More info can be found on the BBChallenge website and the corresponding wiki.
Current 2-symbol records for S(n) (maximum shifts function) and Σ(n) (busy beaver function):
| n | S(n) | Σ(n) | Machine | Link |
|---|---|---|---|---|
| 1 | 1 | 1 | 1RZ1RA | |
| 2 | 6 | 4 | 1RB1LB_1LA1RZ | Radó (1962) |
| 3 | 21 | 5 | 1RB1RZ_1LB0RC_1LC1LA | Shen (1963) |
| 3 | 14 | 6 | 1RB1RZ_0RC1RB_1LC1LA | Shen (1963) |
| 4 | 107 | 13 | 1RB1LB_1LA0LC_1RZ1LD_1RD0RA | Brady (1983) |
| 5 | 47,176,870 | 4098 | 1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA | Marxen & Buntrock (1990) |
| 6 | >10^^15 | >10^^15 | 1RB0LD_1RC0RF_1LC1LA_0LE1RZ_1LF0RB_0RC0RE | Kropitz (2022) |
| 7 | >10^^15 | >10^^15 | Same as above | |
| 8 | >10^^15 | >10^^15 | Same as above | |
| 9 | >10^^30 | >10^^30 | 1LD1LB_1LZ1LA_0LB1LD_0LE0LD_1LF1RC_0LG0LF_1LH1RE_0LI0LH_1RI1RG | Green (1964), Ligocki (2023) |
| 10 | >10^^(10^10^15) | >10^^(10^10^15) | 1LB1RZ_0LC1LC_0LD0LC_1LE1RA_0LF0LE_1LG1RD_0LH0LG_1LI1RF_0LJ0LI_1RJ1RH | Green (1964), Ligocki (2023) |
| 11 | >10^^^(2.16*10^15) | >10^^^(2.16*10^15) | Green (1964), Ligocki (2023) | |
| 12 | >10^^^(10^^^3) | >10^^^(10^^^3) | Green (1964), Ligocki (2023) | |
| 13 | >10{2046}3 | >10{2046}3 | Wythagoras (2016) | |
| 14 | >10{10^18267}3 | >10{10^18267}3 | Wythagoras (2016) | |
| 15 | >10{10^^5}3 | >10{10^^5}3 | Wythagoras (2016) | |
| 16 | >{10,1000,1,2} | >{10,1000,1,2} | Wythagoras (2021), Ligocki (2022) |
Note: 'Z' is used for the halting state in these machines since H needs to be used for any machine with at least 8 states. Also note that for n>5, S(n) and Σ(n) are indistinguishable. This is the case for almost all large busy beavers.
To be more precise, here are the exact best known values of S(n):
| n | S(n) | Hyper-E equivalent |
|---|---|---|
| 1 | 1 | |
| 2 | 6 | |
| 3 | 21 | |
| 4 | 21 | |
| 5 | 47,176,870 | |
| 6 | >10^^15.6046 | E10565.1028#14 |
| 7 | >10^^15.6046 | E10565.1028#14 |
| 8 | >10^^15.6046 | E10565.1028#14 |
| 9 | >10^^30.0410 | E12.5609#29 |
| 10 | >10^^(10^(1.0314*10^15)) | E15.0134#2#2 |
| 11 | >10^^^(2.1619*10^15) | E15.3348#1#1#2 |
| 12 | >10^^^(10^^10^^(2.1619*10^15)) | E15.3348#1#3#2 |
| 13 | >10{2046}3 | E10##2046 |
| 14 | >10{1.7*10^18,267}3 | E18267.2304##1#2 |
| 15 | >10{10^10^10^10^18,705,352}3 | E10##(E7.2719#5) |
| 16 | >{10,{10^^^10^^^7},1,2} | E10##1#(E168#1#1#7#2) |
Current 3-symbol records for S(n) and Σ(n):
| n | S(n) | Σ(n) | Link |
|---|---|---|---|
| 1 | 1 | 1 | |
| 2 | 38 | 9 | |
| 3 | 1.19*10^17 | 374,676,383 | Ligocki (2007) |
| 4 | 10^14,072 | 10^7036 | Ligocki (2008) |
| 5 | 10^14,072 | 10^7036 | Same as above |
| 6 | 10{6}8 | 10{6}8 | Wythagoras (2016) |
| 7 | 10{374,676,381}3 | 10{374,676,381}3 | Wythagoras (2016) |
Current 4-symbol records for S(n) and Σ(n):
| n | S(n) | Σ(n) | Link |
|---|---|---|---|
| 1 | 1 | 1 | |
| 2 | ≥3,932,964 | ≥2050 | Ligocki (2007) |
| 3 | >10{15}4 | >10{15}4 | Kropitz (2024) |
Current 5-symbol records for S(n) and Σ(n):
| n | S(n) | Σ(n) | Link |
|---|---|---|---|
| 1 | 1 | 1 | |
| 2 | >10^10^10^10^6.5 | >10^10^10^10^6.5 | Yuan (2024) |
Current 6-symbol records for S(n) and Σ(n):
| n | S(n) | Σ(n) | Link |
|---|---|---|---|
| 1 | 1 | 1 | |
| 2 | >10^^10^^(10^10^115) | >10^^10^^(10^10^115) | Kropitz (2023) |
As you can see, the tables for 4 symbols onwards have very few entries. The entries for S(2,5) and S(2,6) have been developing since 2005, but besides a few special cases (see below) we have yet to dive into more than 2 states for them.
As of July 2024, BB(6,2), BB(3,3), BB(2,5) and any combination of more states or symbols has a known "cryptid" machine that cannot be proven until some currently unsolved mathematical problem is solved. See the cryptids wiki page for more details.
Additional records: