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
   

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2. erdős discrepancy problem
   Suppose we have a row of Transformers, some of which are Autobots
   and some of which are Decepticons.
   1
   

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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
   

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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
   

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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
   

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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
   

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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
   

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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
   

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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
   

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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
    

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11. erdős discrepancy problem
    Skip counting starts at any number and then “skips forward” by that
    number:
    3
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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24. erdős discrepancy problem
    A torturer promises to free you if you can solve the following
    problem.
    8
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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55. erdős discrepancy problem
    Who Cares?
    14
    

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56. erdős discrepancy problem
    Who Cares?
    ∙ Combinatorial number theory
    14
    

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57. erdős discrepancy problem
    Who Cares?
    ∙ Combinatorial number theory
    ∙ Discrepancy theory
    14
    

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58. erdős discrepancy problem
    Who Cares?
    ∙ Combinatorial number theory
    ∙ Discrepancy theory
    ∙ Hypergraph theory
    14
    

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59. erdős discrepancy problem
    Who Cares?
    ∙ Combinatorial number theory
    ∙ Discrepancy theory
    ∙ Hypergraph theory
    ∙ Wireless communication
    14
    

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60. erdős discrepancy problem
    Who Cares?
    ∙ Combinatorial number theory
    ∙ Discrepancy theory
    ∙ Hypergraph theory
    ∙ Wireless communication
    ∙ Cryptography
    14
    

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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
    

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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
    

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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
    

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