Games of No Strategy and Low-Grade Combinatorics
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1 Games of No Strategy and Low-Grade Combinatorics James Propp (jamespropp.org), UMass Lowell Mathematical Enchantments (mathenchant.org) presented at MOVES 2015 on August 3, 2015 Slides at 1 / 76
2 Interesting math from boring puzzles Boring puzzle: Tile a 2-by-n rectangle (here n = 5) with n dominos. 2 / 76
3 Interesting math from boring puzzles Interesting problem: In how many ways can a 2-by-n rectangle be tiled by dominos? 3 / 76
4 Interesting math from boring puzzles Interesting problem: In how many ways can a 2-by-n rectangle be tiled by dominos? Answer (old result; whose?): The nth Fibonacci number. 4 / 76
5 Interesting math from boring puzzles Interesting problem: In how many ways can a 2-by-n rectangle be tiled by dominos? Answer (old result; whose?): The nth Fibonacci number. 5 / 76
6 More interesting math from boring puzzles Boring puzzle: Tile an Aztec diamond of order n (here n = 2) with n(n + 1) dominos. (Aside: An Aztec diamond of order n consists of rows of length 2, 4, 6,..., 2n 2, 2n, 2n, 2n 2,..., 6, 4, 2, all centered.) 6 / 76
7 More interesting math from boring puzzles More interesting problem: In how many ways can an Aztec diamond of order n be tiled by dominos? 7 / 76
8 More interesting math from boring puzzles More interesting problem: In how many ways can an Aztec diamond of order n be tiled by dominos? Answer (Elkies-Kuperberg-Larsen-Propp): 2 n(n+1)/2. 8 / 76
9 More interesting math from boring puzzles More interesting problem: In how many ways can an Aztec diamond of order n be tiled by dominos? Answer (Elkies-Kuperberg-Larsen-Propp): 2 n(n+1)/2. 9 / 76
10 Grades of enumerative combinatorics These results are part of the branch of mathematics called enumerative combinatorics. I like to split the subject into grades, according to the growth-rates of the functions involved. I say f (n) is of grade k if f (n) grows like exp n k up to smaller correction factors; more precisely, if (log log f (n))/ log n k. The sequence of Fibonacci numbers is of grade 1; the sequence whose nth term is 2 n(n+1)/2 is of grade / 76
11 0,1,2,...? Combinatorics of grade 0 (e.g., figurate numbers) is ancient. Combinatorics of grade 1 (e.g., Fibonacci numbers, Catalan numbers) goes back several centuries. Combinatorics of grade 2 is just over a century old. Aside from a theorem of Linial s on generalized spanning trees in hypergraphs, and a handful of related results, enumerative combinatorics has not progressed beyond grade / 76
12 0,1,2,...? Combinatorics of grade 0 (e.g., figurate numbers) is ancient. Combinatorics of grade 1 (e.g., Fibonacci numbers, Catalan numbers) goes back several centuries. Combinatorics of grade 2 is just over a century old. Aside from a theorem of Linial s on generalized spanning trees in hypergraphs, and a handful of related results, enumerative combinatorics has not progressed beyond grade 2. Is there such a thing as enumerative combinatorics of grade 3? If so, how can we graduate to it? Let s return to these questions later. 12 / 76
13 Interesting math from boring games Boring game: Planted Brussels Sprouts (closely related to Brussels Sprouts and Cloves) 13 / 76
14 Interesting math from boring games Boring game: Planted Brussels Sprouts (closely related to Brussels Sprouts and Cloves) Initial state: n points surrounded by a circle, with each of the points connected to the circle by non-crossing arcs. Call the n points free ends. 14 / 76
15 Interesting math from boring games Boring game: Planted Brussels Sprouts (closely related to Brussels Sprouts and Cloves) Initial state: n points surrounded by a circle, with each of the points connected to the circle by non-crossing arcs. Call the n points free ends. Legal move: Join two free ends with a curve that does not cross any previously drawn curve (including the original circle) and then put a short stroke across the curve to create two new free ends, one on either side of the new curve. Delete the old free ends. 15 / 76
16 Interesting math from boring games Boring game: Planted Brussels Sprouts (closely related to Brussels Sprouts and Cloves) Initial state: n points surrounded by a circle, with each of the points connected to the circle by non-crossing arcs. Call the n points free ends. Legal move: Join two free ends with a curve that does not cross any previously drawn curve (including the original circle) and then put a short stroke across the curve to create two new free ends, one on either side of the new curve. Delete the old free ends. Winner: The last player to make a legal move wins. 16 / 76
17 Interesting math from boring games Example (n = 4): 17 / 76
18 Interesting math from boring games Example (n = 4): First player moves: 18 / 76
19 Interesting math from boring games Example (n = 4): First player moves: 19 / 76
20 Interesting math from boring games Example (n = 4): First player moves: 20 / 76
21 Interesting math from boring games Example (n = 4): First player moves: 21 / 76
22 Interesting math from boring games Example (n = 4): Second player moves: 22 / 76
23 Interesting math from boring games Example (n = 4): Second player moves: 23 / 76
24 Interesting math from boring games Example (n = 4): Second player moves: 24 / 76
25 Interesting math from boring games Example (n = 4): Second player moves: 25 / 76
26 Interesting math from boring games Example (n = 4): First player moves: 26 / 76
27 Interesting math from boring games Example (n = 4): First player moves: 27 / 76
28 Interesting math from boring games Example (n = 4): First player moves: 28 / 76
29 Interesting math from boring games Example (n = 4): First player moves: First player wins! 29 / 76
30 Interesting math from boring games Interesting problem: How short or long can a game be, and how many lines of play are there? 30 / 76
31 Interesting math from boring games Interesting problem: How short or long can a game be, and how many lines of play are there? Theorem (Propp): The game ends after exactly n 1 moves, and the number of lines of play is n n 2 (grade 1). 31 / 76
32 Interesting math from boring games Interesting problem: How short or long can a game be, and how many lines of play are there? Theorem (Propp): The game ends after exactly n 1 moves, and the number of lines of play is n n 2 (grade 1). Question: Is there a bijection between lines of play and spanning trees of the complete graph K n? 32 / 76
33 Interesting math from boring games Interesting problem: How short or long can a game be, and how many lines of play are there? Theorem (Propp): The game ends after exactly n 1 moves, and the number of lines of play is n n 2 (grade 1). Question: Is there a bijection between lines of play and spanning trees of the complete graph K n? Question: Is there an exact formula for the number of end-positions of the game? 33 / 76
34 More interesting math from boring games Boring game: A chip firing game 34 / 76
35 More interesting math from boring games Boring game: A chip firing game Initial state: A row of 2n + 1 bins. The middle bin has 2 chips. Every other bin contains 1 chip, except for the two outermost bins, which are empty / 76
36 More interesting math from boring games Boring game: A chip firing game Initial state: A row of 2n + 1 bins. The middle bin has 2 chips. Every other bin contains 1 chip, except for the two outermost bins, which are empty Legal move: If the kth bin contains two or more chips, you may remove two chips from it, putting one of them in the k 1st bin and the other in the k + 1st bin ( firing the kth bin). 36 / 76
37 More interesting math from boring games Boring game: A chip firing game Initial state: A row of 2n + 1 bins. The middle bin has 2 chips. Every other bin contains 1 chip, except for the two outermost bins, which are empty Legal move: If the kth bin contains two or more chips, you may remove two chips from it, putting one of them in the k 1st bin and the other in the k + 1st bin ( firing the kth bin). Winner: The last player to make a legal move wins. 37 / 76
38 More interesting math from boring games A sample game with n = 2 (bins that are about to get fired are bold): 38 / 76
39 More interesting math from boring games A sample game with n = 2 (bins that are about to get fired are bold): Initial position: / 76
40 More interesting math from boring games A sample game with n = 2 (bins that are about to get fired are bold): Initial position: First player moves: / 76
41 More interesting math from boring games A sample game with n = 2 (bins that are about to get fired are bold): Initial position: First player moves: Second player moves: / 76
42 More interesting math from boring games A sample game with n = 2 (bins that are about to get fired are bold): Initial position: First player moves: Second player moves: First player moves: / 76
43 More interesting math from boring games A sample game with n = 2 (bins that are about to get fired are bold): Initial position: First player moves: Second player moves: First player moves: Second player moves: / 76
44 More interesting math from boring games A sample game with n = 2 (bins that are about to get fired are bold): Initial position: First player moves: Second player moves: First player moves: Second player moves: Second player wins! / 76
45 Interesting math from boring games Interesting problem: How short or long can a game be, and how many lines of play are there? 45 / 76
46 Interesting math from boring games Interesting problem: How short or long can a game be, and how many lines of play are there? Theorem (Propp): The game ends after exactly n 2 moves, and the number of lines of play is (n 2 )!/((1)(2) (n))((2)(3) (n + 1)) ((n)(n + 1) (2n 1)) (grade 2). Note that this number is highly composite: it is on the order of exp n 2, but all its prime factors are less than n / 76
47 Interesting math from boring games Interesting problem: How short or long can a game be, and how many lines of play are there? Theorem (Propp): The game ends after exactly n 2 moves, and the number of lines of play is (n 2 )!/((1)(2) (n))((2)(3) (n + 1)) ((n)(n + 1) (2n 1)) (grade 2). Note that this number is highly composite: it is on the order of exp n 2, but all its prime factors are less than n 2. Enumerative combinatorics guild secret: These games are just square standard Young tableaux in disguise. There s a general theory of chip-firing games, and it predicts that under very general conditions, the number of moves is predestined. But nobody has looked at the number of lines of play. 47 / 76
48 Even more interesting math from boring games Boring game: Another chip firing game 48 / 76
49 Even more interesting math from boring games Boring game: Another chip firing game Initial state: A row of 2 floor(n/2) + 1 bins. The middle bin has n chips. All other bins are empty n / 76
50 Even more interesting math from boring games Boring game: Another chip firing game Initial state: A row of 2 floor(n/2) + 1 bins. The middle bin has n chips. All other bins are empty n Legal move: If the kth bin contains two or more chips, you may remove two chips from it, putting one of them in the k 1st bin and the other in the k + 1st bin ( firing the kth bin). 50 / 76
51 Even more interesting math from boring games Boring game: Another chip firing game Initial state: A row of 2 floor(n/2) + 1 bins. The middle bin has n chips. All other bins are empty n Legal move: If the kth bin contains two or more chips, you may remove two chips from it, putting one of them in the k 1st bin and the other in the k + 1st bin ( firing the kth bin). Winner: The last player to make a legal move wins. 51 / 76
52 More interesting math from boring games A sample game with n = 4 (bins that get fired are bold): 52 / 76
53 More interesting math from boring games A sample game with n = 4 (bins that get fired are bold): Initial position: / 76
54 More interesting math from boring games A sample game with n = 4 (bins that get fired are bold): Initial position: First player moves: / 76
55 More interesting math from boring games A sample game with n = 4 (bins that get fired are bold): Initial position: First player moves: Second player moves: / 76
56 More interesting math from boring games A sample game with n = 4 (bins that get fired are bold): Initial position: First player moves: Second player moves: First player moves: / 76
57 More interesting math from boring games A sample game with n = 4 (bins that get fired are bold): Initial position: First player moves: Second player moves: First player moves: Second player moves: / 76
58 More interesting math from boring games A sample game with n = 4 (bins that get fired are bold): Initial position: First player moves: Second player moves: First player moves: Second player moves: First player moves: / 76
59 More interesting math from boring games A sample game with n = 4 (bins that get fired are bold): Initial position: First player moves: Second player moves: First player moves: Second player moves: First player moves: First player wins! / 76
60 Interesting math from boring games Interesting problem: How short or long can a game be, and how many lines of play are there? 60 / 76
61 Interesting math from boring games Interesting problem: How short or long can a game be, and how many lines of play are there? Theorem (Anderson-Lovász-Shor-Spencer-Tardos-Winograd): The game ends after m(m + 1)(2m + 1)/6 moves, where m=floor(n/2). 61 / 76
62 Interesting math from boring games Interesting problem: How short or long can a game be, and how many lines of play are there? Theorem (Anderson-Lovász-Shor-Spencer-Tardos-Winograd): The game ends after m(m + 1)(2m + 1)/6 moves, where m=floor(n/2). But: how many lines of play? 62 / 76
63 Interesting math from boring games Interesting problem: How short or long can a game be, and how many lines of play are there? Theorem (Anderson-Lovász-Shor-Spencer-Tardos-Winograd): The game ends after m(m + 1)(2m + 1)/6 moves, where m=floor(n/2). But: how many lines of play? The resulting sequence 1, 1, 1, 2, 4, 252, 2304, , ,... just got added to the OEIS a few days ago. 63 / 76
64 Welcome to the third grade! (?) What makes this sequence so intriguing is a combination of properties: (1) It s of the THIRD grade. (2) The terms have lots of small prime factors I can t explain. 252 = = = = times a big prime the next term = times a big prime the next term = times a product of three big primes 64 / 76
65 Welcome to the third grade! (?) Does the product of all (or some) of the small primes have some combinatorial meaning? This is the first hint I ve seen that there could be nontrivial results of grade / 76
66 Welcome to the third grade! (?) Does the product of all (or some) of the small primes have some combinatorial meaning? This is the first hint I ve seen that there could be nontrivial results of grade 3. Statistical physicists have found exact solutions to lots of lattice models (such as tiling models) in 1D and 2D, but not 3D. Perhaps we could break this dimension barrier if we knew more about grade 3 combinatorics. 66 / 76
67 Ending on a sweet note My favorite boring game is Impartial Cut-Cake, also called the Chocolate Bar Game. 67 / 76
68 Ending on a sweet note My favorite boring game is Impartial Cut-Cake, also called the Chocolate Bar Game. Initial state: An a-by-b rectangle is scored into a b unit squares 68 / 76
69 Ending on a sweet note My favorite boring game is Impartial Cut-Cake, also called the Chocolate Bar Game. Initial state: An a-by-b rectangle is scored into a b unit squares Positions: Collections of such rectangles 69 / 76
70 Ending on a sweet note My favorite boring game is Impartial Cut-Cake, also called the Chocolate Bar Game. Initial state: An a-by-b rectangle is scored into a b unit squares Positions: Collections of such rectangles Moves: Break such a rectangle into two rectangles along a score-line 70 / 76
71 Ending on a sweet note My favorite boring game is Impartial Cut-Cake, also called the Chocolate Bar Game. Initial state: An a-by-b rectangle is scored into a b unit squares Positions: Collections of such rectangles Moves: Break such a rectangle into two rectangles along a score-line Winner: Last player to make a legal move That is, the player who uses up all remaining score-lines wins. 71 / 76
72 Ending on a sweet note Example: From a player can move to or or In the first case, we use up 3 lengths of score-line; in the second and third cases, we use up 2 lengths of score-line. 72 / 76
73 Ignore this slide! Claim: If a or b is even, a-by-b is a win for the first player. Proof: Go first, and divide the rectangle into two identical pieces; thereafter, mimic your opponent s move. You can abuse yourself by working out a winning strategy for the second player when a and b are both odd. 73 / 76
74 Ignore this slide! Claim: If a or b is even, a-by-b is a win for the first player. Proof: Go first, and divide the rectangle into two identical pieces; thereafter, mimic your opponent s move. You can abuse yourself by working out a winning strategy for the second player when a and b are both odd. But you shouldn t, because it turns out that EVERY move is a win! To see why, it s best to ignore everything I ve told you in the last two minutes, because it was all designed to distract you from what s really going on. 74 / 76
75 But don t ignore this one There s a one sentence-proof that the duration of the game is independent of the moves that are made. You might want to work on this during the break. If you can t solve the problem, ask a child. As for the number of lines of play: High school students Caleb Ji, Robin Park, and Angela Song, under the supervision of Tanya Khovanova, and with assistance from Pavel Etingof, have studied the case of a 2-by-n bar. Letting B n denote the number of lines of play, they showed that if p is 2, 5, or a prime that is congruent to 1 or 4 mod 5, then B n is divisible by p for all sufficiently large n, whereas if p is ANY other prime, then this is NOT the case. 75 / 76
76 Further reading R.J. Anderson, L. Lovász, P.W. Shor, J. Spencer, E. Tardos, and S. Winograd, Disks, balls, and walls: analysis of a combinatorial game, American Mathematical Monthly, Volume 96, Number 6, June-July 1989, pages Grant Cairns and Korrakot Chartarrayawadee, Brussels Sprouts and Cloves, Mathematics Magazine, Volume 80, Number 1, February 2007, pages 46 58(13). Richard Guy, She Loves Me, She Loves Me Not: Relatives of two games of Lenstra, 1.pdf. Gil Kalai, Enumeration of Q-acyclic simplicial complexes, Israel J. Math. 45 (1983), no. 4, James Propp, Games of no strategy, in preparation. Slides at 76 / 76
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