Codes & the Hat Game

Troy Lynn BullockJohn H. Reagan High School, Houston ISD

Shalini KapoorMcArthur High School, Aldine ISD

Faculty Mentor: Dr. Tie LiuGraduate Assistant: Neeharika Marukala

Outline

An introduction to communication systems Error correction codes The hat game Lesson plan

Introduction

Communications touches the lives of everyone in many ways.

Lets look at some applications of communications in this information age!

An Information Age

The Internet

An Information Age

Deep-space communication

An Information Age

Satellite broadcasting

An Information Age

Cell phone and modem

An Information Age

Data storage

Basic Communication System

Information Source

Transmitter Receiver

Destination

CommunicationChannel

Bits Bits

Waveform WaveformDistortion

Communication Channel

Introduce distortion to the transmit signal As a result, some bits are flipped at the

receiver (e.g., 0→1 or 1→0) Which bits will be flipped are

random/unpredictable Random bit flipping conveys false

information to the destination and is bad for communication

Solution: Error Correction Codes

Repetition Codes

Consider using bit “0” to represent 0 and “1” to represent 1: If the bit is flipped, then we have no idea which bit was sent

Now consider using three bits “000” to represent 0 and “111” to represent 1” If only one of the three bits is flipped, we can still make out

which bit was sent by looking at the majority of the bits More errors can be corrected by making more

repetitions Research question: Can we find codes that are

more efficient than repetition codes?

Coding Efficiency

Can we do better?

Coding Theory A branch of modern mathematics With deep connections to:

Theory of finite field Algebraic geometry Number theory Combinatorics Algorithm Complexity theory Information Theory

With applications from deep-space communications to consumer electronics

A perfect example on how good mathematics can significantly impact our daily life

Achieving Immortalities

Richard HammingIrvine Reed &

Gustave SolomonElwyn Berlekamp

Claude BerrouRobert Gallager

The Hat Game The Setup: One team of three contestants are in a room.

A red or blue hat is randomly put on each contestant;

each contestant can see the hats of everyone else

but not his/her own.

The Game: Each contestant must (simultaneously)

1. Guess the color of his/her hat,

2. Or pass.

To Win: At least one contestant guessed correctly,

and no one guessed incorrectly.

The team can confer on a strategy beforehand.

What Strategy Can Be Used?

A “Naïve” Strategy Pick a team captain.

The captain guesses red/blue randomly.everyone else passes.

This strategy wins 50% of the time.

CAN WE DO BETTER??

A Better Strategy

Each contestant does: If the other two hats are different

colors, pass. If the other two hats are the same

color, guess the opposite color.

Analysis

This strategy wins of the time!

HATS GUESSES WIN?

000 111 no

100 1xx yes

010 x1x yes

001 xx1 yes

110 xx0 yes

101 x0x yes

011 0xx yes

111 000 no

%758

6

Lessons Learned

It’s OK to make a mistake. But when we make mistakes, it’s better to make mistakes together as a team.

When lacking evidence, it’s good to “keep quiet” for the team.

Recording Sheet for Hat Game

Round # Actual Color Guessed Actual Color Guessed Actual Color GuessedOf the Hat Color Of the Hat Color Of the Hat Color

Points earnedPlayer 1 Player 2 Player 3

Geometric Interpretations

For n=3 players

000

001

010100

111

110

101

011

“Bad” sequence

“Good” sequence

Geometric Interpretations

For n=2k-1 players, use Hamming Codes as “bad” sequences Hamming

Ball

“Perfect” Codes

For n=2k-1, all possible 2n binary sequences can be partitioned into Hamming balls of radius 1

Since the Hamming balls are non-overlapping, Hamming codes can correct any single bit flip at the minimum redundancy

For k=2, Hamming codes are the same as repetition codes

For k>2, Hamming codes are much more efficient than repetition codes

Coding Efficiency

Hamming Codes

Repetition Codes

What about n≠2k-1?

Still need to “cover” all binary sequences using Hamming balls of radius 1

The Hamming balls may have to overlap

What about n≠2k-1?

What are the optimum choices for the “bad” sequences?

Answers are known only for n=3~8 despite the effort of many famous mathematicians

A perfect challenge for kids to try a world-class open problem with strong engineering implications!

What Can the Kids Learn?

Permutations/CombinationsProbabilityPercentsCooperative LearningTeam WorkDecision Making

Lesson Plan

Day Topic Instructional Activities

TAKS TEKS Resources

1.

Probability, Combination/ Permutation 3-D figures

Students will be given pre-test on first day.

IPC Obj. 5 Exit level Objective

9,10

(2A.4B), (2A.5B) (P.1/3) (A-D)

Algebra-II book, Previous years’ TAKS test questions.

2.

Trial Play

Students will be motivated to play the game using their own skills. They will be

divided into teams and teacher will just give them the rules of the

game.

IPC Obj. 5 Exit level objective

9,10

(2A.4B), (2A.5B) (P.1/3) (A-D), IPC (5)

IPC (5B)

Hats, White

Boards, Recording

sheets, Post-aids, Markers,

Prizes for winning team.

Lesson Plan cont…

3.

1. Classroom discussion. 2. Lecture

Students will discuss previous day activity and their opinion

about the best strategies to win the game. Teacher will explain BCS and how the logic behind “hat activity” works for coding and decoding in communication

system.

IPC obj 5 Exit level objective

9,10

(2A.4B), (2A.5B) (P.1/3) (A-D), IPC (5)

IPC (5B)

Students’ recording

sheets from previous day.

4.

Actual Play

The game on Day 2 will be repeated and teacher will analyze and discuss

their result.

IPC Obj. 5 Exit level objective

9,10

(2A.4B), (2A.5B) (P.1/3) (A-D), IPC (5)

IPC (5B)

Hats, White

Boards, Recording sheets, Post-aids,

Markers, Prizes for winning

team.

5.

Post test

Students will take the

same test as before and teacher will go over the

test with students.

IPC Obj. 5 Exit level objective

9,10

(2A.4B), (2A.5B) (P.1/3) (A-D), IPC (5)

IPC (5B)

Algebra-II book,

Previous years’ TAKS test questions.

Sample Questions1 . A bag contains 24 green marbles, 22 blue marbles, 14 yellow marbles, and 12 red marbles. Suppose you pick one marble at random. Write if the following statements are True/False.

a. Probability of selecting a yellow marble is 7/36

b. Probability that the marble is not blue is 25/36

c. Probability that the marble is either green or red is ¾

2. A math teacher asks students to recognize the pattern made out of the letters A, B,C, and D. If some of them are as ABCD, ADBC, CDBA, write all the combinations made out of those 4 letters.

3. Matthew ate 5/8 of his dinner. Billy only ate 1/4 of his dinner. What percent more did Matthew eat than Billy?

a. 25% b. 62.5% c. 12.5% d. 37.5%

Acknowledgements

I would like to acknowledge E3 for giving me the opportunity to experience new and different things. Also, Dr. Liu and Neeharika Marukala for enhancing my knowledge in Engineering so that I will be able to bring future students to Texas A&M University that will major in “ENGINEERING” perhaps Electrical Engineering.

Also, I would like to thank National Science Foundation (NSF), Nuclear Power Institute (NPI), Texas Workforce Commission (TWC), and Chevron. The support from these groups have made E3 Program what it is today.