2 ND CLASS WARM-UP 9/9/14
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1 2 ND CLASS WARM-UP 9/9/14 Part 1: (Mild) How many triangles are in this image? Part 2: (Spicy) What if you add more rows of triangles? How many triangles would a 100-row picture have?
2 FIRST CLASS WARM- UP 9/2/14 IS IT POSSIBLE TO COVER THIS 5X5 GRID USING ONLY ONE T YPE OF PENTOMINO? WHAT ABOUT A COMBINATION OF B, X, AND Y PENTOMINOES? b X Y
3 COMBINATORICS I An unsolved tiling puzzle
4 AGENDA Introductions Challenges (warmup) Tiling with b, x, and y pentominoes Tiling with dominoes on a chess board Tiling with dominoes on a plank Counting border patterns An unsolved tiling puzzle Problem Solving Strategies Generalizing/Digging into Challenges
5 HI! I M ZANDRA VINEGAR.
6 HI! I M ZANDRA VINEGAR. Graduate of MIT Major: Mathematics (especially TCS) Cert: Student Another Course to College HS & CRLS HS 1 Year at MoMath (the Museum of Mathematics) Manhattan NY (yes! You should visit!) K-12 Education: teaching 45 minute sessions to school groups, curriculum development, social media This past SPMPS (the Summer Program in Mathematical Problem Solving) Underserved MS Students from NYC public schools Cryptography and Hamming Codes & Sharing Math with the World Currently Math Circles (Berkeley + Stanford) AoPS (Online School) Proof School (Volunteer) Hobbies Making Art Building Furniture Cooking Hiking Reading (audio books)
7 IS IT POSSIBLE TO COVER THIS 5X5 GRID USING ONLY ONE T YPE OF PENTOMINO? WHAT ABOUT A COMBINATION OF B, X, AND Y PENTOMINOES? b X Y
8 HOW MANY DIFFERENT SOLUTIONS ARE THERE? Only x: Only y: Only b: b & x: x & y: y & b: b, x, & y:
9 SOLUTION
10 SOLUTION
11 SOLUTION:
12 SOLUTION:
13 SOLUTION:
14 SOLUTION:
15 SOLUTION:
16 HOW MANY DIFFERENT SOLUTIONS ARE THERE? Only x: 0 Only y: 0 Only b: 0 b & x: 1 x & y: 1 y & b: 8 (2 x 4 rotations) b, x, & y: 0
17 GREAT JOB! I think you deserve a reward. Here is a fuzzy puppy
18 SUMMARY Problem Solving Strategies YOUR IDEAS HERE Solutions & Proofs Digging Deeper: Ways to Generalize the Problem
19 DIGGING DEEPER Game Design (Penguins on Ice) 3D?!
20 Tetrominoes Hexomiones PENTOMIONES also, triangular pieces, Called polyaminoes And polyhexes
21 HABERDASHER S PUZZLE
22 TANGRAMS
23 IS IT POSSIBLE TO COVER AN 8X8 BOARD WITH DOMINOES IF TWO 1X1 SQUARES HAVE BEEN CUT OUT OF OPPOSITE CORNERS OF THE BOARD Each domino covers two adjacent squares: x x
24 SOLUTION No, it s not possible. Each domino covers two adjacent squares: x x
25 DIGGING DEEPER When is it possible? Each domino covers two adjacent squares: x x
26 COMBINATORICS In how many ways is it possible?
27 This is a red panda! BREAK TIME!
28 HOW MANY WAYS ARE THERE TO TILE A 2X8 RECTANGLE WITH 2X1 DOMINOES? 2x8
29 HOW MANY WAYS ARE THERE TO TILE A 2X8 RECTANGLE WITH 2X1 DOMINOES? 2x8 n=2 n=3 n=4
30 SOLUTION FIBONOCCI The Fibonacci Sequence is the series of numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34,... The next number is found by adding up the two numbers before it.
31 WRAP UP Problem Solving Strategies Solutions & Proofs Ways to Generalize the Problem The number of ways to cover an nxm rectangle with dominoes was calculated independently by Temperley & Fisher (1961) and Kasteleyn (1961) Squares are a special case. Aztec diamond Also, Congratulations! You ve all earned this Pink Fairy Armidillo!
32 FIBONACCI Is there a way to jump straight to the n th Fibonacci number? Yes: bonacci-sequence.html
33 COUNTING BORDER PATTERNS How many ways are there to tile this 1x5 rectangle with any of the 5 types of colored tiles below? This puzzle is from the blog Baking and Math The related unsolved problem is from a presentation by Pamela Harris at the annual Midwest Women in Mathematics Symposium om/2014/04/25/openproblem-incombinatorics-tiling-afloor-no-background/
34 1X3 TILING
35 PASCAL S TRIANGLE
36 BINOMIAL COEFFICIENTS
37 BINOMIAL COEFFICIENTS Row n Position k
38 SOLUTION
39 FIBONACCI IN PASCAL!
40 LAST BREAK FOR THE DAY: Great job everybody, that last one was pretty intense!
41 THIS ONE S UNSOLVED Good Luck!
42 NEXT TOPIC More Combinatorics! with Graphs, Stars, and Polytopes
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