Piet Hein and John Nash: BEAUTIFUL MINDS Talk by Bjarne Toft, University of Southern Denmark
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1 Piet Hein and John Nash: BEAUTIFUL MINDS Talk by Bjarne Toft, University of Southern Denmark Piet Hein John Nash
2 High School Graduation 1924 The lost spring
3 Copenhagen Conference 1932 Heisenberg, Werner Karl; Hein, Piet; Bohr, N.; Brillouin, Leon Nicolas; Rosenfeld, Leon; Delbrück, Max; Heitler, Walter; Meitner, Lise; Ehrenfest, Paul; Bloch, Felix; Waller, Ivar; Solomon, Jacques; Fues, Erwin; Strømgren, Bengt; Kronig, Ralph de Laer; Gjelsvik, A; Steensholt, Gunnar; Kramers, Hendrik Anton; Weizsäcker, Carl Friedrich von; Ambrosen, J.P.; Beck, Guido; Nielsen, Harald Herborg; Buch-Andersen; Kalckar, Fritz; Nielsen, Jens Rud; Fowler, Ralph Howard; Hyllerås, Egil Andersen; Lam, Ingeborg; Rindal, Eva; Dirac, Paul Adrian Maurice; N.N.; Darwin, Charles Galton; Manneback, Charles; Lund, Gelius
4 Piet Hein to Martin Gardner (1957)
5 Soma a contradictory surprise
6 SOMA in 1933
7 Piet Hein discovered Hex in 1942 Parentesen, Copenhagen University, December 1942 The Mathematics of games and Games as mathematics
8 1. Just 2. Moving forward 3. Finite 4. Full information 5. Strategic 6. Decisive (no draw)
9 The first can win And this can be proved
10 Politiken Dec. 26, 1942
11 Piet Hein Problems 1-46 from Politiken Dec.1942-June 1943
12 Piet Hein Problem 1 White plays and wins!
13 Life as a game of Hex Life is almost like a game Easy hard Decide your aim With the simplest Rules you start Most easy then To make it hard. (transl. BT)
14
15
16 Piet Hein s two ideas - two theorems - creating the HEX game NOT BOTH CAN WIN NOT BOTH CAN LOSE
17 NOT BOTH CAN WIN 4-COLOUR-THEOREM (1997) : Any map is 4-colourable SIMPEL SPECIAL CASE: No 5 countries can have common borders two and two
18 NOT BOTH CAN LOSE PH: If only 3 faces meet PH: Then local blocking is impossible THEN CLEARLY (?): Global blocking is impossible FIRST PUBLISHED PROOF 1969: (Anatole Beck et al.)
19 The contradiction follows also from SPERNER s SIMPLEX LEMMA
20 Piet Hein (1942): Suddenly in the halflight of dawn a game awoke and demanded to be born BUT AN ARBITRARY PLANAR 3-REGULAR 2-CONNECTED GRAPH IS USEABLE AS BOARD (and Piet Hein s Theorems hold) GENERAL HEX or MUDCRACK HEX
21 John Nash, , (A Beautiful Mind) discovered Hex in 1948
22 Non-cooperative Games John F. Nash Jr. (21 years old)
23 Stockholm 1994
24 Oslo 2015 The Abel Prize ceremony, May 19th Bjarne Toft
25 Deleted scenes from A Beautiful Mind
26 People at Princeton directly involved in discovering/developing/studying Hex John Nash Aage Bohr David Gale Claude Shannon John Milnor Harold Kuhn
27 David Gale: Bridge-It Piet Hein: A version of Hex
28 There is a very simple winning (pairing) strategy for the first player
29 John Nash s Hex theorem The first player has a winning strategy (but a winning first move for the first player in nxn Hex is not known with mathematical certainty not then and not now!) Proof: Strategy stealing.
30 Nash to Gardner 1957
31 Nash s Theorem (strategy stealing)
32 Nash s Theorem (strategy stealing) Anatole Beck s theorem 1969
33 Unsolved problem
34
35
36 Martin Gardner 1957 SCIENTIFIC AMERICAN
37 Martin Gardner Piet Hein: Black earth turned into Yellow Crocus Is undiluted Hocus pocus Persi Diaconis: Pick up anything he wrote. You ll smile and learn something.
38 Claude Berge playing Hex 1974 Claude Berge Jean-Marie Pla Neil Grabois 1974 Michel Las Vergnas
39 Claude Berge and Ryan Hayward in Marseilles 1992
40 Paris juli 2004
41
42 First winning moves for White
43 Variation 1: Rex (Reverse Hex or Misère Hex) Objective: Avoid creating a chain between your two sides! The game cannot end in draw (hence either the first or the second player has a winning strategy) On an nxn board with n even the first player has a winning strategy (first published proof: Evans 1974) On an nxn board with n odd the second player has a winning strategy (first published proof: Lagarias and Sleator 1999). Their proof also covers n even.
44 Hayward, Toft and Henderson 2010
45 Variation 2: Terminated Rex (TRex) : the Rex game stops when there is just one emty field left (i.e. there should always be a choice!)
46 In TRex both players have non-losing strategies
47 Rex on an nxn-board with n odd: Let the second player (Black) play the non-losing strategy from TRex. THIS IS A WINNING STRATEGY FOR THE SECOND PLAYER IN REX: Either the first player (White) creates a white chain or TRex ends with one emty field left. In the Rex game that field has to be chosen by White and a White chain is formed! If also White plays the non-loosing strategy from TRex, then the Rex game will be decided only when the board is full.
48 Rex on an nxn board with odd n: (second player has winning strategy)
49 Variation 3: CYLINDRICAL HEX play on cylinder! THEOREM (Alpern and Belck 1991, Samuel Huneke 2012, Huneke, Hayward and Toft 2014) Cylindrical HEX has a winning strategy for the updown player when the circular dimension n is even (pairing strategy) Cylindrical HEX has a winning strategy for the updown player when the circular dimension is 3 Problem: Circular dimensions 5, 7, 9,.?
50 Discrete Mathematics 331 (2014) 93-97
51 Piet Hein / Bruno Mathsson
52 Superegg at Egeskov on Funen, Denmark
53 Sundial at Egeskov: The object casting its shadow, is the same object as the one on which the shadow is cast (the screen)!
54
55 Piet Hein s new globe of the world Denmark seen from foreign land Looks but like a grain of sand. Denmark as we Danes conceive it Is so big you won t believe it.
56
57 Thank you very much for your attention! Teak Hex board (by Piet Hein 1968) Still available at piethein.com Now also as NEW NORDIC! Price around 200 US$ Super elliptic Hex board (by Piet Hein 1975)
58
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