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Geometry Problem of the Week, October 28 - November 1
What is the maximum number of times six circles of the same size can
intersect? In other words, draw six circles of the same size on a piece of
paper - what is the maximum number of intersections your drawing can
have? Remember to explain how it works - if you send me just a number,
your answer won't be considered correct!
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I think this is a record setting week! 123 correct responses, which I
think is more _correct_ responses than the number of _total_ responses
we've gotten for any other week this year! There were also 30 incorrect
responses, for a grand total of 153. Wow!
Two different answers were considered correct. 30 intersections is the
"conventional" answer, but an answer of "infinity" was also considered
correct. As Adam Ferrell from Cheshire High School points out, it all
depends on how you define "intersection" - he provided both solutions
because he wasn't sure if putting them on top of each other really counted
as intersections. Andrew Rollins of Cheshire High School wrote, "An
intersection, in my opinion, is a point formed by two lines crossing one
another; they must cross. Therefore, I did not consider six lines on top
of one another to be a correct answer." While I'm inclined to agree, I
accepted the other answers anyway. Maybe the next time I use a similar
problem, I should ask for a definition of "intersecting" to go along with
the answers.
I can't find who said this, but someone make the point that if you put
them all on top of each other, do you still have 6 circles? After all,
it's the same set of points no matter what you call it.
Greg Wolfe and Sarah Goyak of Shaler High School make a VERY important
point, related to the problem of accurate definitions: "First, to find
the maximum number of times six circles of the same size can intersect,
you must know what a circle is. Most people, such as we first did, feel
that a circle is the circle itself AND the space inside the circle. This
is not true. The definition of the circle contains only the line that
makes up the circle and NOT the interior region inside of the line."
Superb! This reminded me of my seventh grade math teacher, Mrs.
Scruggs. She gave a bonus question on a test that we all thought we got
right. There was a picture of a triangle - three line segments. The
text said, "Color the triangle". Being intelligent seventh graders
confident that it was an easy 5 points, we all filled in the interior of
the triangle, being careful to stay within the lines. Wrong!
There are also two levels to the possible answers - the brute force
method of drawing and counting and getting 30, and then the "general"
way, where a few cases are tried and then a pattern develops, and a
formula emerges. Then you can just plug in the number of circles. Brian
Kilner and Tim Waldo from Newport High School point out the power of
generalizations - they used their formula to calculate the number of
intersections for 20 circles, which is 380! You sure don't want to do
that by the brute force method!
There is a great variety in the solutions this week - there are a lot of
ways to explain the same thing. I would encourage you to read them all.
I have highlighted three of them.
Justin Lam from Sequoia Middle School went to town this week! He looks
at several different ways to explain it, and you should definitely take a
look at it, as it's pretty impressive! Mick Lorusso from Ignacio High
School provided a very clear explanation that included the general case.
Afrasiab Mirza from CW Jefferys used a formula that one usually learns
when studying probability. In this case, the formula is used to figure
out how many ways you can choose 2 circles from a group of 6. The
solution is highlighted below so that you can learn a bit of probability,
and it certainly is unique among the solutions this week!
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The following students submitted correct solutions this week.
Highlighted solutions are included below. For a full list of solutions,
please check out
http://forum.swarthmore.edu/geopow/fullsolutions/110196.fullsolution.....
Justin Lam, Grade 8, Sequoia Middle School, Pleasant Hill, California
Megan Gauthier, Grade 7, Buffalo Junior High, Buffalo, Minnesota
Mick Lorusso, Grade 9, Ignacio High School, Ignacio, Colorado
Jason Yeung, Grade 10, Iolani School, Honolulu, Hawaii
Jen Baer and Christie Snyder, Grade 10, Shaler Area High School,
Pittsburgh, Pennsylvania
Greg Moore and Melody Miller, Grade 10, Shaler Area High School,
Pittsburgh, Pennsylvania
Gretchen and Christian and Shawn, Grade 10, Shaler Area High School,
Pittsburgh, Pennsylvania
Pranav Shetty and Lynn DeLuca, Grade 10, Shaler Area High School
Brett Bernardo and Anne Rosenstein, Grade 10, Shaler Area High School,
Pittsburgh, Pennsylvania
Jen Wood and Ryan Gallaher, Grade 10, Shaler Area High School, Pittsburgh,
Pennsylvania
Chris Woods and Cori Dernus, Grade 10, Shaler Area High School,
Pittsburgh, Pennsylvania
Don Lawther and Lenny Hufnagel, Grade 10, Shaler Area High School,
Pittsburgh, Pennsylvania
Bob Young and Scott Stromoski, Grade 10, Shaler Area High School,
Pittsburgh, Pennsylvania
Jerry Grech and Alycia Kampetis, Grade 10, Shaler Area High School,
Pittsburgh, Pennsylvania
Nick Jutte and Melissa Antoszewski, Grade 10, Shaler Area High School,
Pittsburgh, Pennsylvania
Erin May and Becky Spinella, Grade 10, Shaler Area High School,
Pittsburgh, Pennsylvania
Keith Dougall and Gretchen Schneider, Grade 10, Shaler Area High School,
Pittsburgh, Pennsylvania
Jenny Booth and Kristen Murray, Grade 10, Shaler Area High School,
Pittsburgh, Pennsylvania
Adam Yacono and Julie Altenbaugh, Grade 10, Shaler Area High School,
Pittsburgh, Pennsylvania
Alexis Sauter and Chris Vendill, Grade 10, Shaler Area High School,
Pittsburgh, Pennsylvania
Jackie Rose, Grade 10, Shaler Area High School, Pittsburgh, Pennsylvania
Andrew Miller and Natalie Navarro, Grade 10, Shaler Area High School,
Pittsburgh, Pennsylvania
Greg Wolfe and Sarah Goyak, Grade 10, Shaler Area High School, Pittsburgh,
Pennsylvania
Sarah Kost and Richard Erb and Katie Behling, Grade 10, Shaler Area High
School, Pittsburgh, Pennsylvania
Ben DeBruin, Grade 10, Oak Park and River Forest High School, Oak Park, Illinois
Kevin Roberts, Grade 10, Southern Trinity High School, Mad River, California
Brian Hansen, Grade 10, Southern Trinity High School, Mad River, California
Kyle Kiser, Grade 10, Southern Trinity High School, Mad River, California
Craig Ditman, Grade 10, Cheshire High School, Cheshire, Connecticut
Tori Eisen, Grade 10, High Technology High School, New Jersey
Christine Landry, Grade 10, High Technology High School, New Jersey
Nicole Bonaparte, Grade 10, High Technology High School, New Jersey
Samir Warty, Grade 10, High Technology High School, New Jersey
Jennifer Lorwey, Grade 10, High Technology High School, New Jersey
Shara Hoffman, Grade 10, High Technology High School, New Jersey
Marisa Rospos, Grade 10, High Technology high School, New Jersey
Pooja Gupta, Grade 10, High Technology High School, New Jersey
Rena Pike, Grade 10, High Technology High School, New Jersey
Josh Kramer, Grade 10, High Technology High School, New Jersey
Christine Blankertz, Grade 10, HIgh Technology High School, New Jersey
Shaun Altneu, Grade 10, High Technology High School, New Jersey
Lindsey Taylor Mazza, Grade 10, High Technology high School, New Jersey
Erin Desfosse, Grade 10, High Tecnology High School, New Jersey
Roger Mong, Grade 8, Zion Heights Junior High School, Richmond Hill,
Ontario, Canada
Kathy Hadden, Grade 8, School of the Holy Child, Drexel Hill, Pennsylvania
Jennifer Althouse and Frank Pendleton, Grade 9, North Cross
Afrasiab Mirza, Grade 12, C.W.Jefferys C.I. , North York, Ontario, Canada
Cyd Cipolla, Grade 9, Millbrook School, Millbrook, New York
Jenny Kaplan, Grade 6, Castilleja Middle School, Palo Alto, California
Kate McTernan, Grade 9, Roselle Park High School, Roselle Park, New Jersey
Jerry Elliott, Grade 10, Roselle Park High School, Roselle Park, New Jersey
Will Kettunen, Grade 9, Cherry Creek High School, Aurora, Colorado
Terry Haslam, Grade 11, Southern Trinity High School, Mad River, California
Giscard Pongon, Grade 12, George Wingate High School, Brooklyn, New York
Joshua Zucker, Grade tutor, Education Program for Gifted Youth, Stanford,
California
Jon Nuger, Grade 9, Longmeadow High School, Longmeadow, Massachusetts
Suzanne Davis, Grade 10, Newport High School, Bellevue, Washington
William Reynolds, Grade 8, Richfield Junior High, Richfield, Minnesota
Karvell Li , Grade 9, Newport High School, Bellevue, Washington
Tomomi Nakajima , Grade 9, Newport High School, Bellevue, Washington
Kate Ueda, Grade 9, Newport High School, Bellevue, Washington
Kristin Dougherty, Grade 10, Germantown Academy, Fort Washington, Pennsylvania
Thomas Kuo, Grade 9, Burroughs High School, Ridgecrest, California
Shameica Edwards and Talesha Clarke, Grade 11::10, George Wingate HS,
Brooklyn, New York
Tetsuya Matsuguchi , Grade 11, Centennial High School, Boise, Idaho
Katie Fredlund and Gena Kerr and Whitney Morris, Grade , Lakeside School,
Seattle, Washington
Joe Fantini and Greg Mellor, Grade 9, Germantown Academy, Fort Washington,
Pennsylvania
Kristin Dougherty, Grade 10, Germantown Academy, Fort Washington, Pennsylvania
Alexa Helsell and Andrew and Matt and Kiara, Grade , Lakeside School,
Seattle, Washington
Nicole Giuliani and Lauren Deal, Grade 10, Lakeside School, Seattle, Washington
Leif Linden, Grade , Germantown Academy, Fort Washington, Pennsylvania
Becky Dunlap and Alison Beste, Grade 9, Germantown Academy, Fort
Washington, Pennsylvania
Gavin Kovite and Josh and Keon and Dougy, Grade , Lakeside School,
Seattle, Washington
Eli Rosenblatt and Adam Childs and Nancy Mikacenic, Grade , Lakeside
School, Seattle, Washington
Tana Kaplan and Marnie, Grade , Lakeside School, Seattle, Washington
Brett Faulds , Grade , Lakeside School, Seattle, Washington
Amy Forster, Grade 7, homeschool, Cygnet, Tasmania, Australia
Addie Faustine, Grade , Southern Trinity High School, California
Danny Frasier, Grade , Southern Trinity High School, California
Michael Savenelli, Grade 10, Cheshire High School, Cheshire, Connecticut
Jake DeGennaro, Grade 9, Cheshire High School, Cheshire, Connecticut
Stephen O'Brien, Grade 10, Cheshire High School, Cheshire, Connecticut
Adam Staffaroni, Grade 10, Cheshire High School, Cheshire, Connecticut
Susie Frasier, Grade , Southern Trinity High School, Mad River, California
Brian Kilner and Tim Waldo, Grade 10, Newport High School, Bellevue, Washington
Tim Neal, Grade 10, Cheshire High School, Cheshire, Connecticut
Ben Falit, Grade 10, Cheshire High School, Cheshire, Connecticut
Andrew Rollins, Grade 10, Cheshire High School, Cheshire, Connecticut
Jeff Gagliardi, Grade 10, Cheshire High School, Cheshire, Connecticut
Jeff Dinatali, Grade 10, Cheshire High School, Cheshire, Connecticut
Shane Chang, Grade 10, Cheshire High School, Cheshire, Connecticut
Paige Golden, Grade 10, Cheshire High School, Cheshire, Connecticut
Leah Brennan, Grade 10, Cheshire High School, Cheshire, Connecticut
Aly Coutts, Grade 10, Cheshire High School, Cheshire, Connecticut
Chris Kennelly, Grade 10, Cheshire High School, Cheshire, Connecticut
Gavin O'Reilly, Grade 10, Cheshire High School, Cheshire, Connecticut
??, Grade ??, Cheshire High School, Cheshire, Connecticut
Adam Ferrell, Grade 10, Cheshire High School, Cheshire, Connecticut
??, Grade ??, Cheshire High School, Cheshire, Connecticut
Jason Palleschi, Grade 11, Cheshire High School, Cheshire, Connecticut
Devin Silberfein, Grade 11, Cheshire High School, Cheshire, Connecticut
Amy Mousaw and Kelly Richburg, Grade 10, Cheshire High School, Cheshire,
Connecticut
Nate Angelo, Grade 11, Cheshire High School, Cheshire, Connecticut
Chris Goodwin, Grade 9, Cheshire High School, Cheshire, Connecticut
Erik Day and Tiffany Forbes, Grade 10, Cheshire High School, Cheshire,
Connecticut
Josh Grochow, Grade 8, Georgetown Day School, Washington, DC
Kate Maxwell, Grade 8, Georgetown Day School, Washington, DC
Oluseyi Ojeifo, Grade 8, Georgetown Day School, Washington, DC
Rachel Winnick, Grade 8, Georgetown Day School, Washington, DC
Sara Constantino, Grade 8, Georgetown Day School, Washington, DC
Chang-xin Fang, Grade 8, Georgetown Day School, Washington, DC
Rob Blair, Grade 8, Georgetown Day School, Washington, DC
Daniel Ornstein, Grade 8, Georgetown Day School, Washington, DC
David Fischer, Grade 8, Georgetown Day School, Washington, DC
Jon Biderman, Grade 8, Georgetown Day School, Washington, DC
Jared Joiner, Grade 8, Georgetown Day School, Washington, DC
Devin Maroney, Grade 8, Georgetown Day School, Washington, DC
Eric Faden, Grade 8, Georgetown Day School, Washington, DC
Jon Cooper, Grade 8, Georgetown Day School, Washington, DC
Lauren Levien, Grade 8, Georgetown Day School, Washington, DC
Scott Lipnick, Grade 8, Georgetown Day School, Washington, DC
Jess Gilburne, Grade 8, Georgetown Day School, Washington, DC
Jessica Carnevale and Chris Wilby, Grade 9, Cheshire High School,
Cheshire, Connecticut
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From: quan....@ucop.edu
Justin Lam
Grade: 8
School: Sequoia Middle School, Pleasant Hill, California
This write-up would be so much understandable if I could just draw the figures.
Anyway, I hope you understand what I am trying to say.
The following table shows the MAXIMUM number of intersections we can have
with n number of circles.
# of circles | Intercepts
-------------|-----------
1 | 0 = 1x0
2 | 2 = 2x1
3 | 6 = 3x2
4 | 12 = 4x3
5 | 20 = 5x4
6 | 30 = 6x5
n | n(n-1)
For n circles (of equal radius), the first circle can have a maximum of 2(n-1)
intersections with the other (n-1) circles. The second circle can have a
maximum of 2(n-2) intersections with the other (n-2) circles (intersections
with the first circle are duplicates). And so on, the (n-1)st can have a
maximum of 2(n-(n-1))=2 intersections (those with the last circle). Therefore,
at the most, the total number of intersections these n circles can have with
each others is
2(n-1)+2(n-2)+2(n-3)+...+2(n-(n-1)) = 2(n(n-1)-(1+2+3+...+(n-1))
= 2(n^2-n-(n-1)n/2)
= 2n^2-2n-n^2+n
= n^2-n
= n(n-1)
Another to derive n(n-1) is to say that the maximum number of intersections
among the n circles is the number of permutations
of 2 numbers from a pool of n numbers (n circles).
For example, (1,2) represents the intersection of Circle 1 and Circle 2
and (2,1) represents the other intersection of Circle 1 and Circle 2.
Since there are n possible numbers for position 1 and (n-1) possible
numbers for position 2 ( (1,1), (2,2), (3,3),...(n,n) are not possible),
the total number of possible pairs is n(n-1).
Now we can construct a diagram so that these n circles of equal radius r
can actually intersect at n(n-1) points.
Let C(1) be the center of the first circle with radius r.
Then place all the other (n-1) centers C(2), C(3),...,C(n)
inside Circle 1 so that all n centers are less than r distance
from each other. This can obviously be done by placing them inside the circle
with radius r/2 and goes through C(1) - the center of Circle 1.
Call this small circle X.
Now we just have to pick points for the centers so that the n(n-1)intersections
are all DIFFERENT.
Let (i,j) be the intercept of Circle i and Circle j.
Obviously, (1,2) and (2,1) are different since C(2) is inside
the small circle X. Now delete all points inside X that are
r distance from (1,2) or (2,1) and pick C(3) - the center of Circle 3
inside X but not one of the deleted points. Then Circle 3
cannot go through (1,2) or (2,1). So the four intercepts Circle 3 with
Circles 1 and 2 are different from (1,2) and (2,1).
Do the samething for Circle 4. That is, delete all points inside Circle X that
are r distance from
(1,2), (2,1), (1,3), (3,1), (2,3), (3,2) and pick the center C(4)
from the remaining points inside X. Similarly, for Circle 6,
delete all points inside X that are r distance from (i,j)
where i=1 to 6 and j=1 to 6) and pick C(6) from the remaining points
inside Circle X. This argument can easily be extended to n circles.
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From: mi...@earthlink.net
Mick Lorusso
Grade: 9
School: Ignacio High School, Ignacio, Colorado
The maximum amount of intersections between 6 circles is 30 times.
To determine this I set up equations which describe the optimal
amount of intersections in groups of 1 circle, 2 circles, 3 circles,
and 4 circles. First I diagramed each of these cases so that each
grouping of circles had the highest number of intersections possible.
Then I made a chart with my data like so:
Number of Circles (n): 1 2 3 4
# of intersections (i): 0 2 6 12
From this information I determined that the number of intersections
is equal to the number of circles multiplied by the number of circles
minus 1.
Thus, the equation was: i= n(n-1). Where (i) is the number of
intersections and (n) is the number of circles.
When 6 circles is plugged into the equation: i= 6(6-1)=30.
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From: rma...@ibm.net
Afrasiab Mirza
Grade: 12
School: C.W.Jefferys C.I. , North York, Ontario, Canada
We know that two circles will intersect each other in at most two points.
Therefore we simply have to find the number of circles that interect each other
and multiply by two.
We have 6 circles. Thus all the possible ways of chosing 2 circles from 6 is
equal to (6 choose 2) = 6!/(2!4!) = 15.
Therefore the maximum possible intersections of two circles would be 15*2 which
equals 30.
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