Projective Geometry Hidden Inside: Can You Spot It?
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1 Digital Dordt Faculty Work: Comprehensive List Projective Geometry Hidden Inside: Can You Spot It? Tom Clark Dordt College, tom.clark@dordt.edu Follow this and additional works at: Part of the Geometry and Topology Commons Recommended Citation Clark, Tom, "Projective Geometry Hidden Inside: Can You Spot It?" (2016). Faculty Work: Comprehensive List. Paper This Conference Presentation is brought to you for free and open access by Digital Dordt. It has been accepted for inclusion in Faculty Work: Comprehensive List by an authorized administrator of Digital Dordt. For more information, please contact ingrid.mulder@dordt.edu.
2 Projective Geometry Hidden Inside: Can You Spot It? Abstract In this talk Dr. Clark shared about a Math Teachers Circle session he recently ran centered around the children s game Spot it! This game has some very interesting mathematics behind it and naturally begs to be explored with inquiry. He described the way he led teachers to ask questions about the game, the way the teachers then explored the topic, and the mathematics behind it all. Keywords projective geometry, games, problem solving, axioms Disciplines Geometry and Topology Comments Presented at the 2016 MAA Mathfest held August 3-6, 2016, in Columbus, Ohio. Talk was based on the paper Analyzing Unique-Matching Games Using Elementary Mathematics, co-authored with Cal Jongsma. This conference presentation is available at Digital Dordt:
3 Projective Geometry Hidden Inside: Can You Spot It? Thomas J. Clark August 4, 2016 Dordt College 1
4 an introduction to spot it!
5 What is Spot It! TM? 3
6 Basic Idea Every card has exactly one match with every other card. No filler symbols. 4
7 My Spot It! TM Story 5
8 Curiosity 6
9 Curiosity Question What mathematical questions come to mind that you might ask about this game? 6
10 Curiosity Question What mathematical questions come to mind that you might ask about this game? How many cards are in the deck? Could there be more/less? 6
11 Curiosity Question What mathematical questions come to mind that you might ask about this game? How many cards are in the deck? Could there be more/less? How many symbols are on each card? Is it the same for each card? 6
12 Curiosity Question What mathematical questions come to mind that you might ask about this game? How many cards are in the deck? Could there be more/less? How many symbols are on each card? Is it the same for each card? How many total symbols are used? Could there be more/less? 6
13 Curiosity Question What mathematical questions come to mind that you might ask about this game? How many cards are in the deck? Could there be more/less? How many symbols are on each card? Is it the same for each card? How many total symbols are used? Could there be more/less? How many times does each symbol appear? Is it the same for each symbol? 6
14 Curiosity Question What mathematical questions come to mind that you might ask about this game? How many cards are in the deck? Could there be more/less? How many symbols are on each card? Is it the same for each card? How many total symbols are used? Could there be more/less? How many times does each symbol appear? Is it the same for each symbol? How many possible decks can one make? 6
15 Curiosity Question What mathematical questions come to mind that you might ask about this game? How many cards are in the deck? Could there be more/less? How many symbols are on each card? Is it the same for each card? How many total symbols are used? Could there be more/less? How many times does each symbol appear? Is it the same for each symbol? How many possible decks can one make? How hard is it to make a Spot It! TM deck of a particular size/type? 6
16 possible mathematical directions
17 Build Your Own Game Problem Solving Theme: Special Cases / Simplify Problem / Patterns 8
18 Build Your Own Game Problem Solving Theme: Special Cases / Simplify Problem / Patterns Attempt to answer the questions by solving the case with 1, 2, 3, and 4 symbols per card. 8
19 Build Your Own Game Problem Solving Theme: Special Cases / Simplify Problem / Patterns Attempt to answer the questions by solving the case with 1, 2, 3, and 4 symbols per card. Look for numerical patterns in that case and attempt to generalize and/or look for explanations of those patterns e.g., s + (s 1) 2 = s 2 s + 1 cards given s symbols per card. 8
20 Build Your Own Game Problem Solving Theme: Special Cases / Simplify Problem / Patterns Attempt to answer the questions by solving the case with 1, 2, 3, and 4 symbols per card. Look for numerical patterns in that case and attempt to generalize and/or look for explanations of those patterns e.g., s + (s 1) 2 = s 2 s + 1 cards given s symbols per card. Analyze the Spot It! TM and Spot It! Junior TM games and look for patterns. When I ran this session at my Math Teachers Circle, this is the direction I started with. I pivoted to projective geometry when we hit questions we couldn t answer and answers we couldn t explain. 8
21 9
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24 Game Rules as Mathematical Axioms Problem Solving Theme: Modeling / Using Precision Question What makes a good game? What makes Spot It! fun? 12
25 Game Rules as Mathematical Axioms Problem Solving Theme: Modeling / Using Precision Question What makes a good game? What makes Spot It! fun? Given any two cards, there is always a match between them. (Function) Each card has same number of symbols. (Fairness) Symbols are equally popular. (Fairness) 12
26 Projective Geometry Axioms Any two points are on exactly one line Any two lines have exactly one point in common. Four points must exist such that no three of them are on the same line. Question How do these axioms relate to Spot It! TM? 13
27 Spot It! TM Axioms Definition We define a point to be a Spot It! TM card and a line to be a set of cards that contain a common symbol. 14
28 Spot It! TM Axioms Definition We define a point to be a Spot It! TM card and a line to be a set of cards that contain a common symbol. Any two cards must must have exactly one symbol in common (are on a line ). For any pair of symbols, there is a card which contains those two symbols. All cards contain the same number of symbols. There is at least one card. Every card contains at least three symbols. Every symbol is on at least three cards. 14
29 Deeper Questions Question Can you make a Spot It! TM deck with 7 symbols on every card? 15
30 Deeper Questions Question Can you make a Spot It! TM deck with 7 symbols on every card? Question What about 9, 10, 11, 12, 13,... symbols per card? 15
31 Secret Code For those interested/skilled with computer programming there are some interesting directions to investigate Spot It! TM Write a program that generates a deck of Spot It! TM cards with a given number of symbols per card. Count the number of possible Spot It! TM decks that can be made with a particular bank of symbols. 16
32 wrapping up...
33 Thanks For Your Attention! To see any of my materials contact me at References 1. Clark, Thomas J. and Jongsma, Cal. Analyzing Unique-Matching Games Using Elementary Mathematics. (Forthcoming) 2. Dietz, Donna A. Spot It! Solitaire Lew. Spot It!...A Combinatoric Cornucopia. spot-it-a-combinatoric-cornucopia/ 4. Sengupta, Deepu. A Mathematical Analysis of Spot It!. resources/materials/dsenguptaspotit.pdf 18
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