Open problems with monetary rewards

There is a history of individuals and organizations offering monetary rewards for solutions, either in the affirmative or negative, to difficult mathematically-oriented problems. For example, the Millennium Prize Problems are seven problems in mathematics that were stated by the Clay Mathematics Institute in 2000. A correct solution to any of the problems results in a $1,000,000 prize being awarded by the institute. To date, only one of the problems has been solved (the Poincaré Conjecture was solved by Grigori Perelman, but he declined the award in 2010). These are hard problems! The renowned mathematician John Conway (Princeton) maintains a list of open problems and for each problem on the list, he is offering $1000 to the first person that provides a correct solution. In this talk, we will explore a few of Conway’s problems, and in the unlikely event we come up with a solution, we’ll split the money.

This talk was given at the Northern Arizona University Friday Afternoon Mathematics Undergraduate Seminar (FAMUS) on Friday, October 24, 2014.

Transcript

1. open problems with monetary rewards
   Friday Afternoon Mathematics Undergraduate Seminar
   Dana C. Ernst
   Northern Arizona University
   October 24, 2014
   

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2. open problems with monetary rewards
   There is a history of individuals and organizations offering monetary
   rewards for solutions—affirmative or negative—to difficult open
   problems.
   Their reasons could be to draw other mathematicians’ attention, to
   express their belief in the magnitude of the difficulty of the problem,
   to challenge others, etc. In the words of the Clay Mathematics
   Institute:
   “…to elevate in the consciousness of the general public the fact
   that in mathematics, the frontier is still open and abounds in
   important unsolved problems; to emphasize the importance of
   working towards a solution of the deepest, most difficult
   problems; and to recognize achievement in mathematics of
   historical magnitude.”
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3. millennium prize problems
   

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4. millennium prize problems
   The Millennium Prize Problems are seven problems in mathematics
   that were stated by the Clay Mathematics Institute in 2000.
   1. Poincaré Conjecture
   2. P versus NP
   3. Hodge Conjecture
   4. Riemann Hypothesis
   5. Yang–Mills Existence and Mass Gap
   6. Navier–Stokes Existence and Smoothness
   7. Birch and Swinnerton–Dyer Conjecture
   A correct solution to any of the problems results in a $1,000,000
   prize. The Poincaré Conjecture was solved by Grigori Perelman, but
   he declined the award in 2010. The other problems remain unsolved.
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5. wolfskehl and fermat’s last theorem
   

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6. wolfskehl and fermat’s last theorem
   Theorem (Fermat’s Last Theorem)
   No three positive integers a, b, and c can satisfy the equation
   an + bn = cn for any integer value of n greater than two.
   “I have discovered a truly marvelous demonstration of this
   proposition that this margin is too narrow to contain.”
   – Fermat (but in Latin)
   Paul Friedrich Wolfskehl (1856–1906), was a physician and
   mathematician. He bequeathed 100,000 marks (about 1,000,000
   pounds in 1997) to the first person to prove Fermat’s Last Theorem.
   On June 27, 1997, the prize was finally won by Andrew Wiles, but due to
   hyperinflation Germany, the award had dwindled to 30,000 pounds.
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7. the beal conjecture
   

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8. beal’s conjecture
   Conjecture
   If ax + by = cz, where a, b, c, x, y, and z are positive integers and x, y
   and z are all greater than 2, then a, b, and c must have a common
   prime factor.
   Texas billionaire D. Andrew Beal stated this conjecture in 1993. Beal
   has been trying to prove his theorem ever since, offering cash
   rewards in steadily increasing amounts:
   ∙ 1997: $5,000
   ∙ 2000: $100,000
   ∙ 2013: $1,000,000
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9. conway’s problems
   

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10. john h. conway
    John H. Conway (born 26 December 1937) is a British mathematician
    active in the theory of finite groups, knot theory, number theory,
    combinatorial game theory, and coding theory. Conway is currently
    Professor of Mathematics at Princeton University.
    Conway maintains a list of open problems and for each problem on
    the list, he is offering $1,000 to the first person that provides a
    correct solution.
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11. the angel problem
    The Angels and Devils game is played by two players called the angel
    and the devil. It is played on an infinite chessboard.
    The angel has a power k, where 1 ≤ k ∈ N, specified before the game
    starts. The board starts empty with the angel at the origin. On each
    turn, the angel jumps to a different empty square that can be
    reached by at most k moves of a chess king.
    The devil, on its turn, may add a block on any single square not
    containing the angel. The angel may leap over blocked squares, but
    cannot land on them. The devil wins if the angel is unable to move.
    The angel wins by surviving indefinitely.
    Problem: Can an angel with high enough power win?
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12. the angel problem
    Conway offered a reward for a
    general solution to this problem:
    ∙ $100 for a winning strategy for
    an angel of sufficiently high
    power,
    ∙ $1000 for a proof that the devil
    can win irrespective of the
    angel’s power.
    In 2006, 4 independent and almost simultaneous proofs emerged
    that the angel has a winning strategy.
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13. sylver coinage game
    The Sylver Coinage Game is a game in which 2 players alternately
    name positive integers that are not the sum of nonnegative
    multiples of previously named integers.
    The person who names 1 is the loser!
    Named after Sylvester, who proved the game terminates.
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14. sylver coinage game
    Sample game between A and B
    ∙ A opens with 5. Now neither player can name 5, 10, 15, . . .
    ∙ B names 4. Now neither player can name 4, 5, 8, 9, 10, or any
    number greater than 11.
    ∙ A names 11. Now the only remaining numbers are 1, 2, 3, 6, and 7.
    ∙ B names 6. Now the only remaining numbers are 1, 2, 3, and 7.
    ∙ A names 7. Now the only remaining numbers are 1, 2, and 3.
    ∙ B names 2. Now the only remaining numbers are 1 and 3.
    ∙ A names 3, leaving only 1.
    ∙ B is forced to name 1 and loses.
    Problem: If player 1 names 16, and both players play optimally
    thereafter, then who wins?
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15. climb to a prime
    Let n be a positive integer. Write the prime factorization in the usual
    way, e.g., 60 = 22 · 3 · 5, in which the primes are written in increasing
    order, and exponents of 1 are omitted. Then drop exponents straight
    down and omit all multiplication signs, obtaining a number f(n).
    Now repeat.
    So, for example, f(60) = f(22 · 3 · 5) = 2235. Next, because
    2235 = 3 · 5 · 149, it maps, under f, to 35149, and since 35149 is prime,
    it maps to itself. Thus 60 → 2235 → 35149 → 35149, so we have
    climbed to a prime, and we stop there forever.
    Problem: Is it true that every number eventually climbs to a prime?
    The number 20 has not been verified to do so. Observe that
    20 → 225 → 3252 → 223271 → · · · , eventually getting to more than
    100 digits without yet reaching a prime!
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16. the thrackle problem
    A doodle on a piece of paper is called a thrackle if it consists of
    certain distinguished points, called spots, and some differentiable
    (i.e., smooth) curves, called paths, ending at distinct spots and so
    that every pair of paths hit exactly once, where hit means having a
    common point at which they have distinct tangents and which is
    either an endpoint of both or an interior point of both.
    The thrackle to the left has 6
    spots and 6 paths.
    Problem: Can a thrackle have
    more paths than spots?
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17. 99-graph
    Problem: Is there a graph with 99 vertices in which every edge (i.e.,
    pair of joined vertices) belongs to a unique triangle and every
    non-edge (pair of unjoined vertices) to a unique quadrilateral?!
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18. sources
    

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19. sources
    http://mathoverflow.net/questions/66084/
    open-problems-with-monetary-rewards
    http://www.claymath.org
    http://en.wikipedia.org/wiki/Millennium_Prize_Problems
    http://en.wikipedia.org/wiki/Paul_Wolfskehl
    http://en.wikipedia.org/wiki/John_Horton_Conway
    http://www.cheswick.com/ches/conway1000.pdf
    http://en.wikipedia.org/wiki/Angel_problem
    http://en.wikipedia.org/wiki/Sylver_coinage
    http://www.thrackle.org
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