# Three Bones

In 2015, Krzysztof P. Jasiutowicz put this image on Google+ with the comment:

Good but untranslatable (probably) đ

Here is my attempt to prove him wrong about the untranslatability:

# Seven Triple Double Checks

I have recently enhanced the chess search engine I write in my spare time with double-check detection. To test it, I made the engine ingest 5.82 million unique over-the-board games from the 15 Million chess games database kept up to date until 2015. 0.85% of these games turned out to contain double checks. More precisely, there are:

• 48261 games with one double check,
• 1184 games with two double checks,
• 76 games with three double checks,
• 3 games with four double checks,
• 3 games with five double checks.

I naively expected that the games with many double checks would shine with mate combinations, yet as a rule they just end with perpetual check, plain and double in turn. Much more spectacular and much less frequent are games where double checks occur in a row. It took my engine 0.75 and 0.55 seconds, respectively, to answer the queries

`???++ K?? ???++`

and

`???++ K?? ???##`

with 129 pairs and 6 triplets of consecutive double checks. I show the triplets below. They all follow the same pattern where rook blocks and discovers the diagonals of a bishop pair. Incidentally, among chess problems, a mate in 13 by StojniÄ and BabiÄ (The Problemist, 2004) sports as many as 13 consecutive double checks based on the same idea.

Holger NormanâHansen vs Erik Andersen
Copenhagen, 1930

1. e4 e5 2. Nf3 Nf6 3. Nxe5 d6 4. Nf3 Nxe4 5. d4 d5 6. Bd3 Bg4 7. O-O Bd6 8. c4 O-O 9. cxd5 f5 10. Nc3 Nd7 11. h3 Bh5 12. Nxe4 fxe4 13. Bxe4 Nf6 14. Bf5 Kh8 15. Be6 Ne4 16. g4 Bg6 17. Kg2 Qf6 18. Be3 Rae8 19. h4

19. âŠ Rxe6 20. dxe6 Nc3 21. bxc3 Be4 22. Kh3 Qxf3+ 23. Qxf3 Rxf3+ 24. Kg2 Rg3++ 25. Kh2 Rg2++ 26. Kh1 Rh2++ 27. Kg1 Rh1# 0-1

Steven Avramidis vs Georgios Alexopoulos
Windsor, 1978

1.d4 Nf6 2.c4 c5 3.dxc5 e5 4.b4 a5 5.Ba3 axb4 6.Bxb4 Na6 7.Ba3 Nxc5 8.Bxc5 Bxc5 9.Nf3 e4 10.Nd4 e3 11.fxe3 d5 12.cxd5 Nxd5 13.Nc2 O-O 14.g3 Nxe3 15. Qxd8 Nxc2+ 16.Kd2 Rxd8+ 17.Kxc2 Bf5+ 18.Kb2 Bd4+ 19.Nc3 Rac8 20.Rc1

20. âŠ Rxc3 21.Rxc3 Rc8 22.Bg2 Rxc3 23.Rc1 Rc2++ 24.Kb1 Rb2++ 25.Ka1 Rb1# 0-1

Pascal TchingâSin vs Edouard Bonnet

1. e4 c5 2. Nf3 d6 3. d4 cxd4 4. Nxd4 Nf6 5. Nc3 a6 6. Be2 e5 7. Nf3 Be7 8. h3 O-O 9. O-O b5 10. Bg5 Nbd7 11. a3 Bb7 12. Bd3 Rc8 13. Re1 h6 14. Bh4 Nb6 15. Qd2 Nc4 16. Bxc4 Rxc4 17. Bxf6 Bxf6 18. Rad1 Be7 19. Qe2 Qa8 20. Nd2 Rcc8 21. Nf1 f5 22. Ng3 fxe4 23. Ncxe4 d5 24. Nd2 Bc5 25. Nf3 Rce8 26. Rf1 Qb8 27. Nh5 e4 28. Nd4 Qe5 29. c3 Bd6 30. g3 Bc8 31. Kg2 Qg5 32. g4 Qh4 33. Ng3 g6 34. Qc2 Kg7 35. Nde2 Bb7 36. Qd2 Be5 37. Qe3 Re7 38. Qb6 Ref7 39. Qe6 Bb8 40. Qb6 Be5 41. Qe3 Rf3 42. Qb6 R8f7 43. Qe6 Bb8 44. Qe8 Bc7 45. Rd4 e3 46. g5 Qxg5 47. Rg4

47. âŠ d4 48. Rxg5 Rxg3++ 49. Kh2 Rg2++ 50. Kh1 Rh2++ 0-1

Guido MĂŒssig vs Natia Engels
Germany, 2003.11.08

1. d4 Nf6 2. c4 c5 3. d5 e6 4. Nc3 exd5 5. cxd5 d6 6. e4 g6 7. f3 Bg7 8. Bg5 a6 9. a4 h6 10. Be3 O-O 11. Qd2 Kh7 12. Nge2 Nbd7 13. Ng3 Qa5 14. Be2 b5 15. O-O b4 16. Nd1 Re8 17. Bf4 Ne5 18. Ne3 Nfd7 19. Nh1 Qc7 20. Bg3 c4 21. f4 Nd3 22. Nxc4 Qxc4 23. Bxd3 Qd4+ 24. Nf2 Nc5 25. f5 Nb3 26. fxg6+ fxg6 27. Qc2 Nxa1 28. Rxa1 Qxb2 29. Qxb2 Bxb2 30. Rb1 Bc3 31. Bxd6 a5 32. Bb5 Rg8 33. g4 h5 34. h3 Bd4 35. Kg2 Ba6 36. Bc6 Raf8 37. Bxf8 Rxf8 38. Nh1 Bd3 39. Re1 b3 40. Bb5 Bc2 41. Bc4 b2 42. Ba2 Bc3 43. Rg1

43.Â âŠ Bxe4+ 44. Kg3 Rf3+ 45. Kg2 Bd4 46. Re1 Re3+ 47. Kf1 Bd3+ 48. Kf2 Re2++ 49. Kf1 Rf2++ 50. Kg1 Rf1++ 0-1

Maria Petsetidi vs Nikolaos Tepelenis
Agios Kirykos, 2010.07.15

1. e4 e6 2. b3 d5 3. Bb2 dxe4 4. Nc3 Nf6 5. Qe2 Bd7 6. Nxe4 Nxe4 7. Qxe4 Bc6 8. Qg4 Nd7 9. O-O-O Nf6 10. Qe2 Qd5 11. Nf3 Qe4 12. d4 Bd6 13. Ne5 Qxe2 14. Bxe2 Bxg2 15. Rhg1 Be4 16. c4 g6 17. d5 exd5 18. Ng4 Bf4+ 19. Ne3 Ke7 20. Ba3+ Ke6 21. Rd4 c6 22. Kd1 Be5 23. cxd5+ Nxd5 24. Nxd5 Bxd5 25. Rd2 Bxh2 26. Re1 Be5 27. Bg4+ Kf6 28. f4 Bxf4 29. Rf2 g5 30. Bb2+ Kg6 31. Rg1 f5 32. Be2 Be3 33. Rh2 h5 34. Rf1 Rh7 35. Bd3 Bf4 36. Re2 Rd8 37. Re5 Be4 38. Re6+ Kf7 39. Rf6+ Ke7 40. Re1

40. âŠ Rxd3+ 41. Kc2 Rd2++ 42. Kc1 Rc2++ 43. Kb1 Rxb2++ {missing a mate in one} 0-1

Kjell BĂžrre Grebstad vs KarlâPetter Jernberg
TromsĂž, 2010.07.31

1. d4 d5 2. c4 e6 3. Nf3 Nf6 4. Nc3 Bb4 5. Bg5 Bxc3+ 6. bxc3 Nbd7 7. e3 c6 8. Qc2 O-O 9. Bd3 Qc7 10. Bh4 Re8 11. h3 c5 12. Bg3 Qc6 13. Rb1 cxd4 14. cxd4 Ne4 15. Bh2 Ndf6 16. Ne5 Qa6 17. O-O Qa5 18. f3 Nd2 19. Rb5 Nxf3+ 20. Nxf3 Qd8 21. Ne5 a6 22. Rb2 Qe7 23. c5 g6 24. Qf2 Kg7 25. Bg3 h6 26. Qf3 Rf8 27. Rbf2 Bd7

28. Qxf6+ Qxf6 29. Rxf6 Be8 30. Nxg6 Rg8 31. Be5 fxg6 32. Rxg6++ Kh7 33. Rg7++ Kh8 34. Rh7# 1-0

The seventh confirmed occurrence of a triple double check comes from Renaud and Kahnâs book The Art of Checkmate. The initial moves of the game are lost.

Victor Place vs N.N.

1. Nxg7 Kxg7 2. d5 Bg4 3. Rxf6 Bxd1 4. Rg6++ Kh7 5. Rg7++ Kh8 6. Rh7++ Kg8 7. Rh8# 1-0

# WisĆa in Fact Likes Cracovia but Doesnât Know How to Start Talking

The relationships between fans of football clubs in Poland can be fourfold: neutrality, friendship (zgoda), enmity (kosa), or pact (ukĆad). The belief that there are two disjointÂ blocs gathered around The Great Triad (Arka, Cracovia, and Lech) and Three Kings of Great Cities (ĆlÄsk, WisĆa, and Lechia) is false. Here is the largest connected component of the graph of friendships.

The graph of enmities would be less clear. For instance, Cracovia has friendships with Tarnovia and Sandecja but Tarnovia and Sandecja are enemies. Or GKS, GĂłrnik, Ruch, and ZagĆÄbie: every two of them are enemies.

# In a triangle ABC, cot(A/2)+cot(B/2)+cot(C/2)=cot(A/2)cot(B/2)cot(C/2)

Here is my proof in Greek style, getting by without trigonometry.

# When Will Earnings in Poland Match Those in Western Europe?

They already did, 450 years ago.

Source of data: Bob Allenâs home page; indirectly also three Polish books: Ceny w Krakowie w latach 1369â1600, 1601â1795, 1796â1914. The earnings at the end of the period would appear less inflated if expressed in gold: the gold/silver price ratio was approximately 15.5 until 1872, whereas in the early 1900s, it grew to 33â40.

# Bad Philosophy: Clark’s Argument against Spectrum Inversion

Many philosophical debates are haunted by Lockeâs thought experiment of spectrum inversion: âif the idea that a violet produced in one man’s mind by his eyes were the same that a marigold produced in another man’s, and vice versaâ [1].

Austen Clark attempted to prove rigorously that spectrum inversion would be detectable by observing the subjectsâ response to colours [2, 3]. He argued as follows: Suppose that two people perceive mutually inverted spectra and agree in their judgements of visual stimuli and the relations between them. Therefore, the colour solid (i.e. the visible colours as co-ordinates in some space) must exhibit symmetry. Since this solid (represented in three-dimensional Euclidean space e.g. as the Munsell colour system [4]) is in fact asymmetric, the premise must be false.

Can you spot the fallacy? It stems from assuming that everyoneâs colour solid is identical. Clark mistook a textbook model for the real thing â a fallacy known as reification. Both the Munsell colour system and other proposed colour solids abstract away from differences in visual perception that occur among observers and even across experiments with the same observer [5â9]. One can easily conceive of a personal colour solid that remains the same (within measurement error) when mirrored or rotated. If there cannot be people who function like us with a mirrored or rotated colour solid, it is not its asymmetry that forbids their existence.

[1] Locke, John. âAn Essay Concerning Human Understandingâ, book II, ch. XXXII, par. 15. http://humanum.arts.cuhk.edu.hk/Philosophy/Locke/echu/lok0042.htm#15

[2] Clark, Austen. Spectrum Inversion and the Color Solid. Southern Journal of Philosophy, vol. 23, no. 4, Winter 1986, pp. 431â443. http://selfpace.uconn.edu/paper/CSOLID.HTM

[3] Clark, Austen. âA Theory of Sentienceâ. Oxford University Press: 2000. pp. 17â18. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.85.5641&rep=rep1&type=pdf#17

[5] Webster, Michael A.; Eriko Miyahara; Gokhan Malkoc; Vincent E. Raker. Variations in normal color vision. I. Cone-opponent axes. Journal of the Optical Society of America, vol. 17, no. 9, September 2000, pp. 1535â1544. http://www.ncbi.nlm.nih.gov/pubmed/10975363

[6] Webster, Michael A.; Eriko Miyahara; Gokhan Malkoc; Vincent E. Raker. Variations in normal color vision. II. Unique hues. Journal of the Optical Society of America, vol. 17, no. 9, September 2000, pp. 1545â1555. http://localhopf.cns.nyu.edu/events/vjclub/archive/webster2000.pdf

[7] Webster, Michael A.; Shernaaz M. Webster; Shrikant Bharadwaj; Richa Verma; Jaikishan Jaikumar; Gitanjali Madan; E. Vaithilingham. Variations in normal color vision. III. Unique hues in Indian and United States observers. Journal of the Optical Society of America, vol. 19, no. 10, October 2002, pp. 1951â1962. http://wolfweb.unr.edu/homepage/mwebster/assets/pdfs/WebsterJOSA2002.pdf

[8] Malkoc, Gokhan; Paul Kay; Michael A. Webster. Variations in normal color vision. IV. Binary hues and hue scaling. Journal of the Optical Society of America, vol. 22, no. 10, October 2005, pp. 2154â2168. http://www1.icsi.berkeley.edu/~kay/mkw.josa.2005.pdf

[9] Juricevic, Igor; Michael A. Webster. Variations in normal color vision. V. Simulations of adaptation to natural color environments. Visual Neuroscience, vol. 26, no. 1, January 2009, pp. 133â145. http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2684467/ï»ż