1 - Busy beavers defined by 4-tuples
2 - Busy beavers whose head can stand still
3 - Two-dimensional busy beavers

• Rado (1963) claimed that Sigma(2,2) = 4, but that S(2,2) was yet unknown.

• The value S(2,2) = 6 was probably set by Lin in 1963.
  See study of the winner by H. Marxen.

• The winner and some other good machines:

  A0	A1	B0	B1	sigma(M)	s(M)
  1RB	1LB	1LA	1RH	4	6
  1RB	1RH	1LB	1LA	3	6
  1RB	0LB	1LA	1RH	3	6

• Soon after the definition of the functions S and Sigma, by Rado (1962), it was conjectured that S(3,2) = 21, and Sigma(3,2) = 6.

• Lin (1963) proved this conjecture and this proof was eventually published by Lin and Rado (1965).
  See studies by Heiner Marxen of the winners for the S function, and the Sigma function.

• The winners and some other good machines:

  A0	A1	B0	B1	C0	C1	sigma(M)	s(M)
  1RB	1RH	1LB	0RC	1LC	1LA	5	21
  1RB	1RH	0LC	0RC	1LC	1LA	5	20
  1RB	1LA	0RC	1RH	1LC	0LA	5	20
  0RB	1RH	0LC	1RA	1RB	1LC	4	17
  0RB	1LC	1LA	1RB	1LB	1RH	5	16
  1RB	1RH	0RC	1RB	1LC	1LA	6	14
  1RB	1RC	1LC	1RH	1RA	0LB	6	13
  1RB	1LC	1LA	1RB	1LB	1RH	6	13
  0RB	1LC	1RC	1RB	1LA	1RH	5	13
  1RB	1RA	1LC	1RH	1RA	1LB	6	12
  1RB	1LC	1RC	1RH	1LA	0LB	6	11

• Brady (1964,1965,1966) found a machine M such that s(M) = 107 and sigma(M) = 13 (see study by H. Marxen), and conjectured that S(4,2) = 107 and Sigma(4,2) = 13.

• Brady (1974,1975) proved this conjecture, and the proof was eventually published in Brady (1983).

• Independently, Machlin and Stout (1990) published another proof of the same result, first reported by Kopp (1981) (Kopp is the maiden name of Machlin).

• The winner and some other good machines:

  A0	A1	B0	B1	C0	C1	D0	D1	sigma(M)	s(M)
  1RB	1LB	1LA	0LC	1RH	1LD	1RD	0RA	13	107
  1RB	1LD	1LC	0RB	1RA	1LA	1RH	0LC	9	97
  1RB	0RC	1LA	1RA	1RH	1RD	1LD	0LB	13	96
  1RB	1LB	0LC	0RD	1RH	1LA	1RA	0LA	6	96
  1RB	1LD	0LC	0RC	1LC	1LA	1RH	0LA	11	84
  1RB	1RH	1LC	0RD	1LA	1LB	0LC	1RD	8	83
  1RB	0RD	1LC	0LA	1RA	1LB	1RH	0RC	12	78

• Green (1964) found a machine M with sigma(M) = 17.

• Lynn (1972) found machines M and N with s(M) = 435 and sigma(N) = 22.

• Lynn, cited by Brady (1983), found in 1974 machines M and N with s(M) = 7,707 and sigma(N) = 112.

• Uwe Schult, cited by Dewdney (1984a), found, in January 1983, a machine M with s(M) = 134,467 and sigma(M) = 501. This machine was analyzed by Robinson, cited by Dewdney (1984b), and independently by Michel (1993).

• George Uhing, cited by Dewdney (1985a,b), found, in December 1984, a machine M with s(M) = 2,133,492 and sigma(M) = 1,915. This machine was analyzed by Michel (1993).

• George Uhing, cited by Brady (1988), found, in February 1986, a machine M with s(M) = 2,358,064 (and sigma(M) = 1,471). This machine was analyzed by Michel (1993). Machine 7 in Marxen's bb-list can be obtained from Uhing's one, as given by Brady (1988), by the permutation of states (A D B E) (see study by H. Marxen).

• Heiner Marxen and Jürgen Buntrock found, in August 1989, a machine M with s(M) = 11,798,826 and sigma(M) = 4,098. This machine was cited by Marxen and Buntrock (1990), and by Machlin and Stout (1990), and was analyzed by Michel (1993). See study by H. Marxen.

• Heiner Marxen and Jürgen Buntrock found, in September 1989, a machine M with s(M) = 23,554,764 (and sigma(M) = 4,097). This machine was cited by Machlin and Stout (1990), and was analyzed by Michel (1993). See study by H. Marxen. See analysis by P. Michel.

• Heiner Marxen and Jürgen Buntrock found, in September 1989, a machine M with s(M) = 47,176,870 and sigma(M) = 4,098. This machine was cited by Marxen and Buntrock (1990), and was analyzed by Michel (1993). See study by H. Marxen. See analysis by P. Michel. It is the current record holder.

• Marxen gives a list of machines M with high values of s(M) and sigma(M).

• The study of Turing machines with 5 states and 2 symbols is still going on. Marxen and Buntrock (1990), Skelet, and Hertel (2009) created programs to detect never halting machines, and manually proved that some machines, undetected by their programs, never halt. In each case, about a hundred holdouts were resisting computer and manual analyses. Daniel Briggs created a website and a forum dedicated to this study.

• Norbert Bátfai, allowing transitions where the head can stand still, found, in August 2009, a machine M with s(M) = 70,740,810 and sigma(M) = 4098. Note that this machine does not follow the current rules of the busy beaver competition.

• The record holder and some other good machines:

(Among the first eleven machines, the five starred ones are from Norbert Bátfai. The six machines without star were discovered by Marxen and Buntrock, the next two ones were by Uhing, and the last one was by Schult. Heiner Marxen says there are no other sigma values within the sigma range above).

• Green (1964) found a machine M with sigma(M) = 35.

• Lynn (1972) found a machine M with s(M) = 522 and sigma(M) = 42.

• Brady (1983) found machines M and N with s(M) = 13,488 and sigma(N) = 117.

• Uwe Schult, cited by Dewdney (1984a), found a machine M with sigma(M) = 2,075.

• Heiner Marxen and Jürgen Buntrock found, in January 1990, a machine M with s(M) = 13,122,572,797 and sigma(M) = 136,612. This machine was cited by Marxen and Buntrock (1990). See study by H. Marxen.

• Heiner Marxen and Jürgen Buntrock found, in January 1990, a machine M with s(M) = 8,690,333,381,690,951 and sigma(M) = 95,524,079. This machine was posted on the web (Google groups) on September 3, 1997. See machine 2 in Marxen's bb-list. See study by H. Marxen. See analysis by R. Munafo in his website, and in the present website.

• Heiner Marxen and Jürgen Buntrock found, in July 2000, a machine M with s(M) > 5.3 × 10⁴² and sigma(M) > 2.5 × 10²¹. This machine was posted on the web (Google groups) on August 5, 2000. See machine 3 in Marxen's bb-list, and machine k in Marxen's bb-6list. See study by H. Marxen.

• Heiner Marxen and Jürgen Buntrock found, in August 2000, a machine M with s(M) > 6.1 × 10¹¹⁹ and sigma(M) > 1.4 × 10⁶⁰. This machine was posted on the web (Google groups) on October 23, 2000. See machine o in Marxen's bb-6list. See study by H. Marxen. See analysis by P. Michel.

• Heiner Marxen and Jürgen Buntrock found, in August 2000, a machine M with s(M) > 6.1 × 10⁹²⁵ and sigma(M) > 6.4 × 10⁴⁶². This machine was posted on the web (Google groups) on October 23, 2000. See machine q in Marxen's bb-6list. See study by H. Marxen. See analyses by R. Munafo, the short one or the long one, and by P. Michel.

• Heiner Marxen and Jürgen Buntrock found, in February 2001, a machine M with s(M) > 3.0 × 10¹⁷³⁰ and sigma(M) > 1.2 × 10⁸⁶⁵. This machine was posted on the web (Google groups) on March 5, 2001. See machine r in Marxen's bb-6list. See study by H. Marxen. See analysis by P. Michel.

• Terry and Shawn Ligocki found, in November 2007, a machine M with s(M) > 8.9 × 10¹⁷⁶² and sigma(M) > 2.5 × 10⁸⁸¹. See study by H. Marxen. See analysis by P. Michel.

• Terry and Shawn Ligocki found, in December 2007, a machine M with s(M) > 2.5 × 10²⁸⁷⁹ and sigma(M) > 4.6 × 10¹⁴³⁹. See study by H. Marxen. See analysis by P. Michel.

• Pavel Kropitz found, in May 2010, a machine M with s(M) > 3.8 × 10²¹¹³² and sigma(M) > 3.1 × 10¹⁰⁵⁶⁶. See study by H. Marxen. See analysis.

• Pavel Kropitz found, in June 2010, a machine M with s(M) > 7.4 × 10³⁶⁵³⁴ and sigma(M) > 3.5 × 10¹⁸²⁶⁷. See study by H. Marxen. See analysis by P. Michel. It is the current record holder.

• Marxen gives a list of machines M with high values of s(M) and sigma(M).

• The record holder and some other good machines:

• Brady (1988) found a machine M with s(M) = 38 (see study by H. Marxen).

• This machine was found independently by Michel (2004), who gave sigma(M) = 9, and conjectured that S(2,3) = 38 and Sigma(2,3) = 9.

• Lafitte and Papazian (2007) proved this conjecture. T. and S. Ligocki (unpublished) proved this conjecture, independently.

• The winner and some other good machines:

  A0	A1	A2	B0	B1	B2	sigma(M)	s(M)
  1RB	2LB	1RH	2LA	2RB	1LB	9	38
  1RB	0LB	1RH	2LA	1RB	1RA	8	29
  0RB	2LB	1RH	1LA	1RB	1RA	6	27
  1RB	2LA	1RH	1LB	1LA	0RA	6	26
  1RB	1LA	1LB	0LA	2RA	1RH	6	26
  1RB	1LB	1RH	2LA	2RB	1LB	7	24

• Michel (2004) found machines M and N with s(M) = 40,737 and sigma(N) = 208.

• Brady found, in November 2004, a machine M with s(M) = 29,403,894 and sigma(M) = 5600 (see study by H. Marxen).

• Brady found, in December 2004, a machine M with s(M) = 92,649,163 and sigma(M) = 13,949 (see study by H. Marxen). See analysis by P. Michel.

• Myron P. Souris found, in July 2005 (M.P. Souris said: actually in 1995, but then no one seemed to care), machines M and N with s(M) = 544,884,219 and sigma(N) = 36,089 (see study of M and study of N by H. Marxen). See analysis of M, and analysis of N, by P. Michel.

• Grégory Lafitte and Christophe Papazian found, in August 2005, a machine M with s(M) = 4,939,345,068 and sigma(M) = 107,900 (see study by H. Marxen). See analysis by P. Michel.

• Grégory Lafitte and Christophe Papazian found, in September 2005, a machine M with s(M) = 987,522,842,126 and sigma(M) = 1,525,688 (see study by H. Marxen). See analysis by P. Michel.

• Grégory Lafitte and Christophe Papazian found, in April 2006, a machine M with s(M) = 4,144,465,135,614 and sigma(M) = 2,950,149 (see study by H. Marxen). See analysis by P. Michel.

• Terry and Shawn Ligocki found, in August 2006, a machine M with s(M) = 4,345,166,620,336,565 and sigma(M) = 95,524,079 (see study by H. Marxen). See analysis.

• Terry and Shawn Ligocki found, in November 2007, a machine M with s(M) = 119,112,334,170,342,540 and sigma(M) = 374,676,383 (see study by H. Marxen). See analysis by P. Michel. It is the current record holder.

• Brady gives a list of machines M with high values of s(M).

• The record holder and some other good machines:

• Terry and Shawn Ligocki found, in April 2005, a machine M with s(M) = 250,096,776 and sigma(M) = 15,008 (see study by H. Marxen).

• This machine was superseded by the machines with 3 states and 3 symbols found in July 2005 by Myron P. Souris.

• Terry and Shawn Ligocki found, in October 2007, a machine M with s(M) > 1.5 × 10¹⁴²⁶ and sigma(M) > 1.1 × 10⁷¹³ (see study by H. Marxen).

• Terry and Shawn Ligocki found successively, in November 2007, machines M with
  • s(M) > 7.7 × 10¹⁶¹⁸ and sigma(M) > 1.6 × 10⁸⁰⁹ (see study by H. Marxen),
  • s(M) > 3.7 × 10¹⁹⁷³ and sigma(M) > 8.0 × 10⁹⁸⁶ (see study by H. Marxen),
  • s(M) > 3.9 × 10⁷⁷²¹ and sigma(M) > 4.0 × 10³⁸⁶⁰ (see study by H. Marxen),
  • s(M) > 3.9 × 10⁹¹²² and sigma(M) > 2.5 × 10⁴⁵⁶¹ (see study by H. Marxen).

• Terry and Shawn Ligocki found successively, in December 2007, machines M with
  • s(M) > 7.9 × 10⁹⁸⁶³ and sigma(M) > 8.9 × 10⁴⁹³¹ (see study by H. Marxen),
  • s(M) > 5.3 × 10¹²⁰⁶⁸ and sigma(M) > 4.2 × 10⁶⁰³⁴ (see study by H. Marxen).

• Terry and Shawn Ligocki found, in January 2008, a machine M with s(M) > 1.0 × 10¹⁴⁰⁷² and sigma(M) > 1.3 × 10⁷⁰³⁶ (see study by H. Marxen). It is the current record holder.

• The record holder and the past record holders:

• Brady (1988) found a machine M with s(M) = 7,195.

• This machine was found independently and analyzed by Michel (2004), who gave sigma(M) = 90. See study by H. Marxen. See analysis by P. Michel.

• Terry and Shawn Ligocki found, in February 2005, a machine M with s(M) = 3,932,964 and sigma(M) = 2,050. See study by H. Marxen. See analysis by P. Michel. It is the current record holder. There is no machine between the first two ones (Ligocki, Brady). There is no machine such that 3,932,964 < s(M) < 200,000,000 (Ligocki, September 2005).

• The record holder and some other good machines:

  A0	A1	A2	A3	B0	B1	B2	B3	sigma(M)	s(M)
  1RB	2LA	1RA	1RA	1LB	1LA	3RB	1RH	2,050	3,932,964
  1RB	3LA	1LA	1RA	2LA	1RH	3RA	3RB	90	7,195
  1RB	3LA	1LA	1RA	2LA	1RH	3LA	3RB	84	6,445
  1RB	3LA	1LA	1RA	2LA	1RH	2RA	3RB	84	6,445
  1RB	2RB	3LA	2RA	1LA	3RB	1RH	1LB	60	2,351

• Terry and Shawn Ligocki found, in April 2005, a machine M with s(M) = 262,759,288 and sigma(M) = 17,323 (see study by H. Marxen).

• This machine was superseded by the machines with 3 states and 3 symbols found in July 2005 by Myron P. Souris.

• Terry and Shawn Ligocki found, in September 2007, a machine M with s(M) > 5.7 × 10⁵² and sigma(M) > 2.4 × 10²⁶ (see study by H. Marxen),

• Terry and Shawn Ligocki found successively, in October 2007, machines M with
  • s(M) > 4.3 × 10²⁸¹ and sigma(M) > 6.0 × 10¹⁴⁰ (see study by H. Marxen),
  • s(M) > 7.6 × 10⁸⁶⁸ and sigma(M) > 4.6 × 10⁴³⁴ (see study by H. Marxen),
  • s(M) > 3.1 × 10¹²⁵⁶ and sigma(M) > 2.1 × 10⁶²⁸ (see study by H. Marxen).

• Terry and Shawn Ligocki found successively, in November 2007, machines M with
  • s(M) > 8.4 × 10²⁶⁰¹ and sigma(M) > 1.7 × 10¹³⁰¹ (see study by H. Marxen),
  • s(M) > 3.4 × 10⁴⁷¹⁰ and sigma(M) > 1.4 × 10²³⁵⁵ (see study by H. Marxen),
  • s(M) > 5.9 × 10⁴⁷⁴⁴ and sigma(M) > 2.2 × 10²³⁷² (see study by H. Marxen).

• Terry and Shawn Ligocki found, in December 2007, a machine M with s(M) > 5.2 × 10¹³⁰³⁶ and sigma(M) > 3.7 × 10⁶⁵¹⁸ (see study by H. Marxen). It is the current record holder.

• The record holder and the past record holders:

• Terry and Shawn Ligocki found, in February 2005, machines M and N with s(M) = 16,268,767 and sigma(N) = 4,099 (see study by H. Marxen).

• Terry and Shawn Ligocki found, in April 2005, a machine M with s(M) = 148,304,214 and sigma(M) = 11,120 (see study by H. Marxen).

• Grégory Lafitte and Christophe Papazian found, in August 2005, a machine M with s(M) = 8,619,024,596 and sigma(M) = 90,604 (see study by H. Marxen).

• Grégory Lafitte and Christophe Papazian found, in September 2005, a machine M with sigma(M) = 97,104 (and s(M) = 7,543,673,517) (see study by H. Marxen).

• Grégory Lafitte and Christophe Papazian found, in October 2005, a machine M with s(M) = 233,431,192,481 and sigma(M) = 458,357 (see study by H. Marxen).

• Grégory Lafitte and Christophe Papazian found, in October 2005, a machine M with s(M) = 912,594,733,606 and sigma(M) = 1,957,771 (see study by H. Marxen). See analysis by P. Michel.

• Grégory Lafitte and Christophe Papazian found, in December 2005, a machine M with s(M) = 924,180,005,181 (and sigma(M) = 1,137,477) (see study by H. Marxen). See analysis by P. Michel.

• Grégory Lafitte and Christophe Papazian found, in May 2006, a machine M with s(M) = 3,793,261,759,791 and sigma(M) = 2,576,467 (see study by H. Marxen). See analysis by P. Michel.

• Grégory Lafitte and Christophe Papazian found, in June 2006, a machine M with s(M) = 14,103,258,269,249 and sigma(M) = 4,848,239 (see study by H. Marxen). See analysis by P. Michel.

• Grégory Lafitte and Christophe Papazian found, in July 2006, a machine M with s(M) = 26,375,397,569,930 (and sigma(M) = 143) (see study by H. Marxen). See comments

• Terry and Shawn Ligocki found, in August 2006, a machine M with s(M) = 7,069,449,877,176,007,352,687 and sigma(M) = 172,312,766,455 (see study by H. Marxen). See analysis.

• Terry and Shawn Ligocki found, in October 2007, a machine M with s(M) > 5.2 × 10⁶¹ and sigma(M) > 9.3 × 10³⁰ (see study by H. Marxen).

• Terry and Shawn Ligocki found, in October 2007, two machines M and N with s(M) = s(N) > 1.6 × 10²¹¹ and sigma(M) = sigma(N) > 5.2 × 10¹⁰⁵ (see study of M, and study of N by H. Marxen).

• Terry and Shawn Ligocki found, in November 2007, a machine M with s(M) > 1.9 × 10⁷⁰⁴ and sigma(M) > 1.7 × 10³⁵² (see study by H. Marxen). See analysis by P. Michel. It is the current record holder.

• The record holder and some other good machines:

• Previous record holders and some other good machines:

• Terry and Shawn Ligocki found, in February 2005, machines M and N with s(M) = 98,364,599 and sigma(N) = 10,574 (see study by H. Marxen).

• Terry and Shawn Ligocki found, in April 2005, a machine M with s(M) = 493,600,387 and sigma(M) = 15,828 (see study by H. Marxen).

• This machine was superseded by the machine with 2 states and 5 symbols found in August 2005 by Grégory Lafitte and Christophe Papazian.

• Terry and Shawn Ligocki found, in September 2007, a machine M with s(M) > 2.3 × 10⁵⁴ and sigma(M) > 1.9 × 10²⁷ (see study by H. Marxen).

• This machine was superseded by the machine with 2 states and 5 symbols found in October 2007 by Terry and Shawn Ligocki.

• Terry and Shawn Ligocki found successively, in November 2007, machines M with
  • s(M) > 4.9 × 10¹⁶⁴³ and sigma(M) > 8.6 × 10⁸²¹ (see study by H. Marxen),
  • s(M) > 2.5 × 10⁹⁸⁶³ and sigma(M) > 6.9 × 10⁴⁹³¹ (see study by H. Marxen).

• Terry and Shawn Ligocki found, in January 2008, a machine M with s(M) > 2.4 × 10⁹⁸⁶⁶ and sigma(M) > 1.9 × 10⁴⁹³³ (see study by H. Marxen). It is the current record holder.

• The record holder and the past record holders:

• Rado (1962) defined S(n) and Sigma(n), that are denoted in these pages by S(n,2) and Sigma(n,2), and showed that they are non-computable functions.
  He proved that
  

  S(n) < (n+1) Sigma(5n)  2^Sigma(5n).
  

• Julstrom (1992) proved that
  

  S(n) < Sigma(20n).

• Wang and Xu (1995) proved that
  

  S(n) < Sigma(10n).

• Yang, Ding and Xu (1997) proved that
  

  S(n) < Sigma(8n),

  and that there is a constant c such that
  

  S(n) < Sigma(3n+c).

• Ben-Amram, Julstrom and Zwick (1996) proved that

  S(n) < Sigma(3n+6),

  and

  S(n) < (2n-1) Sigma(3n+3).

• Ben-Amram and Petersen (2002) proved that there is a constant c such that

  S(n) < Sigma(n + 8n/log₂n + c).

• Ben-Amram A.M., Julstrom B.A. and Zwick U. (1996)
  A note on busy beavers and other creatures
  Mathematical Systems Theory 29 (4), July-August 1996, 375-386.

• Ben-Amram A.M. and Petersen H. (2002)
  Improved bounds for functions related to busy beavers
  Theory of Computing Systems 35 (1), January-February 2002, 1-11.

• Boolos G.S., Burgess J.P. and Jeffrey R.C. (2002)
  Computability and Logic, 4th Ed., Cambridge, 2002.
• Brady A.H. (1964)
  Solutions to restricted cases of the halting problem
  Ph.D. Thesis, Oregon State University, Corvallis, December 1964.

• Brady A.H. (1965)
  Solutions of restricted cases of the halting problem used to determine particular values of a non-computable function (abstract)
  Notices of the AMS 12 (4), June 1965, 476-477.

• Brady A.H. (1966)
  The conjectured highest scoring machines for Rado's Sigma(k) for the value k = 4
  IEEE Transactions on Electronic Computers, EC-15, October 1966, 802-803.

• Brady A.H. (1974)
  UNSCC Technical Report 11-74-1, November 1974.

• Brady A.H. (1975)
  The solution to Rado's busy beaver game is now decided for k = 4 (abstract)
  Notices of the AMS 22 (1), January 1975, A-25.

• Brady A.H. (1983)
  The determination of the value of Rado's noncomputable function Sigma(k) for four-state Turing machines
  Mathematics of Computation 40 (162), April 1983, 647-665.

• Brady A.H. (1988)
  The busy beaver game and the meaning of life
  in: The Universal Turing Machine: A Half-Century Survey, R. Herken (Ed.), Oxford University Press, 1988, 259-277.

• Chaitin G. (1987)
  Computing the busy beaver function
  Open Problems in Communication and Computation, Springer, 1987, 108-112.
  Reprinted in: Information, Randomness and Incompleteness, World Scientifics, 2nd Ed. 1990.

• Dewdney A.K. (1984a)
  Computer recreations
  Scientific American 251 (2), August 1984, 10-17.

• Dewdney A.K. (1984b)
  Computer recreations
  Scientific American 251 (5), November 1984, 27.

• Dewdney A.K. (1985a)
  Computer recreations
  Scientific American 252 (3), March 1985, 19.

• Dewdney A.K. (1985b)
  Computer recreations
  Scientific American 252 (4), April 1985, 16.

• Green M.W. (1964)
  A lower bound on Rado's sigma function for binary Turing machines
  in: Proceedings of the 5th IEEE Annual Symposium on Switching Circuit Theory and Logical Design, November 1964, 91-94.

• Hertel J. (2009)
  Computing the uncomputable Rado sigma function
  The Mathematica Journal 11 (2), 2009, 270-283.

• Julstrom B.A. (1992)
  A bound on the shift function in terms of the busy beaver function
  SIGACT News 23 (3), Summer 1992, 100-106.

• Julstrom B.A. (1993)
  Noncomputability and the busy beaver problem
  The UMAP Journal 14 (1), Spring 1993, 39-74.

• Kopp R.J. (1981)
  The busy beaver problem
  M.A. Thesis, State University of New York, Binghamton, 1981.

• Lafitte G. (2009)
  Busy beavers gone wild
  in: The Complexity of Simple Programs 2008, EPTCS 1, 2009, 123-129.

• Lafitte G. and Papazian C. (2007)
  The fabric of small Turing machines
  in: Computation and Logic in the Real World, Proceedings of the Third Conference on Computability in Europe, June 2007, 219-227.

• Lin S. (1963)
  Computer studies of Turing machine problems
  Ph.D. Thesis, The Ohio State University, Columbus, 1963.

• Lin S. and Rado T. (1965)
  Computer studies of Turing machine problems
  Journal of the ACM 12 (2), April 1965, 196-212.

• Lynn D.S. (1972)
  New results for Rado's sigma function for binary Turing machines
  IEEE Transactions on Computers C-21 (8), August 1972, 894-896.

• Machlin R. and Stout Q.F. (1990)
  The complex behavior of simple machines
  Physica D 42, June 1990, 85-98.

• Marxen H. and Buntrock J. (1990)
  Attacking the Busy Beaver 5
  Bulletin of the EATCS No 40, February 1990, 247-251.

• Michel P. (1993)
  Busy beaver competition and Collatz-like problems
  Archive for Mathematical Logic 32 (5), 1993, 351-367.

• Michel P. (2004)
  Small Turing machines and generalized busy beaver competition
  Theoretical Computer Science 326 (1-3), October 2004, 45-56.

• Obershelp A., Schmidt-Göttsch K. and Todt G. (1988)
  Castor quadruplorum
  Archive for Mathematical Logic 27 (1), 1988, 35-44.

• Petersen H. (2006)
  Computable lower bounds for busy beaver Turing machines
  Studies in Computational Intelligence 25, 2006, 305-319.

• Rado T. (1962)
  On non-computable functions
  Bell System Technical Journal 41 (3), May 1962, 877-884.

• Rado T. (1963)
  On a simple source for non-computable functions
  in: Proceedings of the Symposium on Mathematical Theory of Automata, April 1962, Polytechnic Institute of Brooklyn, April 1963, 75-81.

• Wang K. and Xu S. (1995)
  New relation between the shift function and the busy beaver function
  Chinese Journal of Advanced Software Research 2 (2), 1995, 192-197.

• Yang R., Ding L. and Xu S. (1997)
  Some better results estimating the shift function in terms of busy beaver function
  SIGACT News 28 (1), March 1997, 43-48.

• Heiner Marxen is the top web reference on busy beavers.

• Wikipedia has a busy beaver entry.

• H.J.M. Wijers gives a comprehensive bibliography on busy beavers.

• Allen H. Brady gives an interesting study of machines with 3 states and 3 symbols.

• Skelet (Georgi Georgiev) works on Turing machines with 5 states and 2 symbols, and reversal Turing machines.

• Daniel Briggs created a website and a forum dedicated to the study of Turing machines with 5 states and 2 symbols.

• Robert Munafo gives many informations on busy beavers, and analyzes some machines with 6 states and 2 symbols.

• E.W. Weisstein from MathWorld is also a good reference.

• Hector Zenil gives a demonstration, on the Wolfram Demonstrations Project. See also his website.

• James Harland studies some inverse functions of the busy beaver function: placid platypus and weary wombat.

• P. Gammies gave java applets simulating busy beavers with 2, 3, 4 states and 2 symbols.

• P. Machado and F. Pereira study also the 4-tuple busy beavers.

• B. van Heuveln leads a team searching for 4-tuple busy beavers.

• Tim Hutton studies two-dimensional and higher-dimensional busy beavers.

• Scott Aaronson shows how busy beavers allow to name big numbers.

Link to Pascal Michel's home page