The Liar Game. Mark Wildon
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1 The Liar Game Mark Wildon
2 Guessing Games Ask a friend to thinks of a number between 0 and 15. How many NO/YES questions do you need to ask to find out the secret number?
3 Guessing Games Ask a friend to thinks of a number between 0 and 15. How many NO/YES questions do you need to ask to find out the secret number?
4 Guessing Games Ask a friend to thinks of a number between 0 and 15. How many NO/YES questions do you need to ask to find out the secret number? Q1 YES Is the number 8, 9, 10, 11, 12, 13, 14 or 15?
5 Guessing Games Ask a friend to thinks of a number between 0 and 15. How many NO/YES questions do you need to ask to find out the secret number? Q1 YES Is the number 8, 9, 10, 11, 12, 13, 14 or 15? Q2 Is the number 12, 13, 14 or 15? YES
6 Guessing Games Ask a friend to thinks of a number between 0 and 15. How many NO/YES questions do you need to ask to find out the secret number? Q1 YES Is the number 8, 9, 10, 11, 12, 13, 14 or 15? Q2 Is the number 12, 13, 14 or 15? YES Q3 Is the number 14 or 15? NO
7 Guessing Games Ask a friend to thinks of a number between 0 and 15. How many NO/YES questions do you need to ask to find out the secret number? Q1 YES Is the number 8, 9, 10, 11, 12, 13, 14 or 15? Q2 Is the number 12, 13, 14 or 15? YES Q3 Is the number 14 or 15? NO Q4 Is the number 13? NO
8 Proofs Mathematicians like to give rigorous arguments to justify the claims they make. It is believable that no questioning strategy can guarantee to find the secret number using three or fewer questions. But can we prove this beyond any doubt?
9 Proofs Mathematicians like to give rigorous arguments to justify the claims they make. It is believable that no questioning strategy can guarantee to find the secret number using three or fewer questions. But can we prove this beyond any doubt? At the start of the game, there are 16 possible numbers.
10 Proofs Mathematicians like to give rigorous arguments to justify the claims they make. It is believable that no questioning strategy can guarantee to find the secret number using three or fewer questions. But can we prove this beyond any doubt? At the start of the game, there are 16 possible numbers. Suppose that if the first question is answered YES then there are r possible numbers.
11 Proofs Mathematicians like to give rigorous arguments to justify the claims they make. It is believable that no questioning strategy can guarantee to find the secret number using three or fewer questions. But can we prove this beyond any doubt? At the start of the game, there are 16 possible numbers. Suppose that if the first question is answered YES then there are r possible numbers. Then there are 16 r possible numbers if the first question is answered NO.
12 Proofs Mathematicians like to give rigorous arguments to justify the claims they make. It is believable that no questioning strategy can guarantee to find the secret number using three or fewer questions. But can we prove this beyond any doubt? At the start of the game, there are 16 possible numbers. Suppose that if the first question is answered YES then there are r possible numbers. Then there are 16 r possible numbers if the first question is answered NO. Either r or 16 r is at least 8.
13 Proofs Mathematicians like to give rigorous arguments to justify the claims they make. It is believable that no questioning strategy can guarantee to find the secret number using three or fewer questions. But can we prove this beyond any doubt? At the start of the game, there are 16 possible numbers. Suppose that if the first question is answered YES then there are r possible numbers. Then there are 16 r possible numbers if the first question is answered NO. Either r or 16 r is at least 8. Hence, in the worst case, there are at least 8 possible numbers after Question 1.
14 Proofs Mathematicians like to give rigorous arguments to justify the claims they make. It is believable that no questioning strategy can guarantee to find the secret number using three or fewer questions. But can we prove this beyond any doubt? At the start of the game, there are 16 possible numbers. Suppose that if the first question is answered YES then there are r possible numbers. Then there are 16 r possible numbers if the first question is answered NO. Either r or 16 r is at least 8. Hence, in the worst case, there are at least 8 possible numbers after Question 1. In the worst case, there are at least 4 possible numbers after Question 2.
15 Proofs Mathematicians like to give rigorous arguments to justify the claims they make. It is believable that no questioning strategy can guarantee to find the secret number using three or fewer questions. But can we prove this beyond any doubt? At the start of the game, there are 16 possible numbers. Suppose that if the first question is answered YES then there are r possible numbers. Then there are 16 r possible numbers if the first question is answered NO. Either r or 16 r is at least 8. Hence, in the worst case, there are at least 8 possible numbers after Question 1. In the worst case, there are at least 4 possible numbers after Question 2. In the worst case, there are at least 2 possible numbers after Question 3.
16 Binary and Computers In a modern computer, everything is stored as a lists of the bits (binary digits) 0 and 1.
17 Binary and Computers In a modern computer, everything is stored as a lists of the bits (binary digits) 0 and 1. For example, the number 12 could be stored as 1100, corresponding to the sequence of answers Yes, Yes, No, No. YES NO 0 YES 1 YES NO NO
18 Binary and Computers In a modern computer, everything is stored as a lists of the bits (binary digits) 0 and 1. For example, the number 12 could be stored as 1100, corresponding to the sequence of answers Yes, Yes, No, No. Books, music, videos, computer programs, bitcoins..., are all stored as bits.
19 Binary and Computers In a modern computer, everything is stored as a lists of the bits (binary digits) 0 and 1. For example, the number 12 could be stored as 1100, corresponding to the sequence of answers Yes, Yes, No, No. Books, music, videos, computer programs, bitcoins..., are all stored as bits William Shakespeare (approx 1600)
20 Binary and Computers In a modern computer, everything is stored as a lists of the bits (binary digits) 0 and 1. For example, the number 12 could be stored as 1100, corresponding to the sequence of answers Yes, Yes, No, No. Books, music, videos, computer programs, bitcoins..., are all stored as bits William Shakespeare (approx 1600) To be, or not to be: that is the question: Whether tis nobler in the mind to suffer The slings and arrows of outrageous fortune,
21 Binary and Computers In a modern computer, everything is stored as a lists of the bits (binary digits) 0 and 1. For example, the number 12 could be stored as 1100, corresponding to the sequence of answers Yes, Yes, No, No. Books, music, videos, computer programs, bitcoins..., are all stored as bits William Shakespeare (approx 1600) To be, or not to be: that is the question: Whether tis nobler in the mind to suffer The slings and arrows of outrageous fortune,
22 Binary and Computers In a modern computer, everything is stored as a lists of the bits (binary digits) 0 and 1. For example, the number 12 could be stored as 1100, corresponding to the sequence of answers Yes, Yes, No, No. Books, music, videos, computer programs, bitcoins..., are all stored as bits Anonymous Microsoft Programmer (2010?) Part of the machine code for Microsoft Word 2011.
23 Binary and Computers In a modern computer, everything is stored as a lists of the bits (binary digits) 0 and 1. For example, the number 12 could be stored as 1100, corresponding to the sequence of answers Yes, Yes, No, No. Books, music, videos, computer programs, bitcoins..., are all stored as bits.
24 Why Coding Theory? A bit gives a single piece of information: NO or YES ; on or off ; 0 or 1. A number between 0 and 15: A small QR-code: 4 bits Text on this slide Full text of Hamlet Pictures of Royal Holloway Compact disc of Beethoven 9th Bluray disc of 3 hour film Large Hadron Collider, per second
25 Why Coding Theory? A bit gives a single piece of information: NO or YES ; on or off ; 0 or 1. A number between 0 and 15: 4 bits A small QR-code: 441 bits Text on this slide Full text of Hamlet Pictures of Royal Holloway Compact disc of Beethoven 9th Bluray disc of 3 hour film Large Hadron Collider, per second
26 Why Coding Theory? A bit gives a single piece of information: NO or YES ; on or off ; 0 or 1. A number between 0 and 15: 4 bits A small QR-code: 441 bits Text on this slide Full text of Hamlet Pictures of Royal Holloway Compact disc of Beethoven 9th Bluray disc of 3 hour film Large Hadron Collider, per second bits
27 Why Coding Theory? A bit gives a single piece of information: NO or YES ; on or off ; 0 or 1. A number between 0 and 15: 4 bits A small QR-code: 441 bits Text on this slide Full text of Hamlet Pictures of Royal Holloway Compact disc of Beethoven 9th Bluray disc of 3 hour film Large Hadron Collider, per second bits 1.5 million bits
28 Why Coding Theory? A bit gives a single piece of information: NO or YES ; on or off ; 0 or 1. A number between 0 and 15: 4 bits A small QR-code: 441 bits Text on this slide Full text of Hamlet Pictures of Royal Holloway Compact disc of Beethoven 9th Bluray disc of 3 hour film Large Hadron Collider, per second bits 1.5 million bits 5 million bits each
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32 Why Coding Theory? A bit gives a single piece of information: NO or YES ; on or off ; 0 or 1. A number between 0 and 15: 4 bits A small QR-code: 441 bits Text on this slide Full text of Hamlet Pictures of Royal Holloway Compact disc of Beethoven 9th Bluray disc of 3 hour film Large Hadron Collider, per second bits 1.5 million bits 5 million bits each
33 Why Coding Theory? A bit gives a single piece of information: NO or YES ; on or off ; 0 or 1. A number between 0 and 15: 4 bits A small QR-code: 441 bits Text on this slide Full text of Hamlet Pictures of Royal Holloway Compact disc of Beethoven 9th Bluray disc of 3 hour film Large Hadron Collider, per second bits 1.5 million bits 5 million bits each 6 billion bits
34 Why Coding Theory? A bit gives a single piece of information: NO or YES ; on or off ; 0 or 1. A number between 0 and 15: 4 bits A small QR-code: 441 bits Text on this slide Full text of Hamlet Pictures of Royal Holloway Compact disc of Beethoven 9th Bluray disc of 3 hour film Large Hadron Collider, per second bits 1.5 million bits 5 million bits each 0.7 GB
35 Why Coding Theory? A bit gives a single piece of information: NO or YES ; on or off ; 0 or 1. A number between 0 and 15: 4 bits A small QR-code: 441 bits Text on this slide Full text of Hamlet Pictures of Royal Holloway Compact disc of Beethoven 9th Bluray disc of 3 hour film Large Hadron Collider, per second bits 1.5 million bits 5 million bits each 0.7 GB 50 GB
36 Why Coding Theory? A bit gives a single piece of information: NO or YES ; on or off ; 0 or 1. A number between 0 and 15: 4 bits A small QR-code: 441 bits Text on this slide Full text of Hamlet Pictures of Royal Holloway Compact disc of Beethoven 9th Bluray disc of 3 hour film Large Hadron Collider, per second bits 1.5 million bits 5 million bits each 0.7 GB 50 GB 300 GB
37 Why Coding Theory? A bit gives a single piece of information: NO or YES ; on or off ; 0 or 1. A number between 0 and 15: 4 bits A small QR-code: 441 bits Text on this slide Full text of Hamlet Pictures of Royal Holloway Compact disc of Beethoven 9th Bluray disc of 3 hour film Large Hadron Collider, per second bits 1.5 million bits 5 million bits each 0.7 GB 50 GB 300 GB Errors in reading and writing are inevitable. The 3G specification for mobile phones expects one bit in a thousand to be received wrongly.
38 Mariner 9 The Mariner 9 probe, launched in 1971, took the first images of Mars. The images were grey-scale, with 64 possible shades of grey for each pixel. The pictures were transmitted back to Earth by sending one pixel at a time. Since 64 = , each pixel needs 6 bits to send. The probability of each bit being flipped in the channel was about 5%. Encoding each pixel using 6 bits, about 26% of every image would be wrong.
39 Mariner 9 The Mariner 9 probe, launched in 1971, took the first images of Mars. The images were grey-scale, with 64 possible shades of grey for each pixel. The pictures were transmitted back to Earth by sending one pixel at a time. Since 64 = , each pixel needs 6 bits to send. The probability of each bit being flipped in the channel was about 5%. Encoding each pixel using 6 bits, about 26% of every image would be wrong. Instead each pixel was encoded using 32 bits, increasing the length of the transmitted message over five times.
40 Mariner 9 Image: Improvement Due to Error Correction
41 Mariner 9 Image: Improvement Due to Error Correction
42 The Mariner 9 Code: 32 of the 64 Mariner 9 codewords
43 The Mariner 9 Code: 32 of the 64 Mariner 9 codewords
44 The Liar Game: Dealing with Deliberate Errors Ask a friend to think of a number between 0 and 15. How many NO/YES questions do you need to ask, if your friend is permitted to lie at most once? It is not compulsory to lie.
45 The Liar Game: Dealing with Deliberate Errors Ask a friend to think of a number between 0 and 15. How many NO/YES questions do you need to ask, if your friend is permitted to lie at most once? It is not compulsory to lie. Any interesting strategies?
46 The Liar Game: Dealing with Deliberate Errors Ask a friend to think of a number between 0 and 15. How many NO/YES questions do you need to ask, if your friend is permitted to lie at most once? It is not compulsory to lie. Any interesting strategies? (For example, asking if someone has lied so far?)
47 The Liar Game: Dealing with Deliberate Errors Ask a friend to think of a number between 0 and 15. How many NO/YES questions do you need to ask, if your friend is permitted to lie at most once? It is not compulsory to lie. Any interesting strategies? (For example, asking if someone has lied so far?) Claim. No strategy can guarantee to use fewer than 7 questions.
48 The Liar Game: Dealing with Deliberate Errors Ask a friend to think of a number between 0 and 15. How many NO/YES questions do you need to ask, if your friend is permitted to lie at most once? It is not compulsory to lie. Any interesting strategies? (For example, asking if someone has lied so far?) Claim. No strategy can guarantee to use fewer than 7 questions. Coding theory can be used to find a good strategy. Lies correspond to errors in transmission.
49 The Hamming Code Richard Hamming discovered a one-error correcting binary code of length 7 with 16 codewords. He invented it because he was fed up with the paper tape reader on his early computer misreading his programs. It gives an optimal solution to the Liar Game using 7 questions. Remarkably, it is possible to specify all the questions in advance.
50 The Hamming Code Find the binary codeword corresponding to your secret number The questions are: Is there a 1 in the first position (far left) of the codeword?, Is there a 1 in the second position of the codeword?, and so on. If there is one lie, then the questioner will write down one wrong bit. But because the Hamming code can correct one error, the questioner can still work out what the number is.
51 The Square Code To encode a number between 0 and 15 in the square code Write it in binary as b 1 b 2 b 3 b 4 Make a square with these bits. Put in four check bits around the edges, computed using modulo 2 arithmetic: = 0, = 1, = 1, = 0.
52 The Square Code To encode a number between 0 and 15 in the square code Write it in binary as b 1 b 2 b 3 b 4 Make a square with these bits. Put in four check bits around the edges, computed using modulo 2 arithmetic: = 0, = 1, = 1, = 0. Imagine you receive this What number do you think was sent? (A) 3 = 0011 (B) 5 = 0101 (C) 6 = 0110 (D) 7 = 0111
53 Square Code
54 Square Code
55 Square Code
56 Square Code
57 Double Square Code (16 bits) Versus No Coding (4 bits)
58 Double Square Code (16 bits) Versus No Coding (4 bits)
59 A Hat Game Related to Coding Theory You and two friends are on your way to a party. At the party a black or blue hat will be put on each person s head. You can see your friends hats, but not your own.
60 A Hat Game Related to Coding Theory You and two friends are on your way to a party. At the party a black or blue hat will be put on each person s head. You can see your friends hats, but not your own. When the host shouts go!, you may either say a colour or remain silent. Everyone who speaks must speak at the same time. If everyone who speaks gets the colour of his or her hat correct, you all win some cake. If no-one speaks, or someone gets it wrong, there is no cake.
61 A Hat Game Related to Coding Theory You and two friends are on your way to a party. At the party a black or blue hat will be put on each person s head. You can see your friends hats, but not your own. When the host shouts go!, you may either say a colour or remain silent. Everyone who speaks must speak at the same time. If everyone who speaks gets the colour of his or her hat correct, you all win some cake. If no-one speaks, or someone gets it wrong, there is no cake. Question: What is a good strategy?
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63 Another Hat Game You and four friends are lined up. A black or blue hat is put on each person s head. You can see all the hats in front of you, but not your own, or those behind. So the person at the back of the line can see four hats, the next person can see three, and so on.
64 Another Hat Game You and four friends are lined up. A black or blue hat is put on each person s head. You can see all the hats in front of you, but not your own, or those behind. So the person at the back of the line can see four hats, the next person can see three, and so on. Starting at the back of the line, each person is asked to guess the colour of his or her hat.
65 Another Hat Game You and four friends are lined up. A black or blue hat is put on each person s head. You can see all the hats in front of you, but not your own, or those behind. So the person at the back of the line can see four hats, the next person can see three, and so on. Starting at the back of the line, each person is asked to guess the colour of his or her hat. Question: What is a good strategy?
66 Thank you! Any questions?
67 Thank you! Any questions? Why is maths a good subject to study? What do maths lecturers do all day? How does maths at university differ from A-level maths? Are women just as good as men at maths? (Answer: Yes!)
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