Remark: Every n ∈ N0 has a unique binary representation ∑i ∈ N0 di2i with di ∈ {0,1}. Then holds parity(n) = 1 AND ∑i ∈ N0 di = XORi ∈ N0 di.
Further Links: Combinatorial Game Theory
Sorry, I forgot to document my further progress in computing sg-values for Grundy's Game here. In autum 2012 I tracked more octal-games and Grundy's Game to higher positions.
Known Sparse SGV of Grundy's Game
G( 0)= 0 G( 1)= 0 G( 2)= 0 G( 3)= 1 G( 4)= 0
G( 6)= 1 G( 7)= 0 G( 9)= 1 G( 10)= 0 G( 12)= 1
G( 15)= 1 G( 20)= 0 G( 23)= 0 G( 26)= 0 G( 28)= 1
G( 31)= 1 G( 34)= 1 G( 37)= 1 G( 40)= 1 G( 43)= 1
G( 46)= 1 G( 49)= 1 G( 50)= 0 G( 52)= 1 G( 53)= 0
G( 55)= 1 G( 58)= 1 G( 61)= 1 G( 87)= 7 G( 90)= 7
G( 93)= 7 G( 146)= 6 G( 149)= 6 G( 152)= 6 G( 158)= 11
G( 161)= 11 G( 164)= 11 G( 167)= 11 G( 176)= 7 G( 179)= 7
G( 181)= 12 G( 182)= 7 G( 185)= 7 G( 188)= 7 G( 190)= 12
G( 193)= 12 G( 196)= 10 G( 231)= 10 G( 232)= 1 G( 235)= 1
G( 238)= 1 G( 270)= 0 G( 273)= 0 G( 276)= 0 G( 281)= 1
G( 282)= 0 G( 284)= 1 G( 285)= 0 G( 287)= 1 G( 288)= 0
G( 290)= 1 G( 291)= 13 G( 293)= 1 G( 294)= 13 G( 296)= 1
G( 297)= 13 G( 299)= 1 G( 300)= 13 G( 302)= 1 G( 303)= 13
G( 305)= 1 G( 306)= 13 G( 308)= 1 G( 309)= 13 G( 312)= 13
G( 315)= 13 G( 316)= 0 G( 324)= 1 G( 327)= 1 G( 330)= 1
G( 333)= 1 G( 334)= 0 G( 336)= 1 G( 337)= 0 G( 338)= 12
G( 339)= 1 G( 340)= 0 G( 341)= 12 G( 342)= 1 G( 345)= 1
G( 346)= 0 G( 348)= 1 G( 351)= 1 G( 359)= 0 G( 362)= 0
G( 365)= 0 G( 379)= 12 G( 380)= 7 G( 383)= 7 G( 386)= 0
G( 389)= 0 G( 392)= 0 G( 451)= 11 G( 454)= 11 G( 456)= 7
G( 457)= 11 G( 460)= 11 G( 469)= 7 G( 474)= 7 G( 500)= 11
G( 503)= 11 G( 507)= 6 G( 510)= 6 G( 512)= 13 G( 513)= 6
G( 515)= 13 G( 516)= 6 G( 519)= 6 G( 522)= 6 G( 524)= 10
G( 525)= 1 G( 527)= 10 G( 528)= 1 G( 541)= 21 G( 544)= 21
G( 566)= 0 G( 584)= 13 G( 587)= 13 G( 590)= 13 G( 593)= 13
G( 596)= 13 G( 599)= 13 G( 601)= 1 G( 602)= 13 G( 604)= 1
G( 605)= 13 G( 611)= 21 G( 620)= 18 G( 623)= 18 G( 624)= 13
G( 626)= 18 G( 627)= 13 G( 629)= 18 G( 630)= 0 G( 633)= 0
G( 634)= 12 G( 636)= 0 G( 639)= 0 G( 645)= 24 G( 648)= 21
G( 651)= 13 G( 660)= 21 G( 663)= 24 G( 666)= 24 G( 673)= 0
G( 676)= 0 G( 680)= 12 G( 682)= 0 G( 685)= 0 G( 735)= 11
G( 738)= 11 G( 741)= 11 G( 757)= 12 G( 760)= 12 G( 763)= 12
G( 782)= 10 G( 785)= 10 G( 788)= 24 G( 811)= 13 G( 828)= 10
G( 831)= 10 G( 852)= 24 G( 855)= 11 G( 858)= 11 G( 860)= 20
G( 863)= 20 G( 866)= 20 G( 877)= 24 G( 883)= 13 G( 886)= 13
G( 915)= 12 G( 918)= 12 G( 920)= 13 G( 921)= 12 G( 923)= 0
G( 926)= 0 G( 927)= 12 G( 929)= 0 G( 930)= 12 G( 932)= 0
G( 941)= 13 G( 944)= 13 G( 955)= 1 G( 956)= 24 G( 958)= 1
G( 959)= 21 G( 961)= 1 G( 962)= 21 G( 964)= 1 G( 967)= 12
G( 969)= 10 G( 970)= 12 G( 973)= 12 G( 976)= 12 G( 979)= 12
G( 1005)= 24 G( 1007)= 1 G( 1010)= 1 G( 1013)= 1 G( 1050)= 12
G( 1053)= 12 G( 1063)= 10 G( 1066)= 10 G( 1068)= 6 G( 1069)= 10
G( 1072)= 10 G( 1074)= 6 G( 1075)= 10 G( 1077)= 6 G( 1083)= 11
G( 1086)= 11 G( 1106)= 7 G( 1109)= 7 G( 1112)= 7 G( 1117)= 12
G( 1127)= 10 G( 1188)= 21 G( 1191)= 21 G( 1199)= 12 G( 1202)= 12
G( 1204)= 7 G( 1205)= 12 G( 1207)= 7 G( 1210)= 7 G( 1213)= 13
G( 1216)= 13 G( 1222)= 0 G( 1252)= 21 G( 1255)= 21 G( 1258)= 21
G( 1260)= 1 G( 1261)= 21 G( 1266)= 12 G( 1284)= 20 G( 1290)= 25
G( 1291)= 35 G( 1294)= 35 G( 1297)= 1 G( 1299)= 30 G( 1301)= 13
G( 1304)= 13 G( 1306)= 12 G( 1311)= 10 G( 1317)= 10 G( 1334)= 12
G( 1343)= 1 G( 1347)= 24 G( 1349)= 36 G( 1350)= 24 G( 1379)= 11
G( 1382)= 11 G( 1393)= 37 G( 1396)= 37 G( 1398)= 12 G( 1400)= 19
G( 1403)= 19 G( 1404)= 36 G( 1407)= 12 G( 1410)= 12 G( 1413)= 12
G( 1425)= 11 G( 1442)= 13 G( 1445)= 7 G( 1447)= 24 G( 1458)= 11
G( 1462)= 12 G( 1494)= 13 G( 1506)= 37 G( 1509)= 37 G( 1512)= 37
G( 1529)= 36 G( 1531)= 10 G( 1532)= 36 G( 1534)= 10 G( 1535)= 40
G( 1538)= 40 G( 1539)= 13 G( 1541)= 12 G( 1542)= 13 G( 1544)= 12
G( 1547)= 12 G( 1548)= 21 G( 1551)= 21 G( 1553)= 12 G( 1556)= 12
G( 1557)= 21 G( 1559)= 12 G( 1562)= 12 G( 1565)= 12 G( 1570)= 13
G( 1576)= 13 G( 1579)= 13 G( 1580)= 12 G( 1582)= 13 G( 1583)= 12
G( 1586)= 20 G( 1589)= 20 G( 1595)= 34 G( 1600)= 21 G( 1603)= 13
G( 1605)= 12 G( 1606)= 13 G( 1607)= 10 G( 1610)= 10 G( 1632)= 12
G( 1649)= 13 G( 1683)= 10 G( 1692)= 40 G( 1707)= 10 G( 1718)= 36
G( 1732)= 37 G( 1733)= 6 G( 1739)= 11 G( 1741)= 37 G( 1749)= 12
G( 1752)= 12 G( 1761)= 36 G( 1764)= 12 G( 1801)= 24 G( 1802)= 37
G( 1840)= 40 G( 1848)= 25 G( 1865)= 11 G( 1866)= 41 G( 1869)= 41
G( 1876)= 20 G( 1879)= 20 G( 1888)= 10 G( 1894)= 30 G( 1899)= 13
G( 1960)= 13 G( 1965)= 25 G( 1983)= 19 G( 1986)= 19 G( 1996)= 12
G( 1999)= 31 G( 2002)= 31 G( 2011)= 36 G( 2020)= 36 G( 2021)= 18
G( 2023)= 36 G( 2024)= 18 G( 2027)= 18 G( 2030)= 18 G( 2031)= 30
G( 2032)= 41 G( 2034)= 37 G( 2037)= 37 G( 2045)= 12 G( 2046)= 37
G( 2048)= 12 G( 2056)= 13 G( 2059)= 13 G( 2063)= 36 G( 2072)= 36
G( 2075)= 36 G( 2111)= 20 G( 2138)= 25 G( 2162)= 41 G( 2191)= 36
G( 2215)= 25 G( 2218)= 20 G( 2221)= 20 G( 2248)= 30 G( 2253)= 13
G( 2277)= 31 G( 2314)= 13 G( 2320)= 18 G( 2324)= 30 G( 2326)= 18
G( 2327)= 30 G( 2332)= 12 G( 2335)= 12 G( 2338)= 12 G( 2353)= 31
G( 2356)= 31 G( 2362)= 36 G( 2363)= 18 G( 2368)= 36 G( 2404)= 19
G( 2407)= 20 G( 2465)= 40 G( 2471)= 20 G( 2497)= 21 G( 2533)= 35
G( 2555)= 18 G( 2558)= 18 G( 2576)= 31 G( 2579)= 31 G( 2581)= 20
G( 2582)= 31 G( 2600)= 35 G( 2613)= 18 G( 2614)= 30 G( 2616)= 18
G( 2619)= 18 G( 2629)= 30 G( 2642)= 20 G( 2645)= 19 G( 2648)= 20
G( 2654)= 19 G( 2655)= 35 G( 2657)= 19 G( 2660)= 19 G( 2661)= 35
G( 2664)= 35 G( 2666)= 30 G( 2669)= 30 G( 2672)= 30 G( 2678)= 37
G( 2702)= 37 G( 2704)= 35 G( 2707)= 36 G( 2710)= 35 G( 2716)= 35
G( 2718)= 19 G( 2719)= 36 G( 2731)= 35 G( 2762)= 36 G( 2798)= 30
G( 2805)= 21 G( 2819)= 20 G( 2822)= 20 G( 2842)= 21 G( 2845)= 21
G( 2848)= 18 G( 2851)= 21 G( 2965)= 30 G( 2969)= 36 G( 2971)= 30
G( 2983)= 34 G( 2992)= 37 G( 3000)= 35 G( 3003)= 35 G( 3006)= 35
G( 3009)= 35 G( 3012)= 36 G( 3015)= 36 G( 3021)= 35 G( 3024)= 35
G( 3032)= 59 G( 3038)= 37 G( 3041)= 37 G( 3047)= 37 G( 3050)= 34
G( 3053)= 34 G( 3054)= 19 G( 3056)= 34 G( 3059)= 34 G( 3090)= 37
G( 3093)= 37 G( 3096)= 34 G( 3108)= 34 G( 3124)= 20 G( 3135)= 31
G( 3138)= 31 G( 3155)= 46 G( 3158)= 46 G( 3176)= 20 G( 3179)= 20
G( 3182)= 20 G( 3197)= 60 G( 3224)= 37 G( 3227)= 34 G( 3230)= 34
G( 3238)= 35 G( 3276)= 34 G( 3288)= 37 G( 3290)= 31 G( 3291)= 37
G( 3295)= 19 G( 3298)= 19 G( 3299)= 31 G( 3302)= 10 G( 3305)= 31
G( 3317)= 35 G( 3318)= 18 G( 3320)= 36 G( 3321)= 18 G( 3326)= 36
G( 3328)= 30 G( 3330)= 18 G( 3331)= 34 G( 3334)= 34 G( 3343)= 37
G( 3346)= 34 G( 3352)= 34 G( 3355)= 34 G( 3369)= 35 G( 3372)= 36
G( 3380)= 37 G( 3383)= 37 G( 3386)= 37 G( 3389)= 34 G( 3392)= 34
G( 3395)= 34 G( 3398)= 34 G( 3401)= 34 G( 3418)= 36 G( 3437)= 21
G( 3462)= 12 G( 3500)= 13 G( 3503)= 13 G( 3562)= 21 G( 3574)= 31
G( 3578)= 34 G( 3581)= 37 G( 3586)= 31 G( 3616)= 10 G( 3619)= 36
G( 3630)= 34 G( 3639)= 11 G( 3648)= 34 G( 3651)= 37 G( 3673)= 30
G( 3674)= 36 G( 3675)= 18 G( 3676)= 30 G( 3679)= 34 G( 3682)= 47
G( 3685)= 47 G( 3694)= 34 G( 3714)= 36 G( 3738)= 35 G( 3741)= 13
G( 3760)= 58 G( 3771)= 20 G( 3797)= 21 G( 3874)= 11 G( 3892)= 37
G( 3900)= 35 G( 3932)= 11 G( 3943)= 31 G( 3944)= 11 G( 3952)= 10
G( 3957)= 59 G( 3967)= 10 G( 3979)= 13 G( 4037)= 41 G( 4062)= 46
G( 4068)= 46 G( 4072)= 12 G( 4075)= 12 G( 4086)= 13 G( 4129)= 13
G( 4130)= 12 G( 4132)= 13 G( 4138)= 13 G( 4167)= 12 G( 4170)= 11
G( 4173)= 11 G( 4179)= 12 G( 4189)= 20 G( 4202)= 10 G( 4205)= 10
G( 4206)= 21 G( 4226)= 10 G( 4229)= 10 G( 4240)= 11 G( 4249)= 11
G( 4254)= 10 G( 4257)= 10 G( 4258)= 11 G( 4260)= 10 G( 4263)= 10
G( 4266)= 10 G( 4272)= 10 G( 4277)= 11 G( 4286)= 11 G( 4292)= 47
G( 4315)= 10 G( 4318)= 10 G( 4355)= 46 G( 4359)= 12 G( 4361)= 41
G( 4362)= 12 G( 4365)= 12 G( 4376)= 10 G( 4377)= 12 G( 4391)= 41
G( 4403)= 13 G( 4423)= 47 G( 4440)= 10 G( 4466)= 12 G( 4469)= 12
G( 4484)= 47 G( 4497)= 20 G( 4506)= 60 G( 4519)= 10 G( 4539)= 11
G( 4554)= 58 G( 4565)= 46 G( 4568)= 10 G( 4571)= 10 G( 4574)= 10
G( 4588)= 11 G( 4594)= 11 G( 4597)= 11 G( 4603)= 11 G( 4613)= 47
G( 4614)= 10 G( 4620)= 10 G( 4631)= 60 G( 4640)= 11 G( 4643)= 11
G( 4646)= 11 G( 4649)= 11 G( 4652)= 11 G( 4666)= 10 G( 4699)= 13
G( 4722)= 12 G( 4725)= 12 G( 4728)= 12 G( 4731)= 12 G( 4765)= 12
G( 4838)= 11 G( 4879)= 10 G( 4951)= 11 G( 4954)= 11 G( 4956)= 10
G( 4971)= 10 G( 4983)= 10 G( 4986)= 10 G( 5006)= 68 G( 5029)= 10
G( 5070)= 67 G( 5096)= 10 G( 5107)= 67 G( 5160)= 66 G( 5171)= 68
G( 5180)= 11 G( 5273)= 10 G( 5276)= 13 G( 5285)= 60 G( 5342)= 67
G( 5380)= 66 G( 5383)= 66 G( 5392)= 66 G( 5394)= 67 G( 5397)= 67
G( 5400)= 68 G( 5441)= 69 G( 5444)= 66 G( 5450)= 66 G( 5453)= 66
G( 5455)= 67 G( 5456)= 66 G( 5465)= 13 G( 5496)= 69 G( 5499)= 69
G( 5512)= 48 G( 5574)= 60 G( 5635)= 60 G( 5638)= 67 G( 5676)= 66
G( 5694)= 69 G( 5721)= 54 G( 5743)= 66 G( 5746)= 69 G( 5749)= 69
G( 5767)= 13 G( 5775)= 67 G( 5778)= 67 G( 5781)= 67 G( 5783)= 66
G( 5792)= 66 G( 5829)= 61 G( 5855)= 60 G( 5870)= 60 G( 5875)= 48
G( 5878)= 48 G( 5881)= 48 G( 5928)= 60 G( 5940)= 67 G( 5965)= 49
G( 5968)= 49 G( 6033)= 66 G( 6036)= 66 G( 6042)= 66 G( 6052)= 55
G( 6068)= 67 G( 6074)= 68 G( 6077)= 68 G( 6080)= 68 G( 6092)= 68
G( 6106)= 66 G( 6111)= 68 G( 6117)= 67 G( 6120)= 67 G( 6150)= 41
G( 6172)= 67 G( 6195)= 61 G( 6198)= 61 G( 6200)= 49 G( 6201)= 69
G( 6223)= 48 G( 6232)= 48 G( 6258)= 49 G( 6339)= 48 G( 6370)= 68
G( 6466)= 69 G( 6469)= 69 G( 6494)= 69 G( 6529)= 68 G( 6534)= 48
G( 6554)= 49 G( 6555)= 66 G( 6569)= 49 G( 6574)= 48 G( 6577)= 48
G( 6592)= 48 G( 6609)= 49 G( 6612)= 49 G( 6618)= 49 G( 6794)= 68
G( 6848)= 69 G( 6853)= 49 G( 6891)= 48 G( 6940)= 48 G( 6972)= 54
G( 6983)= 46 G( 6997)= 47 G( 7024)= 54 G( 7067)= 69 G( 7090)= 68
G( 7120)= 48 G( 7129)= 72 G( 7141)= 69 G( 7147)= 69 G( 7170)= 68
G( 7173)= 60 G( 7176)= 60 G( 7179)= 60 G( 7231)= 24 G( 7250)= 47
G( 7257)= 66 G( 7263)= 40 G( 7269)= 25 G( 7273)= 46 G( 7288)= 46
G( 7318)= 25 G( 7335)= 78 G( 7359)= 46 G( 7375)= 72 G( 7392)= 68
G( 7405)= 24 G( 7443)= 66 G( 7469)= 31 G( 7470)= 40 G( 7475)= 67
G( 7498)= 66 G( 7526)= 79 G( 7553)= 40 G( 7561)= 80 G( 7573)= 73
G( 7585)= 41 G( 7605)= 79 G( 7613)= 79 G( 7614)= 66 G( 7616)= 80
G( 7619)= 80 G( 7623)= 66 G( 7625)= 80 G( 7633)= 46 G( 7642)= 73
G( 7660)= 79 G( 7672)= 66 G( 7675)= 79 G( 7694)= 73 G( 7697)= 73
G( 7700)= 73 G( 7723)= 40 G( 7724)= 81 G( 7756)= 73 G( 7767)= 81
G( 7778)= 40 G( 7800)= 30 G( 7811)= 73 G( 7814)= 73 G( 7817)= 68
G( 7820)= 68 G( 7822)= 81 G( 7823)= 68 G( 7826)= 68 G( 7846)= 72
G( 7857)= 79 G( 7861)= 69 G( 7872)= 73 G( 7890)= 41 G( 7909)= 79
G( 7933)= 78 G( 7935)= 46 G( 7936)= 78 G( 7962)= 66 G( 7964)= 47
G( 7965)= 66 G( 7983)= 47 G( 7991)= 78 G( 8000)= 67 G( 8003)= 67
G( 8055)= 78 G( 8064)= 46 G( 8066)= 81 G( 8099)= 47 G( 8107)= 73
G( 8127)= 72 G( 8133)= 72 G( 8136)= 72 G( 8148)= 69 G( 8153)= 73
G( 8189)= 78 G( 8205)= 80 G( 8226)= 78 G( 8237)= 78 G( 8247)= 41
G( 8250)= 41 G( 8275)= 78 G( 8281)= 78 G( 8310)= 79 G( 8322)= 79
G( 8325)= 79 G( 8331)= 79 G( 8345)= 78 G( 8350)= 73 G( 8360)= 46
G( 8363)= 46 G( 8382)= 72 G( 8423)= 72 G( 8435)= 69 G( 8455)= 73
G( 8473)= 68 G( 8545)= 81 G( 8548)= 81 G( 8568)= 80 G( 8580)= 78
G( 8597)= 79 G( 8603)= 79 G( 8635)= 78 G( 8638)= 78 G( 8641)= 78
G( 8653)= 78 G( 8673)= 79 G( 8693)= 78 G( 8699)= 78 G( 8792)= 79
G( 8819)= 37 G( 8836)= 97 G( 8842)= 36 G( 8905)= 79 G( 8954)= 81
G( 8960)= 79 G( 8978)= 96 G( 8986)= 80 G( 8989)= 78 G( 9013)= 35
G( 9184)= 36 G( 9187)= 36 G( 9224)= 80 G( 9521)= 37 G( 9550)= 36
G( 9762)= 37 G( 9966)= 73 G( 9981)= 78 G( 10113)= 96 G( 10167)= 58
G( 10186)= 90 G( 10204)= 73 G( 10238)= 96 G( 10288)= 72 G( 10326)= 78
G( 10374)= 12 G( 10399)= 61 G( 10402)= 61 G( 10418)= 78 G( 10445)= 67
G( 10482)= 73 G( 10512)= 67 G( 10547)= 59 G( 10550)= 59 G( 10570)= 58
G( 10573)= 58 G( 10596)= 59 G( 10597)= 12 G( 10599)= 59 G( 10602)= 59
G( 10605)= 59 G( 10606)= 12 G( 10650)= 96 G( 10653)= 96 G( 10680)= 67
G( 10686)= 67 G( 10695)= 78 G( 10714)= 10 G( 10730)= 60 G( 10750)= 61
G( 10773)= 66 G( 10778)= 67 G( 10791)= 60 G( 10794)= 60 G( 10796)= 67
G( 10797)= 60 G( 10815)= 13 G( 10819)= 79 G( 10822)= 69 G( 10839)= 78
G( 10840)= 72 G( 10842)= 73 G( 10859)= 12 G( 10885)= 13 G( 10894)= 13
G( 10905)= 12 G( 10908)= 11 G( 10909)= 68 G( 10917)= 72 G( 10918)= 68
G( 10921)= 78 G( 10943)= 13 G( 10976)= 67 G( 10998)= 13 G( 11000)= 61
G( 11030)= 12 G( 11052)= 61 G( 11062)= 78 G( 11075)= 86 G( 11084)= 60
G( 11135)= 78 G( 11140)= 11 G( 11153)= 68 G( 11174)= 58 G( 11176)= 72
G( 11180)= 58 G( 11181)= 13 G( 11182)= 72 G( 11196)= 73 G( 11203)= 59
G( 11207)= 12 G( 11208)= 68 G( 11213)= 12 G( 11217)= 93 G( 11223)= 58
G( 11226)= 78 G( 11229)= 58 G( 11262)= 12 G( 11264)= 59 G( 11275)= 78
G( 11284)= 58 G( 11296)= 61 G( 11323)= 12 G( 11325)= 60 G( 11348)= 61
G( 11358)= 78 G( 11364)= 96 G( 11370)= 67 G( 11407)= 13 G( 11442)= 58
G( 11443)= 67 G( 11446)= 67 G( 11463)= 66 G( 11468)= 13 G( 11471)= 13
G( 11482)= 61 G( 11483)= 13 G( 11489)= 13 G( 11494)= 12 G( 11511)= 59
G( 11513)= 67 G( 11514)= 59 G( 11530)= 69 G( 11535)= 10 G( 11536)= 72
G( 11542)= 69 G( 11548)= 69 G( 11550)= 68 G( 11556)= 68 G( 11562)= 68
G( 11565)= 68 G( 11592)=107 G( 11600)= 79 G( 11612)=109 G( 11619)= 12
G( 11642)= 13 G( 11644)=117 G( 11645)= 13 G( 11650)=107 G( 11651)= 10
G( 11656)= 61 G( 11660)= 10 G( 11669)= 67 G( 11677)= 12 G( 11686)= 12
G( 11759)= 69 G( 11768)= 69 G( 11771)= 69 G( 11785)= 13 G( 11800)= 68
G( 11806)= 73 G( 11811)= 12 G( 11817)= 12 G( 11823)= 69 G( 11843)= 13
G( 11890)= 69 G( 11893)= 69 G( 11908)=106 G( 11910)= 68 G( 11914)=106
G( 11934)=107 G( 11937)=108 G( 11950)= 60 G( 11961)= 12 G( 11973)= 12
G( 12004)=107 G( 12013)= 61 G( 12020)= 67 G( 12029)= 67 G( 12037)=127
G( 12056)=102 G( 12068)=107 G( 12087)= 68 G( 12117)=107 G( 12129)=108
G( 12140)=109 G( 12154)= 68 G( 12165)= 79 G( 12171)= 12 G( 12175)=108
G( 12186)= 66 G( 12213)=109 G( 12216)=109 G( 12220)= 72 G( 12229)= 69
G( 12233)=107 G( 12253)=106 G( 12262)=106 G( 12268)=106 G( 12271)=109
G( 12280)= 60 G( 12282)= 97 G( 12309)=108 G( 12364)=107 G( 12385)= 12
G( 12437)= 12 G( 12455)= 12 G( 12458)= 12 G( 12460)=109 G( 12479)=106
G( 12480)=108 G( 12516)= 12 G( 12565)= 92 G( 12599)=108 G( 12625)=106
G( 12626)= 81 G( 12683)=106 G( 12724)= 92 G( 12727)= 91 G( 12736)= 79
G( 12779)=108 G( 12828)=108 G( 12904)= 91 G( 12919)= 91 G( 12948)= 80
G( 13009)= 80 G( 13090)= 79 G( 13096)= 81 G( 13137)= 93 G( 13168)=121
G( 13169)= 86 G( 13313)=127 G( 13334)= 81 G( 13337)= 79 G( 13357)= 80
G( 13398)= 86 G( 13430)= 80 G( 13438)= 92 G( 13476)= 80 G( 13543)= 87
G( 13586)= 90 G( 13615)=108 G( 13679)=133 G( 13688)= 81 G( 13700)= 81
G( 13715)= 96 G( 13778)= 80 G( 13803)= 25 G( 13908)=127 G( 13917)= 91
G( 13955)= 93 G( 13963)=130 G( 13972)=133 G( 13975)=133 G( 13998)=132
G( 14022)= 80 G( 14034)= 97 G( 14035)= 25 G( 14059)= 93 G( 14061)= 24
G( 14222)= 92 G( 14239)=132 G( 14270)= 25 G( 14271)=133 G( 14300)=132
G( 14305)= 24 G( 14360)= 24 G( 14383)= 25 G( 14416)= 93 G( 14421)= 24
G( 14451)=106 G( 14505)= 25 G( 14507)=127 G( 14601)= 24 G( 14802)=106
G( 14935)= 25 G( 15422)=130 G( 15425)=130 G( 15603)=108 G( 15620)=126
G( 15689)=132 G( 15692)=132 G( 15721)=130 G( 15727)=130 G( 15773)=130
G( 15800)=106 G( 15936)=127 G( 15963)=108 G( 15980)=115 G( 15992)= 96
G( 16011)=130 G( 16017)=130 G( 16020)=130 G( 16049)=131 G( 16064)=117
G( 16098)=131 G( 16204)= 11 G( 16230)= 10 G( 16276)=109 G( 16320)= 97
G( 16339)=132 G( 16348)=131 G( 16368)=130 G( 16369)=117 G( 16371)=130
G( 16394)=131 G( 16398)=116 G( 16752)=106 G( 16771)= 58 G( 16848)=107
G( 16851)=107 G( 16935)=109 G( 16960)= 61 G( 16999)=109 G( 17095)=142
G( 17193)=107 G( 17310)=131 G( 17505)=137 G( 17676)=137 G( 17760)=136
G( 17893)=142 G( 17911)=137 G( 17934)=136 G( 17940)=136 G( 17943)=136
G( 18082)=137 G( 18088)=137 G( 18100)=131 G( 18117)=136 G( 18236)=136
G( 18280)=132 G( 18291)=136 G( 18312)=133 G( 18335)=131 G( 18384)=137
G( 18521)=132 G( 18570)=132 G( 18573)=132 G( 18617)= 96 G( 18625)=132
G( 18628)=132 G( 18663)=130 G( 18695)=108 G( 18721)=130 G( 18735)=142
G( 18915)=132 G( 18918)=132 G( 19035)=106 G( 19582)=130 G( 19611)= 47
G( 19687)=132 G( 20455)=136 G( 20507)=130 G( 20774)=131 G( 20775)=130
G( 20777)=131 G( 20818)=130 G( 20829)=137 G( 20922)=136 G( 20931)=130
G( 21123)=130 G( 21164)=131 G( 21204)=131 G( 21210)=131 G( 21335)=131
G( 21352)=133 G( 21355)=133 G( 21375)=132 G( 21401)=133 G( 21654)=133
G( 21671)=132 G( 21926)=133 G( 21972)=136 G( 22036)=136 G( 22068)=131
G( 22324)=132 G( 22349)= 60 G( 22401)= 60 G( 22753)=133 G( 22770)=131
G( 23341)= 72 G( 23423)=131 G( 23478)=132 G( 23693)=133 G( 23710)=132
G( 23719)=132 G( 23774)=132 G( 23849)= 78 G( 24044)=133 G( 24050)=133
G( 24175)=133 G( 24285)=133 G( 24327)= 79 G( 24337)=130 G( 24344)= 73
G( 24383)=133 G( 24386)=133 G( 24465)=133 G( 24893)=156 G( 24978)=133
G( 25079)=136 G( 25692)=130 G( 26227)= 80 G( 26230)= 80 G( 26285)= 80
G( 26305)= 92 G( 26317)= 81 G( 26444)=161 G( 26526)= 80 G( 26598)= 92
G( 26613)= 81 G( 27559)=157 G( 27936)=156 G( 28456)=132 G( 29448)=157
G( 30877)= 92 G( 32487)=192 G( 32986)=193 G( 33195)=199 G( 33465)=193
G( 33468)=193 G( 33523)=193 G( 33578)=193 G( 33642)=193 G( 33697)=193
G( 33842)=192 G( 36041)=193 G( 36184)=157 G( 82860)=108 G(47461861)=261
G(47482266)=258 G(47595779)=259 G(47844890)=260 G(47846178)=260 G(47850055)=260
G(47879108)=261 G(47912346)=258 G(48042117)=258 G(48046310)=261 G(48142092)=258
G(48142376)=265 G(48399022)=259
Known Maximal SGV of Grundy's Game
G( 3)= 1
G( 5)= 2
G( 13)= 3
G( 18)= 4
G( 41)= 5
G( 87)= 7
G( 138)= 8
G( 158)= 11
G( 181)= 12
G( 283)= 14
G( 321)= 16
G( 541)= 21
G( 614)= 22
G( 645)= 24
G( 791)= 26
G( 939)= 28
G( 1016)= 29
G( 1056)= 32
G( 1291)= 35
G( 1349)= 36
G( 1393)= 37
G( 1399)= 39
G( 1535)= 40
G( 1742)= 42
G( 1882)= 50
G( 2035)= 51
G( 3029)= 52
G( 3032)= 59
G( 3197)= 60
G( 4019)= 64
G( 4985)= 65
G( 5006)= 68
G( 5441)= 69
G( 5744)= 75
G( 7091)= 82
G( 7384)= 85
G( 8337)=101
G( 11261)=113
G( 11621)=118
G( 11885)=119
G( 12037)=127
G( 13133)=128
G( 13146)=129
G( 13679)=133
G( 16118)=135
G( 16375)=139
G( 17095)=142
G( 17751)=147
G( 21831)=148
G( 21918)=149
G( 22133)=152
G( 24090)=153
G( 24565)=159
G( 26444)=161
G( 27344)=174
G( 27492)=175
G( 27649)=176
G( 27875)=186
G( 28232)=187
G( 28304)=195
G( 33195)=199
G( 42251)=219
G( 42515)=221
G( 45668)=230
G( 763622)=231
G( 3653177)=234
G( 7250049)=256
G( 47339607)=257
G( 47342322)=262
G( 47342972)=263
G( 47879404)=266
G( 48142611)=267
G( 50662338)=268
G( 50662846)=275
G( 50666285)=280
G( 475094020)=281
G(3677482157)=287
G(4563802297)=291
G(21544358444)= 293
G(21544358589)= 297
Frequencies of Sprague-Grundy-Values of Grundy's Game: Heapsizes 0 - 235
0. 42 1222 1. 54 1343 2 235895639 34359732094 3 235922007 34359732639
4 280693375 34359737272 5 280895124 34359736956 6. 13 1733 7. 20 1445
8 211258329 34359738127 9 211373705 34359738077 10. 62 16230 11. 51 16204
12. 97 12516 13. 77 11843 14 358751606 34359738300 15 359055301 34359738222
16 1503739408 34359738117 17 1503699812 34359738204 18. 21 3675 19. 13 3298
20. 25 4497 21. 26 4206 22 825636790 34359738326 23 825656992 34359738178
24. 19 14601 25. 13 14935 26 319548008 34359738340 27 319478576 34359737908
28 185149233 34359737645 29 185115516 34359737790 30. 18 7800 31. 17 7469
32 1432455778 34359738366 33 1432555345 34359738355 34. 26 3694 35. 23 9013
36. 32 9550 37. 32 9762 38 811959642 34359738367 39 812035154 34359738332
40. 10 7778 41. 12 8250 42 289917958 34359737987 43 289907373 34359737888
44 207300631 34359738187 45 207296837 34359737633 46. 15 8363 47. 12 19611
48. 14 7120 49. 10 6853 50 1227313476 34359738358 51 1227222541 34359738314
52 693170502 34359738347 53 693248697 34359738364 54. 3 7024 55. 1 6052
56 290176385 34359738346 57 290145505 34359738198 58. 12 16771 59. 12 11514
60. 22 22401 61. 14 16960 62 226602151 34359738140 63 226630568 34359738004
64 478307463 34359738342 65 478250973 34359738359 66. 28 12186 67. 32 12029
68. 31 12154 69. 28 12229 70 465913616 34359738341 71 465941487 34359738360
72. 16 23341 73. 20 24344 74 368783312 34359738336 75 368840406 34359738365
76 275915447 34359738310 77 275950754 34359738287 78. 31 23849 79. 24 24327
80. 19 26526 81. 14 26613 82 1029607524 34359738325 83 1029525291 34359738357
84 556334066 34359738226 85 556295814 34359738316 86. 3 13398 87. 1 13543
88 146328314 34359737626 89 146315485 34359737713 90. 2 13586 91. 4 13917
92. 7 30877 93. 5 14416 94 90236028 34359737063 95 90231908 34359737580
96. 9 18617 97. 4 16320 98 528576852 34359738363 99 528530815 34359738334
100 311894496 34359738322 101 311836535 34359738290 102. 1 12056 103. 0 0
104 86393927 34359736389 105 86379232 34359736653 106. 13 19035 107. 11 17193
108. 13 82860 109. 9 16999 110 60399562 34359735810 111 60369087 34359733918
112 360684819 34359738276 113 360658391 34359736534 114. 0 0 115. 1 15980
116. 1 16398 117. 3 16369 118 198913741 34359738263 119 198923485 34359738142
120. 0 0 121. 1 13168 122 79408249 34359736914 123 79405905 34359738333
124 56607685 34359738345 125 56608268 34359733686 126. 1 15620 127. 5 15936
128 1806678461 34359738318 129 1804727844 34359737636 130. 21 25692 131. 16 23423
132. 23 28456 133. 20 24978 134 398394608 34359738350 135 396379620 34359737500
136. 12 25079 137. 7 20829 138 129299929 34359738339 139 129279728 34359738362
140 65196398 34359737782 141 65290631 34359738330 142. 3 18735 143. 0 0
144. 0 0 145. 0 0 146 160992400 34359738353 147 160880683 34359738095
148 84017750 34359737704 149 83903099 34359738017 150. 0 0 151. 0 0
152 39980149 34359735508 153 39967095 34359736337 154. 0 0 155. 0 0
156. 2 27936 157. 3 36184 158 23986135 34359738031 159 23921450 34359736577
160. 0 0 161. 1 26444 162 54651853 34359736737 163 54533116 34359736699
164 34139852 34359735261 165 34015582 34359735406 166. 0 0 167. 0 0
168 16494072 34359735165 169 16428793 34359735374 170. 0 0 171. 0 0
172. 0 0 173. 0 0 174 11700980 34359734663 175 11666639 34359730547
176 19859043 34359707349 177 19776458 34359730077 178. 0 0 179. 0 0
180. 0 0 181. 0 0 182 13506532 34359734158 183 13401304 34359729647
184. 0 0 185. 0 0 186 19134971 34359734370 187 18985055 34359733273
188 13300262 34359732838 189 13197488 34359714470 190. 0 0 191. 0 0
192. 2 33842 193. 8 36041 194 37114406 34359734097 195 37055191 34359733250
196 16231617 34359733305 197 16184771 34359707381 198. 0 0 199. 1 33195
200 6080484 34359636546 201 6064567 34359679501 202. 0 0 203. 0 0
204. 0 0 205. 0 0 206 3504299 34359676809 207 3497464 34359676893
208 7485000 34359643423 209 7458701 34359615616 210. 0 0 211. 0 0
212. 0 0 213. 0 0 214 4317144 34359613809 215 4291256 34359643957
216. 0 0 217. 0 0 218 3608878 34359646806 219 3542173 34359734128
220 2514717 34359644218 221 2468474 34359676571 222. 0 0 223. 0 0
224 5882328 34359679080 225 5836201 34359608991 226. 0 0 227. 0 0
228. 0 0 229. 0 0 230 2709648 34359578753 231 2679408 34359676423
232. 0 0 233. 0 0 234 1640426 34359544242 235 1599664 34359517632
236 1031424 34357900336 237 1008700 34359520728 238. 0 0 239. 0 0
240. 0 0 241. 0 0 242 1322586 34358452291 243 1287974 34359520853
244 791603 34357867597 245 763807 34343051344 246. 0 0 247. 0 0
248 602026 34343054320 249 574824 34347226378 250. 0 0 251. 0 0
252. 0 0 253. 0 0 254 367766 34343053940 255 352104 34358416564
256 2797051 34359543952 257 731801 34297682812 258. 4 48142092 259. 2 48399022
260. 3 47850055 261. 3 48046310 262 327878 34297683645 263 83584 34297673829
264. 0 0 265. 1 48142376 266 38690 34297672985 267 13314 34297672901
268 4397 34290991669 269 2685 34294514393 270. 0 0 271. 0 0
272. 0 0 273. 0 0 274 1565 34297673658 275 796 34142944789
276 327 34143265842 277 222 33993629629 278. 0 0 279. 0 0
280 167 33519727886 281 65 34134481987 282. 0 0 283. 0 0
284. 0 0 285. 0 0 286 14 31213011028 287 7 33165138416
288. 0 0 289. 0 0 290 2 21544358850 291 2 19626619612
292 0 0 293 1 21544358444 294. 0 0 295. 0 0
296 0 0 297 1 21544358589
Each of the four stripes of parallel displayed triplets consists of the running SG-value, its number of appearance and the index of its last occurrence.
Calculated until January 1992: all G(n) with n <= 80*106 which yielded the discovery of 13 new sparse values to the previously 1264 known ones.
Calculated November and December 1993: all G(n) with n <= 502*106.
From August 1995 until March 1996 Dan Hoey computed all values G(n)
up to 11*109. All common values up to 287 have occured in this range.
The last maximal value is G(4563802297)=291, but no further sparse values appeared.
Calculated in December 2000: all G(n) with n <= 5*232.
On average 3130 common values were to check to verify the next ``Nim-value candidate'' and the farest to-iterate index=16746922727 was 50248198 apart.
Calculated until mid of May 2002: all G(n) with n <= 235
at most 3 digit Octal-Games
Starting on November 1st 2012, I made two runs finally reached G(284944236543) on 2012-11-25 and abort the job. I collected the data from the protocol files and even made the statistics correct (several 32-bit wrap arounds in counters had occured), but I did not put the result into this web-page only among the octal games in stat_grundys as usually.
Finally yesterday I noticed this gap and thus I added the new frequencies for Grundys Game here:
Frequencies of Sprague-Grundy-Values of Grundy's Game: Heapsizes 0 - 238
0 42 1 54 2 1937371169 3 1937791847 4 2271413993 5 2272956896 6 13 7 20
8 1715851419 9 1716692598 10 62 11 51 12 97 13 77 14 2864992436 15 2867420227
16 11880267438 17 11879817069 18 21 19 13 20 25 21 26 22 6537674341 23 6537662588
24 19 25 13 26 2602944767 27 2602631805 28 1528466393 29 1528354394 30 18 31 17
32 11339856342 33 11339836157 34 26 35 23 36 32 37 32 38 6443776137 39 6444001521
40 10 41 12 42 2363015135 43 2363152671 44 1696390166 45 1696560915 46 15 47 12
48 14 49 10 50 9772034621 51 9771416446 52 5535022881 53 5535692650 54 3 55 1
56 2324745544 57 2324731742 58 12 59 12 60 22 61 14 62 1814589480 63 1814732624
64 3825663932 65 3825516720 66 28 67 32 68 31 69 28 70 3707987655 71 3707543402
72 16 73 20 74 2972284039 75 2972364290 76 2222613933 77 2222780447 78 31 79 24
80 19 81 14 82 8196800444 83 8196762817 84 4436062291 85 4435936416 86 3 87 1
88 1190767181 89 1190646795 90 2 91 4 92 7 93 5 94 736243025 95 736208333
96 9 97 4 98 4235864640 99 4235796665 100 2503539474 101 2503129865 102 1 103 0
104 707086742 105 707061447 106 13 107 11 108 13 109 9 110 494230345 111 494209457
112 2901226760 113 2901307137 114 0 115 1 116 1 117 3 118 1601553884 119 1601614202
120 0 121 1 122 648919383 123 648907344 124 462185551 125 462240014 126 1 127 5
128 14274867196 129 14258948285 130 21 131 16 132 23 133 20 134 3175304916 135 3159393344
136 12 137 7 138 1051168655 139 1051153837 140 538494002 141 539230451 142 3 143 0
144 0 145 0 146 1319214690 147 1318346728 148 696484304 149 695332967 150 0 151 0
152 328298542 153 328225216 154 0 155 0 156 2 157 3 158 199152117 159 198588457
160 0 161 1 162 465504582 163 464400174 164 291731049 165 290707996 166 0 167 0
168 140673791 169 140152647 170 0 171 0 172 0 173 0 174 100037833 175 99771741
176 170771977 177 170010188 178 0 179 0 180 0 181 0 182 116146256 183 115271572
184 0 185 0 186 165110118 187 163806296 188 114992648 189 114055106 190 0 191 0
192 2 193 8 194 320231783 195 319831546 196 140290801 197 139891031 198 0 199 1
200 52566211 201 52449781 202 0 203 0 204 0 205 0 206 30266265 207 30194418
208 64774630 209 64590485 210 0 211 0 212 0 213 0 214 37364652 215 37149284
216 0 217 0 218 31130896 219 30584723 220 21743157 221 21328957 222 0 223 0
224 50908619 225 50526991 226 0 227 0 228 0 229 0 230 23487755 231 23236338
232 0 233 0 234 14208040 235 13864913 236 8931532 237 8735884 238 0 239 0
240 0 241 0 242 11480651 243 11164319 244 6887643 245 6623781 246 0 247 0
248 5226640 249 4997798 250 0 251 0 252 0 253 0 254 3183362 255 3052190
256 24263253 257 6355881 258 4 259 2 260 3 261 3 262 2845945 263 723571
264 0 265 1 266 336062 267 117636 268 39187 269 24471 270 0 271 0
272 0 273 0 274 13521 275 6875 276 3074 277 1950 278 0 279 0
280 1628 281 531 282 0 283 0 284 0 285 0 286 166 287 71
288 0 289 0 290 21 291 9 292 3 293 2 294 0 295 0
296 0 297 1
sparse=1287 last=48399022 depth=50657340 (3129.46) largest=21544358589
For those who are interested in the statistics until 238 + 150*226.
To discover a possible periodicity in its sg-values, I protocoled all n with G(n) >= 286. But there shows none.
1.6.96 18:50 updated 2002-05-23
last updated 2021-04-10 16:17 UTC+2