primes circle: high school math enrichment at...
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PRIMES Circle: High School MathEnrichment at MIT
Isabel VogtDepartment of Mathematics, MIT
June 25th, 2015
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PRIMES
Umbrella organization: research and outreach
Four programs:
• MIT PRIMES: research on MIT campus for local students
• PRIMES-USA: research for students not in the Boston area
• PRIMES Circle: enrichment program for high schoolstudents from urban Boston area public high schools
• PRIMES STEP: new enrichment program for middle schoolstudents
Isabel Vogt PRIMES Circle at MIT
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PRIMES Circle
MATHEMATICS
• Pair 2-3 high school students with anundergraduate mentor to do directedreading for a semester
• Requirements:
• Individual groups meet once a weekfor 2 hours
• Three whole group meetings• A 20 minute talk at a
“mini-conference”• A 5-7 page paper on a topic related
to their reading
Isabel Vogt PRIMES Circle at MIT
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Students presenting at the conference
Emily, CRLS
Wilkin, BLS
Natalia,Saugus High Sheinya,
Saugus High
Isabel Vogt PRIMES Circle at MIT
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Philosophy of the program
• Much of the math the students are taught in school is rotememorization, and bears little resemblance to themathematical process experienced by mathematicians
• In high school, if you don’t know how to solve a problem, youprobably weren’t paying attention in class
• In research mathematics, if you can solve a problemimmediately, it means that it is not very interesting, and thusnot worth working on
Isabel Vogt PRIMES Circle at MIT
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Philosophy of the program
• The reading topic is meant to be a vehicle for learning aboutthe mathematical process:
• Reading formal exposition• Chipping away at a problem you don’t immediately know how
to solve• Communicating mathematics with your peers and mentors• Formally presenting in oral and written form
Isabel Vogt PRIMES Circle at MIT
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What does this actually entail?
• Example subject: combinatorial and geometric game theory
• Let’s play a game:
The game starts with N “crosses” on a page (for concreteness,say N = 3)Two players, player 1 and player 2 , alternate making movesOn a move, a player draws a line between two “free ends” thatdoes not cross any existing lines and draws a new “cross” inthe middleThe game ends when there are no more available moves
Isabel Vogt PRIMES Circle at MIT
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The game of Brussels Sprouts
For example... (with N = 2)
Isabel Vogt PRIMES Circle at MIT
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The game of Brussels Sprouts
For example... (with N = 2)
So this game was a win for player 2 !
Isabel Vogt PRIMES Circle at MIT
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The game of Brussels Sprouts
Some natural questions that come up:
1. Does the game always end?
2. If so, can we bound how long a game will last?
3. Can we determine who will a game with N starting crosses?
4. Given a finished game, can we tell who won the game?
5. ... etc...
Isabel Vogt PRIMES Circle at MIT
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The game of Brussels Sprouts
...but don’t just let me tell you! Let’s try it!
Record your answers for the following games:
N = 2 3 4
Number of moves
Winner
Isabel Vogt PRIMES Circle at MIT
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The game of Brussels Sprouts
N = 2 3 4
Number of moves 8 13 18
Winner player 2 player 1 player 2
• An odd number of moves means player 1 wins, and aneven number of moves means player 2 wins
• The number of moves (and hence the winner) seems to bedetermined by N ... in fact it looks like
number of moves = 5N − 2.
• ...but, can we prove this?
Isabel Vogt PRIMES Circle at MIT
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The Euler Characteristic
Definition
A planar graph is a collection of vertices (dots) and edgesbetween vertices such that no two edges cross.
The graph is connected if it is possible to follow a sequence ofedges from any one vertex to another.
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Vertices
Edges
Isabel Vogt PRIMES Circle at MIT
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The Euler Characteristic
Which of these graphs do you think are planar?
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Isabel Vogt PRIMES Circle at MIT
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The Euler Characteristic
Which of these graphs do you think are planar?
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Yes!
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NoBeing planar is equivalent to being able to be drawn on thesurface of a sphere (with no overlapping edges).
Isabel Vogt PRIMES Circle at MIT
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The Euler Characteristic
Besides vertices and edges, we’ll need one more concept: thenumber of enclosed faces of the graph.
• This is counted as if the graph lay on a surface... e.g. there isone more face than the “obvious” ones, the “outside of thegraph”
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• V = # Vertices = 7E = # Edges = 9F = # Faces = 4
Isabel Vogt PRIMES Circle at MIT
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The Euler Characteristic
Definition
The Euler characteristic of a graph G is the integer
χ(G) = V − E + F.
So, let’s do another game: everyone draw a connected planargraph and calculate its Euler characteristic.
Isabel Vogt PRIMES Circle at MIT
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The Euler Characteristic
What you’ve discovered is known as “Euler’s Formula”, and is afundamental first step in the area of topology!
Theorem (Euler)
If G is a connected planar graph, then χ(G) = 2.
You might guess that this really says something about the “typeof surface” the graph can be drawn on.
Isabel Vogt PRIMES Circle at MIT
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The Euler Characteristic
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χ = 8− 12 + 6 = 2.•
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χ = 16− 32 + 16 = 0 6= 2!
The surfaces that these can be drawn on are:
spheretorus
Isabel Vogt PRIMES Circle at MIT
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The Euler Characteristic
In fact, you can define the Euler characteristic of a (compact,oriented) surfaces in terms of the Euler characteristic of thegraphs that lie on its surface.
sphereχ = 2
torusχ = 0
2-hole torusχ = −2
3-hole torusχ = −4
In fact, it is a theorem that if the number of holes is g, thenχ = 2− 2g. And further this is the only topological invariant – itclassifies (compact, oriented) surfaces!
Isabel Vogt PRIMES Circle at MIT
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Back to Sprouts
So what does this have to do with Brussels Sprouts?
• At the end of the game, we’ve produced a connected, planargraph
• So if we can find formulas for V , E, F in terms of the numberof moves in the game m, then we can use
V − E + F = 2
to determine m.
Isabel Vogt PRIMES Circle at MIT
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Solution of Brussels Sprouts
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• Vertices:
• begin with N
• add 1 on every move
So at the end of the gamethere are N +m vertices.
Isabel Vogt PRIMES Circle at MIT
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Solution of Brussels Sprouts
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• Edges:
• begin with 0
• add 2 on every move
So at the end of the gamethere are 2m edges.
Isabel Vogt PRIMES Circle at MIT
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Solution of Brussels Sprouts
Faces:
• By definition, the gameends where there isexactly 1 “free end” inevery face
• And on every move thenumber of “free ends”stays constant
So at the end of the gamethere are 4N faces.
Isabel Vogt PRIMES Circle at MIT
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Solution of Brussels Sprouts
...so putting it all together:
χ = V − E + F
= (N +m)− (2m) + (4N)
= 5N −m= 2.
To as we guessed at the beginning, the number of moves is 5N − 2!
Isabel Vogt PRIMES Circle at MIT
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Why this problem?
• Can be explored by playing many games
• The fact that the result does not depend on the particulars ofthe game makes seeing patterns easier
• It relates to some deep and fundamental mathematics – theEuler characteristic and topology more generally
• After understanding Brussels Sprouts, students can move onto more difficult games where skill is involved in winning –such as Sprouts
Isabel Vogt PRIMES Circle at MIT
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2015 PRIMES Circle
Isabel Vogt PRIMES Circle at MIT
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Thank you!
If you want further information:
http://math.mit.edu/research/highschool/primes/circle/
index.php
Isabel Vogt PRIMES Circle at MIT