very hasty notes about math by ~pgadey rss
String Figures
2017-11-27

Pictures:

[2016-05-16] I put together a small photo setup for taking shots of string figures. It is still a work in progress, but will be helpful in making a shots of figures. Eventually I'll put together an album of the figures that I know. The setup consists of putting my laptop on my desk chair, with a dark shirt over the back of the chair for a backdrop. The lamp points towards the black backdrop and lights up the figure. The lighting is a bit too strong and the photos come out overexposed.

Brokhos Brokhos via the pindiki extension

[2016-06-11] The figure below is a slight reworking of the Bolivian figure Footprint from SFM March 2002. The presentation of the Storer code together with the water colouring painting of the final figure is appealing. The colouring of the figure is supposed to mimic the effect of a tie dye string. One can trace through the crossings easily because of the colour.

[2017-11-27]

Bibliography:

  1. Andersen--maori
  2. Ashley--Book-of-Knots
  3. Balducci--Toba-Taksek
  4. bishop-museum
  5. Braunstein--Figuras-y-juegos-de-hildo-de-los-indios-Maka
  6. Braunstein--Las-figuras-de-hilo-del-Gran-Chaco-I
  7. Braunstein--Las-figuras-de-hilo-del-Gran-Chaco-II
  8. Braunstein--Las-figuras-de-hilo-del-Gran-Chaco-III
  9. Braunstein--Las-figuras-de-hilo-del-Gran-Chaco-IV
  10. Compton-String-Figures-from-New-Caledonia-and-the-Loyalty-Islands
  11. Cunnington-String-Figures-and-Tricks-from-Central-Africa
  12. Davidson-Aboriginal-Australian-String-Figures
  13. Davidson--Virginia-Indians
  14. Devonshire-Rarotongan-String-Figures
  15. Dickey--Hawaii
  16. Emerson--Hawaiian-String-Games
  17. Evans-Pritchard-Zande-String-Figures
  18. fifth-thule-expedition
  19. Gibbs-Sihlabela-String_Figures
  20. Griffith--Gold-Coast-String-Games
  21. Gryski--Cats-Cradle-Red
  22. Gryski--Many-Stars-Blue
  23. Gryski--Super-String-Games-White
  24. Haddon--A-Few-American-String-Figures-and-Tricks
  25. Haddon--Artists-in-String
  26. Haddon-Hornbostel-Tessmann--Chama-String-Games
  27. Haddon--String-Figures-from-South-Africa
  28. Handy--Marquesas
  29. Holbrook--String-Stories
  30. Hornell--Fiji
  31. Hornell-Rosser--String-Figures-from-British-New-Guinea
  32. Hornell--String-Figures-from-Anglo-Egyptian-Sudan
  33. Hornell--String-Figures-from-Sierra-Leone-Liberia-and-Zanzibar
  34. Jenness--Papuan
  35. Lane--StringFiguresProtest
  36. Lantis--Nunivak-Eskimo
  37. Lietzmann--Visual-Topology
  38. Lutz--Patomana
  39. Martinez-Crovetto--Distribucion
  40. Mary-Rousseliere--Arviligjuarmiut
  41. Maude--gilbert
  42. Maude--puka-puka
  43. Maude--solomon
  44. Maude--tikopia
  45. Maude--tuamotus
  46. Maude-Wedgwood-String-Figures-from-Northern-New-Guinea
  47. Mcilwraith--Bella-Coola
  48. Moez--Mathematics-and-String
  49. Noble--Shorthand
  50. Parkinson--Yoruba-String-Figures
  51. Paterson--Eskimo-Figures
  52. Rivers-Haddon--Recording-String-Figures
  53. Stewart--Cats-Cradle-Calculus-Challenge
  54. Storer--StringFigures
  55. Taylor--Pull-the-Other-One
  56. Tylor--Remarks-on-the-Geographical-Distribution-of-Games
  57. Victor--Angmagssalik
  58. Walker-Cats-Cradle-and-Other-Topologies
  59. Wedgewood-Schapera--Bechuanaland-Protectorate


Hex Theory
2016-09-11

There is a lot of informaition on the Hex Theory Page.

It is really amazing!

Hex Opening Theory

Hex and Topology

Hex and Percolation Criticality

[2017-03-04] At long last, the first batch of percolation criticality numbers are in!

 # one hundred trials on a 7x7 board with coin-toss colouring
 :! python HexMonteCarlo.py 

 [[  1.   3.   4.   4.   5.   8.   5.]
  [  1.   0.   7.   7.  10.   7.   7.]
  [  4.   6.   4.  10.  10.  12.   7.]
  [  4.   6.   9.   9.   4.   8.   8.]
  [  3.  12.  12.  12.   8.   6.   2.]
  [  4.   9.   7.   7.   9.   5.   4.]
  [ 15.   6.   5.   5.   5.   0.   2.]]


  # ten thousand trials on a 7x7 board

  Simulating 7x7 with N=10000 trials
  [[  128.   232.   363.   478.   582.   704.  1065.]
   [  236.   500.   624.   846.  1001.  1077.   806.]
   [  358.   637.   899.  1076.  1120.   982.   554.]
   [  471.   817.  1114.  1165.  1064.   806.   466.]
   [  572.  1028.  1081.  1086.   908.   636.   350.]
   [  767.  1005.   990.   833.   605.   424.   250.]
   [ 1014.   751.   555.   448.   323.   226.   116.]]

This shows that percolation criticality follows the main diagonal. It is minimized in the acute corners of the board (A. Beck)

Observe that percolation criticality is connected with "draws". If x is critical then x=1 is a Player 1 win and x=2 is Player 2 win. Thus, the position is a "draw" unless we know the value of x.

Hex and Monte-Carlo

[2017-11-11]

The code for the simulator is now available here

[2017-11-27]

Simulator can see when the first player has won.

   Given board position:
   [[ 0.  1.  0.  0.  0.]
    [ 0.  1.  0.  0.  0.]
    [ 0.  1.  1.  0.  0.]
    [ 0.  1.  0.  0.  0.]
    [ 0.  1.  0.  0.  0.]]
   Simulating 5x5 with N=500 trials using bernoulli sampling
   [[ 500.    0.  500.  500.  500.]
    [ 500.    0.  500.  500.  500.]
    [ 500.    0.    0.  500.  500.]
    [ 500.    0.  500.  500.  500.]
    [ 500.    0.  500.  500.  500.]]

Simulator can see when the second player has won.

   Given board position:
   [[ 0.  0.  0.  0.  0.]
    [ 2.  2.  2.  2.  2.]
    [ 0.  0.  1.  0.  0.]
    [ 0.  0.  0.  0.  0.]
    [ 0.  0.  0.  0.  0.]]
   Simulating 5x5 with N=500 trials using bernoulli sampling
   [[ 0.  0.  0.  0.  0.]
    [ 0.  0.  0.  0.  0.]
    [ 0.  0.  0.  0.  0.]
    [ 0.  0.  0.  0.  0.]
    [ 0.  0.  0.  0.  0.]]

Yay! -- The simulator can see a necessary move. The board template has Player 2 with an almost complete crossing.

   Given the board position:
   [[ 0.  0.  0.  0.  0.]
    [ 2.  2.  0.  2.  2.]
    [ 0.  0.  1.  0.  0.]
    [ 0.  0.  0.  0.  0.]
    [ 0.  0.  0.  0.  0.]]
   Simulating 5x5 with N=500 trials using bernoulli sampling
   [[ 139.  138.  182.  184.  147.]
    [   0.    0.  277.    0.    0.]
    [ 154.  138.    0.  170.  156.]
    [ 168.  146.  178.  156.  135.]
    [ 150.  162.  143.  137.  169.]]

The simulator can solve Hayward Puzzle 1 Note: This puzzle requires a large number (N~10000) trials to solve. Lower numbers of trials tend give incorrect solutions. Notice that tere are several cells close the P(win|x=1) maximizer: [4, 3, 4415.0]

  4390.  4314.
  4415.  4146.

  Given board position:
  [[ 0.  0.  0.  0.  0.  2.  0.]
   [ 0.  0.  0.  0.  1.  2.  0.]
   [ 0.  2.  0.  1.  0.  0.  0.]
   [ 0.  0.  0.  0.  0.  2.  0.]
   [ 0.  0.  2.  0.  0.  2.  0.]
   [ 0.  2.  0.  0.  1.  0.  0.]
   [ 0.  0.  0.  0.  0.  0.  0.]]
  Simulating 7x7 with N=10000 trials using bernoulli sampling
  The P(win|x=1) maximizer:[4, 3, 4415.0]
  [[ 3397.  3495.  3506.  3817.  4330.     0.  3595.]
   [ 3590.  3494.  3666.  3741.     0.     0.  3622.]
   [ 3561.     0.  3773.     0.  3827.  3560.  3627.]
   [ 3514.  3597.  3588.  4390.  4314.     0.  3554.]
   [ 3587.  3642.     0.  4415.  4146.     0.  3613.]
   [ 3930.     0.  3568.  3802.     0.  3527.  3492.]
   [ 4038.  3434.  3516.  3590.  3617.  3580.  3396.]]

The simulator can solve Hayward Puzzle 2

  Given board position:
  [[ 0.  0.  0.  0.  0.  0.]
   [ 0.  0.  0.  0.  0.  0.]
   [ 0.  0.  0.  0.  0.  0.]
   [ 1.  0.  0.  0.  0.  0.]
   [ 2.  2.  0.  0.  0.  0.]
   [ 1.  0.  0.  0.  0.  0.]]
  Simulating 6x6 with N=10000 trials using bernoulli sampling
  The P(win|x=1) maximizer:[4, 2, 4758.0]
  [[ 3830.  3884.  3833.  4054.  4151.  4390.]
   [ 3847.  3846.  4083.  4290.  4460.  4235.]
   [ 3854.  3921.  4203.  4496.  4523.  4137.]
   [    0.  3882.  4493.  4628.  4441.  4154.]
   [    0.     0.  4758.  4406.  4189.  4002.]
   [    0.  4120.  4296.  4069.  3952.  3850.]]

The simulator can solve Hayward Puzzle 4

  Given board position:
  [[ 0.  1.  0.  0.  0.  0.]
   [ 0.  0.  2.  2.  2.  0.]
   [ 1.  0.  1.  0.  2.  0.]
   [ 0.  0.  2.  0.  0.  0.]
   [ 1.  0.  1.  0.  0.  0.]
   [ 0.  0.  2.  0.  0.  0.]]
  Simulating 6x6 with N=10000 trials using bernoulli sampling
  The P(win|x=1) maximizer:[3, 0, 5694.0]
  [[ 3925.     0.  3989.  3927.  3927.  4400.]
   [ 5060.  4949.     0.     0.     0.  4341.]
   [    0.  4745.     0.  4212.     0.  4351.]
   [ 5694.  4394.     0.  4323.  4283.  4119.]
   [    0.  4572.     0.  4259.  4218.  4073.]
   [ 5269.  4434.     0.  4139.  4018.  3915.]]

The simulator cannot solve Hein 1 from Politiken 1942. Hein puzzles available here.

Given board position:
[[ 0.  0.  0.  0.  0.  0.  0.  0.  0.  0.  0.]
 [ 0.  0.  0.  0.  0.  0.  0.  0.  0.  0.  0.]
 [ 0.  1.  0.  0.  0.  0.  0.  0.  0.  0.  0.]
 [ 2.  2.  2.  2.  2.  2.  2.  1.  2.  2.  2.]
 [ 0.  0.  0.  0.  0.  0.  0.  0.  0.  0.  0.]
 [ 0.  0.  0.  0.  0.  0.  0.  0.  0.  0.  0.]
 [ 0.  0.  0.  0.  0.  0.  0.  0.  0.  0.  0.]
 [ 0.  0.  0.  0.  0.  2.  2.  0.  1.  0.  0.]
 [ 0.  0.  0.  1.  0.  0.  0.  0.  0.  0.  0.]
 [ 0.  0.  0.  0.  0.  0.  0.  0.  0.  0.  0.]
 [ 0.  0.  0.  0.  0.  0.  0.  0.  0.  0.  0.]]
Simulating 11x11 with N=10000 trials using bernoulli sampling
The P(win|x=1) maximizer:[2, 7, 3176.0]

[[ 2226.  2095.  2210.  2169.  2216.  2239.  2230.  2265.  2454.  2480.  2445.]
 [ 2245.  2219.  2130.  2200.  2244.  2187.  2246.  2445.  2675.  2474.  2264.]
 [ 2257.     0.  2173.  2164.  2266.  2302.  2387.  3176.  2937.  2180.  2245.]
 [    0.     0.     0.     0.     0.     0.     0.     0.     0.     0.  0.]
 [ 2176.  2137.  2227.  2212.  2308.  2397.  3032.  3072.  2338.  2217.  2210.]
 [ 2161.  2216.  2281.  2237.  2386.  2511.  2604.  2399.  2466.  2238.  2293.]
 [ 2169.  2222.  2321.  2297.  2397.  2449.  2241.  2436.  2310.  2234.  2126.]
 [ 2212.  2278.  2198.  2376.  2505.     0.     0.  2345.     0.  2233.  2194.]
 [ 2223.  2179.  2331.     0.  2262.  2257.  2259.  2302.  2303.  2271.  2182.]
 [ 2232.  2302.  2351.  2297.  2372.  2271.  2237.  2211.  2306.  2295.  2216.]
 [ 2316.  2303.  2236.  2288.  2226.  2236.  2207.  2276.  2266.  2251.  2230.]]

Jack van Rijswijck's Probabilistic Y-Reduction

See Jack's paper: Search and Evaluation in Hex

Conjecturing

   Suppose x and y are neighbours. 
      Q. Is it possible that P(x) = P(y)?
      Q. If x is a winning opening and P(x) < P(y) then is y also a winning opening? 

   Suppose x, y, and z are neighbours.
     Q. Do we always get a better "pair of neighbours"?

   Can P(x) gradient ascent have periodic points?

   What "should" the P(x) flow lines look like?

   If x_0 is a winning opening and x_n is its forward orbit under P(x)-ascent
     then do we get x_n is a winning opening?

   The Role of Central Symmetry P(x) = P(-x)
adding
2016-12-08

Adding

It is often handy to add numbers.

For example, 2+2=4.

Bongard Problems
2016-08-01-1 at 13h

"Given examples of x such that P(x) and examples such that not P(x), what is P?"

The puzzle above is the proto-type for all Bongard type puzzles. I'm a fancier of these kind of puzzles, because I think that they really get at an important skill that doesn't get practiced much: inductive thinking. One has to dream up a solution to the problems, there is no formal method for attacking them.

The original canon of Bongard's own Bongard problems and some additional problems, copied from the appendix in Harry Foundalis' thesis, are available without solutions here.

There is also an easy to print edition with six problems per page available: here.

For more information see:

  1. Wikipedia for historical information about Bongard problems.

  2. Linhares, A. (2000). A glimpse at the metaphysics of Bongard problems. Artificial Intelligence, Volume 121, Issue 1-2, pp. 251–270. Available here (mirror). This is a nice paper. It uses the concrete, printed on the page, sense of certain well chosen Bongard problems to talk about metaphysics. It reminds me of the good in Hofstader.

Low Complexity Art
2016-04-24

This is a working note about Jürgen Schmidhuber's notion of low-complexity art (wp).

Consider a piece of music, or a text, and call it X. Now -- Fix a programming language, and call it P. We say that the complexity of X relative to P is the length of the shortest program written in P that outputs X. We will call the length of that program K(X,P). This notion is called the Kolmogorov complexity of X measured relative to the programming language P.

It is interesting to note that the language P is irrelevant to definition of K(X,P). Suppose that you wanted work in some other language Q. Now you want to evaluate K(X,Q). Suppose it is possible to implement a Q compiler in P using at most N characters. Over all strings X we know that K(X,Q) is at most K(X,P) + N. Therefore, switching languages only costs a fixed amount of length. And so, we can really ingore the language we work in and think of K(X) in terms of one idealized language.

Kolmogorov complexity expresses a really fundamental idea about information. For example, K(X,P) is not computable. Proof?

Speed Prior (wp) is a nice alternative to K(X,P) since Speed Prior is computable.

Freudenthal Problems
2016-04-23-6 at 01h

"Given a message, what does it mean?"

See: Wikipedia or Three Compositions for Nash Camp

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