One step forward, one step back: A puzzle approach to Erdős’ discrepancy problem

In the 1930s, the famous Hungarian mathematician, Paul Erdős, thought that for any infinite sequence of +1 and -1, it would always be possible to find a finite subsequence consisting of every nth term up to some point summing to a number larger than any you choose. But Erdős could not prove his conjecture, which he referred to as the "discrepancy problem." About a year ago, Lisitsa and Konev published a proof that is a significant step towards proving Erdős' problem. Their computer-assisted proof resulted in headlines such as "Wikipedia-size maths proof too big for humans to check" because their proof is as large as the entire content of Wikipedia, making it unlikely that it will ever be checked by a human being. In this episode of FAMUS, we will tinker with Erdős’ puzzle and attempt to wrap our heads around its difficulty.

This talk was given at the Northern Arizona University Friday Afternoon Mathematics Undergraduate Seminar (FAMUS) on Friday, February 13, 2015.

Transcript

1. one step forward, one step back: a puzzle approach to erdős’ discrepancy problem Friday Afternoon Mathematics Undergraduate Seminar Dana C. Ernst Northern Arizona University February 13, 2015
2. erdős discrepancy problem Suppose we have a row of Transformers, some of which are Autobots and some of which are Decepticons. 1
3. erdős discrepancy problem Suppose we have a row of Transformers, some of which are Autobots and some of which are Decepticons. 1 2 3 4 5 6 1
4. erdős discrepancy problem Suppose we have a row of Transformers, some of which are Autobots and some of which are Decepticons. 1 2 3 4 5 6 Starting with the first one and moving to the right, keep track of how many of each type we have up until that point. 1
5. erdős discrepancy problem Suppose we have a row of Transformers, some of which are Autobots and some of which are Decepticons. 1 2 3 4 5 6 A 1 1 2 3 3 4 D 0 1 1 1 2 2 Starting with the first one and moving to the right, keep track of how many of each type we have up until that point. 1
6. erdős discrepancy problem Suppose we have a row of Transformers, some of which are Autobots and some of which are Decepticons. 1 2 3 4 5 6 A 1 1 2 3 3 4 D 0 1 1 1 2 2 Starting with the first one and moving to the right, keep track of how many of each type we have up until that point. In addition, keep track of the absolute value of the difference between the two types at each step. 1
7. erdős discrepancy problem Suppose we have a row of Transformers, some of which are Autobots and some of which are Decepticons. 1 2 3 4 5 6 A 1 1 2 3 3 4 D 0 1 1 1 2 2 1 0 1 2 1 2 Starting with the first one and moving to the right, keep track of how many of each type we have up until that point. In addition, keep track of the absolute value of the difference between the two types at each step. 1
8. erdős discrepancy problem Suppose we have a row of Transformers, some of which are Autobots and some of which are Decepticons. 1 2 3 4 5 6 A 1 1 2 3 3 4 D 0 1 1 1 2 2 1 0 1 2 1 2 Starting with the first one and moving to the right, keep track of how many of each type we have up until that point. In addition, keep track of the absolute value of the difference between the two types at each step. In this case, the largest difference is 2. 1
9. erdős discrepancy problem Problem 1 How could you line up n Transformers and keep the largest difference at 1 (that is, the difference between Autobots and Decepticons is always 1 or 0)? 2
10. erdős discrepancy problem Problem 1 How could you line up n Transformers and keep the largest difference at 1 (that is, the difference between Autobots and Decepticons is always 1 or 0)? 1 2 3 4 5 6 … A 1 1 2 2 3 3 … D 0 1 1 2 2 3 … 1 0 1 0 1 0 … 2
11. erdős discrepancy problem Skip counting starts at any number and then “skips forward” by that number: 3
12. erdős discrepancy problem Skip counting starts at any number and then “skips forward” by that number: 1, 2, 3, 4, 5, . . . 2, 4, 6, 8, 10, . . . 3, 6, 9, 12, 15, . . . 4, 8, 12, 16, 20, . . . 3
13. erdős discrepancy problem Skip counting starts at any number and then “skips forward” by that number: 1, 2, 3, 4, 5, . . . 2, 4, 6, 8, 10, . . . 3, 6, 9, 12, 15, . . . 4, 8, 12, 16, 20, . . . In more mathematical terms, it’s listing all the multiples of a given number. 3
14. erdős discrepancy problem Instead of keeping track at every Transformer, we could use skip counting and only include particular Transformers in the row. 4
15. erdős discrepancy problem Instead of keeping track at every Transformer, we could use skip counting and only include particular Transformers in the row. Let’s take our original example and skip count by 2s: 4
16. erdős discrepancy problem Instead of keeping track at every Transformer, we could use skip counting and only include particular Transformers in the row. Let’s take our original example and skip count by 2s: 1 2 3 4 5 6 A 0 1 2 D 1 1 1 1 0 1 4
17. erdős discrepancy problem Instead of keeping track at every Transformer, we could use skip counting and only include particular Transformers in the row. Let’s take our original example and skip count by 2s: 1 2 3 4 5 6 A 0 1 2 D 1 1 1 1 0 1 In this case, the largest difference is 1. 4
18. erdős discrepancy problem Problem 2 How could you line up 6 Transformers so that for both counting by 1s and counting by 2s, the largest difference is 1? 5
19. erdős discrepancy problem Problem 2 How could you line up 6 Transformers so that for both counting by 1s and counting by 2s, the largest difference is 1? 1 2 3 4 5 6 A 1 1 1 2 2 3 D 0 1 2 2 3 3 1 0 1 0 1 0 A 0 1 2 D 1 1 1 1 0 1 5
20. erdős discrepancy problem Definition If every kind of skip counting is used (1s, 2s, 3s, 4s, etc.), then the largest difference possible is called the discrepancy. 6
21. erdős discrepancy problem Problem 3 Design a row of 11 Transformers so that the discrepancy is 1. That is, for any kind of skip counting, the difference between the number of Autobots and Decepticons is always 1 or 0. 7
22. erdős discrepancy problem Problem 3 Design a row of 11 Transformers so that the discrepancy is 1. That is, for any kind of skip counting, the difference between the number of Autobots and Decepticons is always 1 or 0. 1 2 3 4 5 6 7 8 9 10 11 7
23. erdős discrepancy problem Problem 3 Design a row of 11 Transformers so that the discrepancy is 1. That is, for any kind of skip counting, the difference between the number of Autobots and Decepticons is always 1 or 0. 1 2 3 4 5 6 7 8 9 10 11 Problem 4 What about 12 Transformers? This problem is a bit harder. Let’s rephrase it. 7
24. erdős discrepancy problem A torturer promises to free you if you can solve the following problem. 8
25. erdős discrepancy problem A torturer promises to free you if you can solve the following problem. Two steps to your right is a flesh eating zombie. 8
26. erdős discrepancy problem A torturer promises to free you if you can solve the following problem. Two steps to your right is a flesh eating zombie. Two steps to your left is something even scarier… 8
27. erdős discrepancy problem A torturer promises to free you if you can solve the following problem. Two steps to your right is a flesh eating zombie. Two steps to your left is something even scarier…like Taylor Swift! 8
28. erdős discrepancy problem A torturer promises to free you if you can solve the following problem. Two steps to your right is a flesh eating zombie. Two steps to your left is something even scarier…like Taylor Swift! 8
29. erdős discrepancy problem A torturer promises to free you if you can solve the following problem. Two steps to your right is a flesh eating zombie. Two steps to your left is something even scarier…like Taylor Swift! Can you write a set of 12 instructions that you must follow so that you avoid contact with the zombie and Taylor subject to the following constraints? ∙ Each step is either a step right (+1) or a step left (-1). ∙ Your solution should still work even if your torturer decides you must follow every kth step of instructions. 8
30. erdős discrepancy problem Without loss of generality, assume that your first step is to the right. It turns out that the rest of your moves are forced. Step 1 2 3 4 5 6 7 8 9 10 11 12 Move 1 9
31. erdős discrepancy problem Without loss of generality, assume that your first step is to the right. It turns out that the rest of your moves are forced. Step 1 2 3 4 5 6 7 8 9 10 11 12 Move 1 -1 9
32. erdős discrepancy problem Without loss of generality, assume that your first step is to the right. It turns out that the rest of your moves are forced. Step 1 2 3 4 5 6 7 8 9 10 11 12 Move 1 -1 1 9
33. erdős discrepancy problem Without loss of generality, assume that your first step is to the right. It turns out that the rest of your moves are forced. Step 1 2 3 4 5 6 7 8 9 10 11 12 Move 1 -1 1 -1 9
34. erdős discrepancy problem Without loss of generality, assume that your first step is to the right. It turns out that the rest of your moves are forced. Step 1 2 3 4 5 6 7 8 9 10 11 12 Move 1 -1 -1 1 -1 9
35. erdős discrepancy problem Without loss of generality, assume that your first step is to the right. It turns out that the rest of your moves are forced. Step 1 2 3 4 5 6 7 8 9 10 11 12 Move 1 -1 -1 1 1 -1 9
36. erdős discrepancy problem Without loss of generality, assume that your first step is to the right. It turns out that the rest of your moves are forced. Step 1 2 3 4 5 6 7 8 9 10 11 12 Move 1 -1 -1 1 1 -1 -1 9
37. erdős discrepancy problem Without loss of generality, assume that your first step is to the right. It turns out that the rest of your moves are forced. Step 1 2 3 4 5 6 7 8 9 10 11 12 Move 1 -1 -1 1 -1 1 -1 -1 9
38. erdős discrepancy problem Without loss of generality, assume that your first step is to the right. It turns out that the rest of your moves are forced. Step 1 2 3 4 5 6 7 8 9 10 11 12 Move 1 -1 -1 1 -1 1 -1 1 -1 9
39. erdős discrepancy problem Without loss of generality, assume that your first step is to the right. It turns out that the rest of your moves are forced. Step 1 2 3 4 5 6 7 8 9 10 11 12 Move 1 -1 -1 1 -1 1 1 -1 1 -1 9
40. erdős discrepancy problem Without loss of generality, assume that your first step is to the right. It turns out that the rest of your moves are forced. Step 1 2 3 4 5 6 7 8 9 10 11 12 Move 1 -1 -1 1 -1 1 1 -1 -1 1 -1 9
41. erdős discrepancy problem Without loss of generality, assume that your first step is to the right. It turns out that the rest of your moves are forced. Step 1 2 3 4 5 6 7 8 9 10 11 12 Move 1 -1 -1 1 -1 1 1 -1 -1 1 1 -1 9
42. erdős discrepancy problem Without loss of generality, assume that your first step is to the right. It turns out that the rest of your moves are forced. Step 1 2 3 4 5 6 7 8 9 10 11 12 Move 1 -1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 9
43. erdős discrepancy problem Without loss of generality, assume that your first step is to the right. It turns out that the rest of your moves are forced. Step 1 2 3 4 5 6 7 8 9 10 11 12 Move 1 -1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 So, we can’t survive if we are required 12 steps. However, we can survive if only 11 are required. 9
44. erdős discrepancy problem Problem 5 What if the zombie and Taylor are 3 steps away? How long of a list of instructions should your torturer force you to write in order to guarantee that at least one skip-counting subsequence kills you? 10
45. erdős discrepancy problem Our puzzle is a special case of a problem posed by the Hungarian mathematician Paul Erdős in the 1930s. Erdős offered $500 to anyone that could answer the following question in either the affirmative or negative. 11
46. erdős discrepancy problem Our puzzle is a special case of a problem posed by the Hungarian mathematician Paul Erdős in the 1930s. Erdős offered $500 to anyone that could answer the following question in either the affirmative or negative. The Erdős Discrepancy Problem (EDP) Let {xn} be an infinite ±1 sequence and let C be a positive integer. Must there exist positive integers d and k such that the following holds? k ∑ i=1 xi·d > C 11
47. erdős discrepancy problem Our puzzle is a special case of a problem posed by the Hungarian mathematician Paul Erdős in the 1930s. Erdős offered $500 to anyone that could answer the following question in either the affirmative or negative. The Erdős Discrepancy Problem (EDP) Let {xn} be an infinite ±1 sequence and let C be a positive integer. Must there exist positive integers d and k such that the following holds? k ∑ i=1 xi·d > C We’ve already shown that EDP is true for C = 1 using any sequence of length 12 or more. Problem 5 is the C = 2 case. 11
48. erdős discrepancy problem In 2010, the Polymath project starting working on EDP. They generated sequences of length 1124 having discrepancy 2, prompting many to believe this was an appropriate bound. 12
49. erdős discrepancy problem In 2010, the Polymath project starting working on EDP. They generated sequences of length 1124 having discrepancy 2, prompting many to believe this was an appropriate bound. Theorem (Lisitsa & Konev, 2014) Every ±1 sequence of 1161 or more elements has a skip-counting subsequence that sums to more than 2 in absolute value. Bound of 1161 is minimal! 12
50. erdős discrepancy problem In 2010, the Polymath project starting working on EDP. They generated sequences of length 1124 having discrepancy 2, prompting many to believe this was an appropriate bound. Theorem (Lisitsa & Konev, 2014) Every ±1 sequence of 1161 or more elements has a skip-counting subsequence that sums to more than 2 in absolute value. Bound of 1161 is minimal! This settles EDP for C ≤ 2. The rest of the cases remain open! 12
51. erdős discrepancy problem In 2010, the Polymath project starting working on EDP. They generated sequences of length 1124 having discrepancy 2, prompting many to believe this was an appropriate bound. Theorem (Lisitsa & Konev, 2014) Every ±1 sequence of 1161 or more elements has a skip-counting subsequence that sums to more than 2 in absolute value. Bound of 1161 is minimal! This settles EDP for C ≤ 2. The rest of the cases remain open! They utilized a computer-based proof that relies on encoding the problem into Boolean satisfiability and applying state of the art SAT solvers. Amount of data generated for proof was 13GB, which is larger than the size of the entire written contents of Wikipedia. 12
52. erdős discrepancy problem The fact that a sequence of length 1160 has discrepancy 2 can be easily checked. Establishing the bound of 1161 is “probably one of longest proofs of a non-trivial mathematical result ever produced.” 13
53. erdős discrepancy problem The fact that a sequence of length 1160 has discrepancy 2 can be easily checked. Establishing the bound of 1161 is “probably one of longest proofs of a non-trivial mathematical result ever produced.” Result prompted headlines such as: “Wikipedia-size maths proof too big for humans to check” “Computer cracks Erdős puzzle – but no human brain can check the answer” “Erdős puzzle solved? Computer works out elusive maths riddle (but its calculations are so complex, no human can check if it’s correct)” 13
54. erdős discrepancy problem The fact that a sequence of length 1160 has discrepancy 2 can be easily checked. Establishing the bound of 1161 is “probably one of longest proofs of a non-trivial mathematical result ever produced.” Result prompted headlines such as: “Wikipedia-size maths proof too big for humans to check” “Computer cracks Erdős puzzle – but no human brain can check the answer” “Erdős puzzle solved? Computer works out elusive maths riddle (but its calculations are so complex, no human can check if it’s correct)” In their paper, they also showed that you can survive the zombie and Taylor Swift if they are 4 steps away (C = 3 case) if you only need a list 13,700 instructions. 13
55. erdős discrepancy problem Who Cares? 14
56. erdős discrepancy problem Who Cares? ∙ Combinatorial number theory 14
57. erdős discrepancy problem Who Cares? ∙ Combinatorial number theory ∙ Discrepancy theory 14
58. erdős discrepancy problem Who Cares? ∙ Combinatorial number theory ∙ Discrepancy theory ∙ Hypergraph theory 14
59. erdős discrepancy problem Who Cares? ∙ Combinatorial number theory ∙ Discrepancy theory ∙ Hypergraph theory ∙ Wireless communication 14
60. erdős discrepancy problem Who Cares? ∙ Combinatorial number theory ∙ Discrepancy theory ∙ Hypergraph theory ∙ Wireless communication ∙ Cryptography 14
61. erdős discrepancy problem Who Cares? ∙ Combinatorial number theory ∙ Discrepancy theory ∙ Hypergraph theory ∙ Wireless communication ∙ Cryptography But really many mathematicians just want to slay dragons! 14
62. erdős discrepancy problem Who Cares? ∙ Combinatorial number theory ∙ Discrepancy theory ∙ Hypergraph theory ∙ Wireless communication ∙ Cryptography But really many mathematicians just want to slay dragons! Methods used to solve this problem can give us new tools for solving other problems. 14
63. erdős discrepancy problem References ∙ Erdős’s discrepancy problem now less of a problem at Aperiodical ∙ Jason Dyer’s blog post A gentle introduction to the 5th Polymath project ∙ Singing Banana’s YouTube video New Wikipedia sized proof explained with a puzzle ∙ The Erdős discrepancy problem page at PolyMath ∙ Lisitsa and Konev’s arXiv article A SAT Attack on the Erdős Discrepancy Conjecture 15