto Milani, age 19 - Liceo Scientifico Gandini,
Lodi, Italy
Jeffrey Mo, age 9 - University Elementary
School, Calgary, Alberta, Canada
Gabe Monteleone, age 16
- Shaler Area High School, Pittsburgh, PA
Alexandra
Morosan, age 13 - Bayview Middle School, North York, Ontario,
Canada
Michael Moyer, age 17 - The Way Home School,
Carlisle, PA
Preeti Nalavade, age 12 - Wissahickon
Middle School, Ambler, PA
Tien Nguyen, age 14 - Johnson
Middle School, Westminster, CA
Scott Niekum, age 16 -
Shaler Area High School, Pittsburgh, PA
Jennifer North,
age 13 - West Essex Jr. High, North Caldwell, NJ
Joseph
O'Connor, age 25+ - none, Pleasant Grove, UT
Tyler
Pabst, age 14 - Long Trail School, Dorset, VT
Katerina
Papacosma, age 11 - Old Trail School, Bath, OH
Giridhar
Parameswaran, age 16 - Carine Senior High School, Perth,
Australia
Jason Pass, age 17 - Unity High School, Balsam
Lake, WI
Jeremy Pass, age 17 - Unity High School, Balsam
Lake, WI
Andriy Pazuniak, age 14 - Wilmington Friends,
Wilmington, DE
Nick Petry, age 15 - Long Trail School,
Dorset, VT
Michael Pizer, age 10 - University School of
Milwaukee, Milwaukee, WI
Bill Puhse, age 25+ - Granite
City High School, Granite City, IL
Yogesh Raghunathan,
age 13 - Shelton Intermediate, Shelton, CT
Jimmy Reade,
age 14 - West Morris Central High School, Chester,
NJ
Danielle Resick, age 15 - Watertown Senior High
School, Watertown, SD
Olga Rodina, age 14 - Winman
Junior High School, Warwick, RI
Daniel Roehrig, age 15 -
Long Trail School, Dorset, VT
Matthew Roitstein, age 15
- William S. Hart High School, Newhall, CA
Adam
Rosenthal, age 15 - Brophy College Preparatory, Phoenix,
AZ
Mohamed Salameh, age 14 - Nicolaus Copernicus
P.S.#25, Jersey City, NJ
Shari Scalone, age 13 -
Sawgrass Springs Middle, Coral Springs, FL
Kevin
Schieber, age 15 - Jefferson C-123, Conception Jct.,
MO
Emily Schindler, age 15 - Shaler Area High School,
Pittsburgh, PA
Dominic Schirrippa, age 25+ - none, East
Hampton, NY
Kim Schneider, age 16 - Shaler Area High
School, Pittsburgh, PA
Michael Selick, age 14 -
Cresskill Jr/Sr High School, Cresskill, NJ
Sagar Shah,
age 14 - Nicolaus Copernicus P.S.#25, Jersey City, NJ
Nicole
Sharpe, age 10 - Wilderness School, Walkerville, SA 5081,
Australia
Danielle Sheptow, age 15 - Redmond High
School, Redmond, OR
Todd Smith, age 25+ - Long Trail
School, Dorset, VT
Ainie Soetanto, age 14 - Marple
Newtown Senior High School, Newtown Square, PA
Ana
Sovic, age 16 - XV. Gimnazija, Zagreb, Croatia
Jamie
Stark, age 16 - Shaler Area High School, Pittsburgh,
PA
Nathan Strauss, age 11 - Forsyth School, St. Louis,
MO
David Sumpter, age 15 - South Medford High School,
Medford, OR
Josh Sztorc, age 14 - Sawgrass Springs
Middle, Coral Springs, FL
Bryant Taylor, age 16 - Edgren
High School, Misawa, Japan
Melissa Tengowski, age 15 -
Shaler Area High School, Pittsburgh, PA
Danelle
Thompson, age 16 - Redmond High School, Redmond, OR
Bob
Tobin, age 16 - Shaler Area High School, Pittsburgh,
PA
Peter Tran, age 10 - Summer Hill Public School,
Sydney, Australia
Alex Tuchel-Veyhl, age 11 - Forsyth
School, St. Louis, MO
David Turner, age 15 - Moreau
Catholic High school, Hayward, CA
Craig Vetter, age 15 -
Marple Newtown Senior High School, Newtown Square, PA
Asher
Walkover, age 16 - Rambam Mesivta, Lawrence, NY
Jason
Walther, age 16 - Unity High School, Balsam Lake,
WI
Diane Wang, age 15 - Champlain Valley Union High
School, Hinesburg, VT
Theresa Wang, age 15 - Marple
Newtown Senior High School, Newtown Square, PA
Peter
Westergaard, age 25+ - University of Waterloo, Ontario,
Canada, Waterloo, Canada
James Western, age 15 -
Wallkill Valley High School, Hamburg, NJ
Michael White,
age 16 - Shaler Area High School, Pittsburgh, PA
Nathan
White, age 15 - South Medford High School, Medford,
OR
Brian Wineberg, age 16 - Shaler Area High School,
Pittsburgh, PA
Doug Wolfe, age 16 - Shaler Area High
School, Pittsburgh, PA
Lindsay Wolfson, age 16 - Shaler
Area High School, Pittsburgh, PA
Amy Yarnell, age 16 -
Marple Newtown Senior High School, Newtown Square, PA
Chan
Zheng Ming, age 13 - The Chinese High School, Singapore,
Singapore
Xiao Zhu, age 15 - Queen Elizabeth High
School, Halifax, Canada
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