Big numbers, graph coloring, and Herculesʼ battle with the hydra
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1 Big numbers, graph coloring, and Herculesʼ battle with the hydra Tim Riley March 12, 2011 K 12 Education and Outreach
2 Challenge You have two minutes Using standard math notation, English words, or both, name a single whole number not an infinity on a blank index card Be precise enough for any reasonable modern mathematician to determine exactly what number you ve named, by consulting only your card and, if necessary, the published literature
3
4
5
6 Pindar: ca BC Archimedes c 287 BC c 212 BC
7
8 Exponential growth
9
10
11 Bozorgmehr, King Anushirvan of Persia's grand vizier, challenges the Indian envoy to a game of chess
12 Graph coloring and chromatic number Petersen Graph Groetzsch Graph Clebsch Graph
13 Petersen Graph chromatic number = 3 Groetzsch Graph chromatic number = 4 Clebsch Graph chromatic number = 4
14 Is there a polynomial time algorithm that will tell you whether a graph is 3-colorable?
15 Subset sum Hamiltonian cycles Travelling salesman problem
16 Graham s number and Ramsey Theory
17 Ramsey Theory game K 5 Sim Guatsavo Simmons 1969 K 6 K 7
18 On K 5, there can be draws K 6 At a gathering of any six people, some three of them are either mutual acquaintances or complete strangers There are no draws On K 6, with perfect strategy, the second player always wins
19 R(m) is the minimal n such that however one 2 colors K n, it will always contain a monochrome subgraph K m R(1) = 1 R(4) = 18 R(2) = 2 R(3) = 6 43 R(5) R(6) 165 Erdős asks us to imagine an alien force, vastly more powerful than us, landing on Earth and demanding the value of R(5) or they will destroy our planet In that case, he claims, we should marshal all our computers and all our mathematicians and attempt to find the value But suppose, instead, that they ask for R(6) In that case, he believes, we should attempt to destroy the aliens Paul Erdős,
20 Consider an n dimensional hypercube, and connect each pair of vertices to obtain K 2 n What is the smallest value of n such that every 2 coloring contains at least one monochrome planar subgraph? K 4
21 Knuth s arrow notation evaluated right to left Example
22 is an upper bound for the solution to the hypercube problem
23 General belief is that the answer to Graham s problem is 6 The infinite we shall do right away The finite may take a little longer Stanislaw Ulam
24
25 A hydra is a word on letters a and b Hercules strikes off the first letter The hydra regenerates by: a a b b b Repeat Who wins? How long does it take? Generalisations? Questions? Example bab bab ab b bab b b b bab b b b bab b b b bab b b b b ab b b b b b bbbbbb bbbbb bbbb bbb is b b Hercules victorious in b 12 strikes
26 a 1 strike a a a b b 3 strikes a a a a b a b b a b b a b b b b b b b b b 7 strikes a a a a a b a b a b b a b b a b b a b b b a b b b b b b a b b b b b b a b b b b b b a b b b b b b a b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b 15 strikes a a a n Hercules wins against all hydra 2 n 1 strikes
27 Hydra using three letters a, b and c Regeneration: a ab, b bc, c c a 1 strike H(w) := 4 a a a b b c c strikes # strikes it takes to kill w H(a n+1 )=3 2 H(an) 2 a a a a b a b b c a b b c c a b b c b c c a b b c b c c b c c c b c b c c b c c c b c c c c c b c c c b c c c c b c c c c c b c c c c b c c c c c b c c c c c c c c c c b c c c c c c b c c c c c c c c c c b c c c c c c c b c c c c c c c c c c b c c c c c c c c b c c c c c c c c c c b c c c c c c c c c b c c c c c c c c c c b c c c c c c c c c c b c c c c c c c c c c c c c c c c c c c c c b c c c c c c c c c c c c c c c c c c c c c b c c c c c c c c c c c c c c c c c c c c c b c c c c c c c c c c c c c c c c c c c c c b c c c c c c c c c c c c c c c c c c c c c b c c c c c c c c c c c c c c c c c c c c c b c c c c c c c c c c c c c c c c c c c c c b c c c c c c c c c c c c c c c c c c c c c b c c c c c c c c c c c c c c c c c c c c c b c c c c c c c c c c c c c c c c c c c c c b c c c c c c c c c c c c c c c c c c c c c b c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c 46 strikes a a a a a b a b a b b c a b b c a b b c c a b b c b c c a b b c b c c a b b c b c c b c c c a b b c b c c b c c c strikes Hercules still wins against all hydra eventually!
28 a 1 strike d a ab c c c d d d Hydra using four letters a, b, c and Regeneration:, b b,, 5 a a a b b c cd d strikes a a a a b a b b c a b b c b c cd a b b c cd cd cdd a b b c b c cd cdd strikes Hercules still always wins!
29 a1, a2, ai ai ai 1, i > 1 a1 a1 Hk (n) := H(ak n ) < n n n k 1 n
30 Ackermann s function For integers, k, n > 0 A 1 (n) := 2n and A k+1 (n) := A (n) k (1) k n n n n 3 2 n A 3 (65536) k 3,n 2, H k (n) A k (n) k 1,n 0, H k (n) A k (n + k)
31 Theorem a 1,,a k, t, p t 1 a 1 t = a 1, t 1 a i t = a i a i 1 (i>1) [p, a i t]=1 (i>0) = G k,p [p, H k ] has Dehn function A k when k>1 p p
32 Going faster! n A n (n) is recursive but not primitive recursive Wilhelm Ackermann,
33 Bump the base and subtract = = = = = =? ?
![Goodstein sequences [Goodstein, 1944] 266 = 2 8 +2 3 +2 = 2 22+1 +2 2+1 +2 443426488243037769948249630619149892886 = 3 33+1 +3 3+1 +2](/docs-images/93/111631955/images/34-0.jpg "323170060713110073007148766886699519604441026697154840321303454275246551388678 908931972014115229134636887179609218980194941195591504909210950881523864482831")
34 Goodstein sequences [Goodstein, 1944] 266 = = = = Reuben Louis Goodstein
35 Raymond Smullyan Balls in a box A game of bounded height, but unbounded width
36 Gödel's First Incompleteness Theorem Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true, but not provable in the theory (Kleene 1967, p 250) Kurt Gödel, Kirby & Paris (1982) It is unprovable in Peano Arithmetic that all Goodstein sequences terminate at zero
37 Turing Machines and Busy Beaver Functions Alan Turing,
38 Radó s Busy Beaver function: BB(n) is the maximum halting time of all halting Turing machines of size at most n Tibor Radó,
39
40 Bertrand Russell, The Berry Paradox G G Berry,
41 References / Acknowledgements / Further reading Scott Aaronson, Who can name the bigger number? wwwscottaaronsoncom/writings/bignumbershtml Archimedes, The Sand Reckoner Will Dison and Timothy Riley, Hydra groups, frontmathucdavisedu/ Martin Gardener, Mathematical Games, Scientific American, November 1977 Wikipedia especially the articles on Ramsey Theory, Ramsey s Theorem, Graham s number and Knuth s arrow notation Slides available at: wwwmathcornelledu/~riley/talkshtml
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