2226:
1743:
1825:
to anything: it is simply a game involving meaningless strings. Working within the system, an algorithm could successively generate every valid string of symbols in an attempt to generate MU, and though it would never succeed, it would search forever, never deducing that the quest was futile. For a
1837:
The inability of the MIU system to express or deduce facts about itself, such as the inability to derive MU, is a consequence of its simplicity. However, more complex formal systems, such as systems of mathematical logic, may possess this ability. This is the key idea behind
1820:
It also demonstrates the contrast between interpretation on the "syntactic" level of symbols and on the "semantic" level of meanings. On the syntactic level, there is no knowledge of the MU puzzle's insolubility. The system does not
33:
involving a simple formal system called "MIU". Hofstadter's motivation is to contrast reasoning within a formal system (i.e., deriving theorems) against reasoning about the formal system itself. MIU is an example of a
781:
434:
891:
1834:
divisibility by three. This is the "semantic" level of the system — a level of meaning that the system naturally attains. On this level, the MU puzzle can be seen to be impossible.
1048:
809:
678:
1146:
1826:
human player, however, after a number of attempts, one soon begins to suspect that the puzzle may be impossible. One then "jumps out of the system" and starts to reason
842:
303:
by repeatedly applying the given rules. In other words, MU is not a theorem of the MIU formal system. To prove this, one must step "outside" the formal system itself.
1251:
1218:
986:
950:
705:
617:
590:
1303:
1277:
1934:
1080:
1006:
915:
725:
2125:
317:
in a string. Only the second and third rules change this number. In particular, rule two will double it while rule three will reduce it by 3. Now, the
1973:." As the Example column illustrates, a rule may be applied only to an entire MIU-number, not to an arbitrary part of its decimal representation.
2181:
1729:
in the first rule. Also, in this rendering, the arrangement of factors in this rule was made consistent with that of the other three rules.)
1470:
gives a mapping of the MIU system to arithmetic, as follows. First, every MIU-string can be translated to an integer by mapping the letters
2142:
1839:
2030:
1782:
2118:
730:
1093:
count can always be made even, by applying the first rule once, if necessary. Applying the fourth rule sufficiently often, all
1907:
are variables, standing for strings of symbols. A rule may be applied only to the whole string, not to an arbitrary substring.
380:
2321:
1813:
The MU string and the impossibility of its derivation is then analogous to a statement of mathematical logic which cannot be
1764:
1713:
The rendering of the first rule above differs superficially from that in the book, where it is written as "f we have made 10
2376:
2386:
2371:
2111:
1760:
1753:
2381:
2259:
62:
which can be combined to produce strings of symbols. The MU puzzle asks one to start with the "axiomatic" string
2083:
847:
1875:
307:
39:
2211:
2366:
2151:
1466:
1011:
29:
1802:— an encapsulation of mathematical and logical concepts using symbols. The MI string is akin to a single
35:
2225:
2062:
786:
2316:
2191:
1880:
1859:
634:
2327:
2264:
2201:
1830:
the system, rather than working within it. Eventually, one realises that the system is in some way
1108:
2254:
2171:
2134:
2003:
1814:
1807:
363:
24:
814:
2342:
2310:
2269:
2026:
343:
Subtracting 3 from a number that is not divisible by 3 does not make it divisible by 3 either.
1223:
1190:
958:
2294:
2279:
928:
683:
595:
568:
2284:
2249:
464:
2161:
2048:, Third Edition, Brooks/Cole, 2004. Chapter 8.4, "General Recursive Definitions," p. 501.
1282:
1256:
1919:
1855:
1065:
991:
900:
710:
2360:
2019:
1799:
1795:
The MIU system illustrates several important concepts in logic by means of analogy.
2289:
310:; that is, some quantity or property that doesn't change while applying the rules.
1936:
always exists, since the powers of 2 alternatingly evaluate to 1 and 2, modulo 3.
1742:
1725:
was reserved for use in exponents of 10 only, and therefore it was replaced by
1995:
306:
In order to prove assertions like this, it is often beneficial to look for an
340:
Doubling a number that is not divisible by 3 does not make it divisible by 3.
326:
2244:
1767: in this section. Unsourced material may be challenged and removed.
2103:
2087:
1482:
to the numbers 3, 1, and 0, respectively. (For example, the string
295:
The puzzle cannot be solved: it is impossible to change the string
1803:
2100:
An online JavaScript implementation of the MIU production system.
2079:
An online interface for producing derivations in the MIU System.
2107:
1496:
Third, the four formal rules given above become the following:
1957:
is any natural number smaller than 10. Each rule of the form "
1736:
1489:
Second, the single axiom of the MIU system, namely the string
70:
using in each step one of the following transformation rules:
2066:
776:{\displaystyle N\equiv 1{\text{ or }}N\equiv 2{\pmod {3}}}
16:
Puzzle in
Douglas Hofstadter's book "Gödel, Escher, Bach"
1717: + 1, then we can make 10 × (10
429:{\displaystyle n\equiv 2^{a}\not \equiv 0{\pmod {3}}.\,}
1922:
1285:
1259:
1226:
1193:
1111:
1068:
1014:
994:
961:
931:
903:
850:
817:
789:
733:
713:
686:
637:
598:
571:
383:
2347:
By
Hofstadter and the Fluid Analogies Research Group
1862:, and begins the relevant chapter with a quote from
2303:
2233:
2141:
1490:
1483:
1479:
1475:
1471:
1452:
1439:
1426:
1413:
1400:
1387:
1374:
1361:
1348:
1335:
1322:
1309:
1184:
1168:
1156:
1152:
1148:
1102:
1098:
1094:
1090:
1086:
1082:
1059:
1055:
1051:
952:
922:
918:
894:
624:
620:
548:
540:
536:
532:
528:
524:
504:
497:
487:
483:
479:
456:
371:
352:
348:
334:
322:
314:
300:
296:
280:
272:
267:
250:
237:
229:
224:
220:
211:
197:
184:
176:
171:
162:
151:
141:
133:
128:
124:
118:
107:
67:
63:
59:
55:
51:
2018:
1928:
1541:× 10 + 1)
1297:
1271:
1245:
1212:
1140:
1074:
1042:
1000:
980:
944:
909:
885:
836:
803:
775:
719:
699:
672:
611:
584:
428:
313:In this case, one can look at the total number of
1806:, and the four transformation rules are akin to
1151:. Applying the third rule to reduce triplets of
1858:uses the MU puzzle to introduce the concept of
1721: + 1)". Here, however, the variable
2119:
1187:, which respects properties 1 to 3, leads to
443:counts how often the second rule is applied.
8:
2021:Gödel, Escher, Bach: An Eternal Golden Braid
451:More generally, an arbitrarily given string
1998:Gödel, Escher, Bach: A Mental Space Odyssey
2126:
2112:
2104:
1798:It can be interpreted as an analogy for a
1953:stand for arbitrary natural numbers, and
1921:
1783:Learn how and when to remove this message
1284:
1258:
1231:
1225:
1198:
1192:
1132:
1116:
1110:
1067:
1022:
1015:
1013:
993:
966:
960:
936:
930:
902:
867:
855:
849:
822:
816:
797:
796:
788:
757:
743:
732:
712:
691:
685:
664:
648:
636:
603:
597:
576:
570:
471:respects the three following properties:
425:
406:
394:
382:
1514:
1509:
886:{\displaystyle 2^{n}\equiv N{\pmod {3}}}
1994:Justin Curry / Curran Kelleher (2007).
1986:
1892:
2046:Discrete Mathematics with Applications
1852:Discrete Mathematics with Applications
1305:; it can be hence derived as follows:
1179:To illustrate the construction in the
988:is divisible by 3, by construction of
551:respects properties 1 and 2. As shown
447:A decidable criterion for derivability
2182:Fluid Concepts and Creative Analogies
1965:" should be read as "if we have made
531:, or introduces any character out of
7:
1765:adding citations to reliable sources
1097:can then be deleted, thus obtaining
1043:{\displaystyle {\frac {2^{n}-N}{3}}}
1817:or disproven by the formal system.
1507:
875:
765:
414:
337:s is 1 which is not divisible by 3.
127:to the end of any string ending in
81:
14:
804:{\displaystyle n\in \mathbb {N} }
707:cannot be divisible by 3, hence,
552:
66:and transform it into the string
2224:
2017:Hofstadter, Douglas R. (1999) ,
1741:
565:respects properties 1 to 3, let
333:In the beginning, the number of
1752:needs additional citations for
1159:in the right spots will obtain
868:
758:
555:, it also respects property 3.
460:
407:
2322:Indiana University Bloomington
1840:Gödel's Incompleteness Theorem
1183:part of the proof, the string
879:
869:
769:
759:
673:{\displaystyle N=N_{I}+3N_{U}}
418:
408:
50:Suppose there are the symbols
1:
1085:, followed by some number of
355:cannot be achieved because 0
38:and can be reformulated as a
1612:× 10 + 111 × 10 +
1141:{\displaystyle N_{I}+3N_{U}}
727:cannot be, either. That is,
680:. By property 3, the number
170:Double the string after the
1679:
1658:
1629:
1608:
1583:
1564:
1543:
1527:
1486:would be mapped to 31010.)
897:, applying the second rule
893:. Beginning from the axiom
271:
246:
228:
193:
175:
150:
132:
103:
2403:
1688:
1638:
1592:
1552:
1008:, applying the third rule
837:{\displaystyle 2^{n}>N}
478:is only composed with one
2341:Edited by Hofstadter and
2337:
2222:
2063:"Hofstadter's MIU System"
1574:10 × (3 × 10 +
1493:, becomes the number 31.
93:
83:
631:, respectively, and let
527:, changes the number of
1876:Invariant (mathematics)
1246:{\displaystyle N_{U}=1}
1213:{\displaystyle N_{I}=4}
981:{\displaystyle 2^{n}-N}
40:string rewriting system
2260:Hofstadter's butterfly
1930:
1299:
1273:
1247:
1214:
1167:has been derived from
1142:
1076:
1044:
1002:
982:
946:
911:
887:
838:
805:
777:
721:
701:
674:
613:
586:
511:is not divisible by 3.
430:
321:is that the number of
23:is a puzzle stated by
2212:Surfaces and Essences
1931:
1860:recursive definitions
1733:Relationship to logic
1300:
1274:
1248:
1215:
1143:
1077:
1045:
1003:
983:
947:
945:{\displaystyle 2^{n}}
912:
888:
839:
806:
778:
722:
702:
700:{\displaystyle N_{I}}
675:
614:
612:{\displaystyle N_{U}}
587:
585:{\displaystyle N_{I}}
431:
374:obeys the congruence
36:Post canonical system
2377:Independence results
2317:Egbert B. Gebstadter
2192:Le Ton beau de Marot
1920:
1881:Unrestricted grammar
1761:improve this article
1283:
1257:
1224:
1191:
1109:
1066:
1012:
992:
959:
929:
901:
848:
815:
787:
731:
711:
684:
635:
596:
569:
455:can be derived from
381:
90:Informal explanation
2328:Victim of the Brain
2202:I Am a Strange Loop
2152:Gödel, Escher, Bach
1467:Gödel, Escher, Bach
1298:{\displaystyle n=4}
1272:{\displaystyle N=7}
543:. Therefore, every
362:In the language of
30:Gödel, Escher, Bach
2387:1979 introductions
2372:Unsolvable puzzles
2172:Metamagical Themas
2135:Douglas Hofstadter
2004:MIT OpenCourseWare
1926:
1808:rules of inference
1295:
1269:
1243:
1210:
1138:
1072:
1050:times will obtain
1040:
998:
978:
942:
907:
883:
834:
801:
773:
717:
697:
670:
609:
582:
523:No rule moves the
482:and any number of
426:
364:modular arithmetic
347:Thus, the goal of
319:invariant property
25:Douglas Hofstadter
2354:
2353:
2343:Daniel C. Dennett
2311:Robert Hofstadter
2270:Hofstadter points
1929:{\displaystyle n}
1850:In her textbook,
1793:
1792:
1785:
1705:
1704:
1362:MIIIIIIIIIIIIIIII
1075:{\displaystyle N}
1038:
1001:{\displaystyle n}
910:{\displaystyle n}
746:
720:{\displaystyle N}
619:be the number of
286:
285:
2394:
2382:Formal languages
2295:Superrationality
2280:Platonia dilemma
2265:Hofstadter's law
2228:
2217:
2207:
2197:
2187:
2177:
2167:
2157:
2128:
2121:
2114:
2105:
2099:
2097:
2095:
2086:. Archived from
2078:
2076:
2074:
2065:. Archived from
2049:
2043:
2037:
2036:Here: Chapter I.
2035:
2024:
2014:
2008:
2007:
1991:
1974:
1943:
1937:
1935:
1933:
1932:
1927:
1914:
1908:
1897:
1846:Pedagogical uses
1788:
1781:
1777:
1774:
1768:
1745:
1737:
1501:
1500:
1492:
1485:
1481:
1477:
1473:
1454:
1451:
1450:
1449:
1446:
1441:
1438:
1437:
1436:
1433:
1428:
1425:
1424:
1423:
1420:
1415:
1412:
1411:
1410:
1407:
1402:
1399:
1398:
1397:
1394:
1389:
1386:
1385:
1384:
1381:
1376:
1373:
1372:
1371:
1368:
1363:
1360:
1359:
1358:
1355:
1350:
1347:
1346:
1345:
1342:
1337:
1334:
1333:
1332:
1329:
1324:
1321:
1320:
1319:
1316:
1311:
1304:
1302:
1301:
1296:
1278:
1276:
1275:
1270:
1252:
1250:
1249:
1244:
1236:
1235:
1219:
1217:
1216:
1211:
1203:
1202:
1186:
1170:
1158:
1154:
1150:
1147:
1145:
1144:
1139:
1137:
1136:
1121:
1120:
1104:
1100:
1096:
1092:
1088:
1084:
1081:
1079:
1078:
1073:
1061:
1057:
1053:
1049:
1047:
1046:
1041:
1039:
1034:
1027:
1026:
1016:
1007:
1005:
1004:
999:
987:
985:
984:
979:
971:
970:
954:
951:
949:
948:
943:
941:
940:
924:
920:
916:
914:
913:
908:
896:
892:
890:
889:
884:
882:
860:
859:
843:
841:
840:
835:
827:
826:
810:
808:
807:
802:
800:
782:
780:
779:
774:
772:
747:
744:
726:
724:
723:
718:
706:
704:
703:
698:
696:
695:
679:
677:
676:
671:
669:
668:
653:
652:
626:
622:
618:
616:
615:
610:
608:
607:
591:
589:
588:
583:
581:
580:
550:
542:
538:
534:
530:
526:
506:
499:
489:
485:
481:
458:
435:
433:
432:
427:
421:
399:
398:
373:
359:divisible by 3.
354:
350:
336:
324:
316:
302:
298:
282:
274:
269:
252:
239:
231:
226:
222:
213:
199:
186:
178:
173:
164:
153:
143:
135:
130:
126:
120:
109:
75:
74:
69:
65:
61:
57:
53:
2402:
2401:
2397:
2396:
2395:
2393:
2392:
2391:
2357:
2356:
2355:
2350:
2333:
2299:
2285:Six nines in pi
2250:BlooP and FlooP
2238:
2236:
2229:
2220:
2215:
2205:
2195:
2185:
2175:
2165:
2155:
2137:
2132:
2093:
2091:
2082:
2072:
2070:
2069:on 4 March 2016
2061:
2058:
2053:
2052:
2044:
2040:
2033:
2025:, Basic Books,
2016:
2015:
2011:
1993:
1992:
1988:
1983:
1978:
1977:
1944:
1940:
1918:
1917:
1915:
1911:
1898:
1894:
1889:
1872:
1848:
1789:
1778:
1772:
1769:
1758:
1746:
1735:
1464:Chapter XIX of
1462:
1460:Arithmetization
1447:
1444:
1443:
1442:
1434:
1431:
1430:
1429:
1421:
1418:
1417:
1416:
1408:
1405:
1404:
1403:
1395:
1392:
1391:
1390:
1382:
1379:
1378:
1377:
1375:MIIIIIIIUIIIIII
1369:
1366:
1365:
1364:
1356:
1353:
1352:
1351:
1343:
1340:
1339:
1338:
1330:
1327:
1326:
1325:
1317:
1314:
1313:
1312:
1281:
1280:
1255:
1254:
1227:
1222:
1221:
1194:
1189:
1188:
1177:
1128:
1112:
1107:
1106:
1064:
1063:
1062:, with exactly
1018:
1017:
1010:
1009:
990:
989:
962:
957:
956:
932:
927:
926:
899:
898:
851:
846:
845:
818:
813:
812:
785:
784:
729:
728:
709:
708:
687:
682:
681:
660:
644:
633:
632:
599:
594:
593:
572:
567:
566:
518:
465:if, and only if
449:
390:
379:
378:
293:
48:
17:
12:
11:
5:
2400:
2398:
2390:
2389:
2384:
2379:
2374:
2369:
2359:
2358:
2352:
2351:
2349:
2348:
2345:
2338:
2335:
2334:
2332:
2331:
2324:
2319:
2314:
2307:
2305:
2301:
2300:
2298:
2297:
2292:
2287:
2282:
2277:
2272:
2267:
2262:
2257:
2252:
2247:
2241:
2239:
2234:
2231:
2230:
2223:
2221:
2219:
2218:
2208:
2198:
2188:
2178:
2168:
2158:
2147:
2145:
2139:
2138:
2133:
2131:
2130:
2123:
2116:
2108:
2102:
2101:
2090:on 14 May 2018
2080:
2057:
2056:External links
2054:
2051:
2050:
2038:
2031:
2009:
1985:
1984:
1982:
1979:
1976:
1975:
1938:
1925:
1909:
1891:
1890:
1888:
1885:
1884:
1883:
1878:
1871:
1868:
1856:Susanna S. Epp
1847:
1844:
1791:
1790:
1749:
1747:
1740:
1734:
1731:
1707:
1706:
1703:
1702:
1687:
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1681:
1678:
1669:
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1657:
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1652:
1637:
1634:
1631:
1628:
1619:
1616:
1607:
1603:
1602:
1591:
1588:
1585:
1582:
1572:
1569:
1565:3 × 10 +
1563:
1559:
1558:
1551:
1548:
1545:
1542:
1535:
1532:
1526:
1522:
1521:
1518:
1513:
1508:
1461:
1458:
1457:
1456:
1401:MIIIIIIIUUIIIU
1294:
1291:
1288:
1268:
1265:
1262:
1242:
1239:
1234:
1230:
1209:
1206:
1201:
1197:
1176:
1173:
1163:. Altogether,
1135:
1131:
1127:
1124:
1119:
1115:
1071:
1037:
1033:
1030:
1025:
1021:
997:
977:
974:
969:
965:
939:
935:
917:times obtains
906:
881:
878:
874:
871:
866:
863:
858:
854:
833:
830:
825:
821:
799:
795:
792:
771:
768:
764:
761:
756:
753:
750:
745: or
742:
739:
736:
716:
694:
690:
667:
663:
659:
656:
651:
647:
643:
640:
606:
602:
579:
575:
517:
514:
513:
512:
503:the number of
501:
491:
448:
445:
437:
436:
424:
420:
417:
413:
410:
405:
402:
397:
393:
389:
386:
345:
344:
341:
338:
292:
289:
288:
287:
284:
283:
278:
275:
270:
264:
259:
256:
245:
241:
240:
235:
232:
227:
217:
206:
203:
192:
188:
187:
182:
179:
174:
168:
160:
157:
149:
145:
144:
139:
136:
131:
121:
113:
110:
102:
98:
97:
92:
87:
82:
47:
44:
15:
13:
10:
9:
6:
4:
3:
2:
2399:
2388:
2385:
2383:
2380:
2378:
2375:
2373:
2370:
2368:
2367:Logic puzzles
2365:
2364:
2362:
2346:
2344:
2340:
2339:
2336:
2330:
2329:
2325:
2323:
2320:
2318:
2315:
2312:
2309:
2308:
2306:
2302:
2296:
2293:
2291:
2288:
2286:
2283:
2281:
2278:
2276:
2273:
2271:
2268:
2266:
2263:
2261:
2258:
2256:
2253:
2251:
2248:
2246:
2243:
2242:
2240:
2232:
2227:
2214:
2213:
2209:
2204:
2203:
2199:
2194:
2193:
2189:
2184:
2183:
2179:
2174:
2173:
2169:
2164:
2163:
2159:
2154:
2153:
2149:
2148:
2146:
2144:
2140:
2136:
2129:
2124:
2122:
2117:
2115:
2110:
2109:
2106:
2089:
2085:
2081:
2068:
2064:
2060:
2059:
2055:
2047:
2042:
2039:
2034:
2032:0-465-02656-7
2028:
2023:
2022:
2013:
2010:
2005:
2001:
2000:
1997:
1990:
1987:
1980:
1972:
1968:
1964:
1961: →
1960:
1956:
1952:
1948:
1942:
1939:
1923:
1913:
1910:
1906:
1902:
1896:
1893:
1886:
1882:
1879:
1877:
1874:
1873:
1869:
1867:
1865:
1861:
1857:
1853:
1845:
1843:
1841:
1835:
1833:
1829:
1824:
1818:
1816:
1811:
1809:
1805:
1801:
1800:formal system
1796:
1787:
1784:
1776:
1766:
1762:
1756:
1755:
1750:This section
1748:
1744:
1739:
1738:
1732:
1730:
1728:
1724:
1720:
1716:
1712:
1700:
1696:
1692:
1685:
1683: →
1682:
1677:
1673:
1670:
1668: →
1667:
1665:
1661:
1655:
1654:
1650:
1646:
1642:
1635:
1633: →
1632:
1627:
1623:
1620:
1618: →
1617:
1615:
1611:
1605:
1604:
1600:
1596:
1589:
1587: →
1586:
1581:
1577:
1573:
1571: →
1570:
1568:
1561:
1560:
1556:
1549:
1547: →
1546:
1540:
1536:
1534: →
1533:
1531:× 10 + 1
1530:
1524:
1523:
1519:
1517:
1512:
1506:
1503:
1502:
1499:
1498:
1497:
1494:
1487:
1469:
1468:
1459:
1388:MIIIIIIIUUIII
1308:
1307:
1306:
1292:
1289:
1286:
1266:
1263:
1260:
1240:
1237:
1232:
1228:
1207:
1204:
1199:
1195:
1182:
1174:
1172:
1166:
1162:
1133:
1129:
1125:
1122:
1117:
1113:
1069:
1035:
1031:
1028:
1023:
1019:
995:
975:
972:
967:
963:
937:
933:
904:
876:
872:
864:
861:
856:
852:
831:
828:
823:
819:
793:
790:
766:
762:
754:
751:
748:
740:
737:
734:
714:
692:
688:
665:
661:
657:
654:
649:
645:
641:
638:
630:
604:
600:
577:
573:
564:
560:
556:
554:
547:derived from
546:
522:
515:
510:
502:
495:
492:
477:
474:
473:
472:
470:
466:
462:
454:
446:
444:
442:
422:
415:
411:
403:
400:
395:
391:
387:
384:
377:
376:
375:
369:
366:, the number
365:
360:
358:
342:
339:
332:
331:
330:
328:
320:
311:
309:
304:
290:
279:
276:
265:
263:
260:
257:
255:
249:
243:
242:
236:
233:
218:
216:
210:
207:
204:
202:
196:
190:
189:
183:
180:
169:
167:
161:
158:
156:
147:
146:
140:
137:
122:
117:
114:
111:
106:
100:
99:
96:
91:
88:
86:
80:
77:
76:
73:
72:
71:
45:
43:
41:
37:
32:
31:
27:and found in
26:
22:
2326:
2290:Strange loop
2274:
2235:Concepts and
2210:
2200:
2190:
2180:
2170:
2162:The Mind's I
2160:
2150:
2092:. Retrieved
2088:the original
2071:. Retrieved
2067:the original
2045:
2041:
2020:
2012:
1999:
1996:
1989:
1970:
1969:we can make
1966:
1962:
1958:
1954:
1950:
1946:
1941:
1912:
1904:
1900:
1895:
1863:
1851:
1849:
1836:
1831:
1827:
1822:
1819:
1812:
1797:
1794:
1779:
1770:
1759:Please help
1754:verification
1751:
1726:
1722:
1718:
1714:
1710:
1708:
1698:
1694:
1690:
1675:
1674:× 10 +
1671:
1663:
1662:× 10 +
1659:
1648:
1644:
1640:
1625:
1624:× 10 +
1621:
1613:
1609:
1598:
1594:
1579:
1575:
1566:
1554:
1538:
1528:
1515:
1510:
1504:
1495:
1488:
1465:
1463:
1414:MIIIIIIIUUUU
1180:
1178:
1164:
1160:
628:
562:
558:
557:
544:
520:
519:
508:
496:begins with
493:
475:
468:
452:
450:
440:
438:
367:
361:
356:
346:
318:
312:
305:
294:
261:
253:
247:
219:Replace any
214:
208:
200:
194:
165:
154:
115:
104:
94:
89:
84:
78:
49:
28:
20:
18:
2084:"MU Puzzle"
2073:29 November
1537:10 × (
1511:Formal rule
463:four rules
266:Remove any
85:Formal rule
2361:Categories
1981:References
1427:MIIIIIIIUU
811:such that
351:with zero
46:The puzzle
2275:MU puzzle
1773:July 2013
1349:MIIIIIIII
1029:−
973:−
862:≡
794:∈
752:≡
738:≡
388:≡
327:divisible
308:invariant
21:MU puzzle
2313:(father)
2245:Ambigram
2237:projects
1916:Such an
1870:See also
1440:MIIIIIII
955:. Since
521:Only if:
401:≢
291:Solution
2304:Related
2255:Copycat
1689: (
1639: (
1630:3111011
1593: (
1553: (
1520:
1516:Example
1448:→
1435:→
1422:→
1409:→
1396:→
1383:→
1370:→
1357:→
1344:→
1331:→
1318:→
1175:Example
1155:into a
459:by the
325:is not
223:with a
95:Example
2216:(2013)
2206:(2007)
2196:(1997)
2186:(1995)
2176:(1985)
2166:(1981)
2156:(1979)
2094:13 May
2029:
1945:Here,
1899:Here,
1815:proven
1701:= 11)
1651:= 11)
1636:30011
1601:= 10)
1590:31010
1478:, and
1453:MIIUII
1185:MIIUII
1089:. The
783:. Let
553:before
439:where
329:by 3:
230:MUIIIU
123:Add a
58:, and
2143:Books
1887:Notes
1832:about
1828:about
1823:refer
1804:axiom
1697:= 2,
1693:= 3,
1680:30011
1647:= 3,
1643:= 3,
1597:= 2,
1557:= 3)
1484:MIUIU
1336:MIIII
1105:with
925:with
516:Proof
500:, and
461:above
299:into
185:MIUIU
2096:2018
2075:2016
2027:ISBN
1949:and
1903:and
1686:311
1578:) +
1550:310
1099:MIII
1052:MIII
919:MIII
844:and
829:>
623:and
592:and
486:and
273:MUUU
238:MUUU
19:The
1864:GEB
1763:by
1711:NB:
1656:4.
1606:3.
1584:310
1562:2.
1525:1.
1505:Nr.
1323:MII
1101:...
1058:...
1054:...
921:...
873:mod
763:mod
627:in
561:If
559:If:
507:in
412:mod
370:of
244:4.
221:III
198:III
191:3.
177:MIU
148:2.
142:MIU
101:1.
79:Nr.
2363::
2002:.
1866:.
1854:,
1842:.
1810:.
1544:31
1491:MI
1474:,
1310:MI
1279:,
1253:,
1220:,
1181:If
1171:.
1169:MI
1056:IU
895:MI
549:MI
539:,
535:,
467:,
457:MI
357:is
349:MU
301:MU
297:MI
281:MU
277:to
268:UU
262:xy
251:UU
234:to
181:to
166:xx
138:to
134:MI
119:IU
68:MU
64:MI
54:,
42:.
2127:e
2120:t
2113:v
2098:.
2077:.
2006:.
1971:y
1967:x
1963:y
1959:x
1955:n
1951:m
1947:k
1924:n
1905:y
1901:x
1786:)
1780:(
1775:)
1771:(
1757:.
1727:k
1723:m
1719:m
1715:m
1709:(
1699:n
1695:m
1691:k
1676:n
1672:k
1664:n
1660:k
1649:n
1645:m
1641:k
1626:n
1622:k
1614:n
1610:k
1599:n
1595:m
1580:n
1576:n
1567:n
1555:k
1539:k
1529:k
1480:U
1476:I
1472:M
1455:.
1445:3
1432:4
1419:4
1406:3
1393:1
1380:3
1367:3
1354:2
1341:2
1328:2
1315:2
1293:4
1290:=
1287:n
1267:7
1264:=
1261:N
1241:1
1238:=
1233:U
1229:N
1208:4
1205:=
1200:I
1196:N
1165:x
1161:x
1157:U
1153:I
1149:I
1134:U
1130:N
1126:3
1123:+
1118:I
1114:N
1103:I
1095:U
1091:U
1087:U
1083:I
1070:N
1060:U
1036:3
1032:N
1024:n
1020:2
996:n
976:N
968:n
964:2
953:I
938:n
934:2
923:I
905:n
880:)
877:3
870:(
865:N
857:n
853:2
832:N
824:n
820:2
798:N
791:n
770:)
767:3
760:(
755:2
749:N
741:1
735:N
715:N
693:I
689:N
666:U
662:N
658:3
655:+
650:I
646:N
642:=
639:N
629:x
625:U
621:I
605:U
601:N
578:I
574:N
563:x
545:x
541:U
537:I
533:M
529:M
525:M
509:x
505:I
498:M
494:x
490:,
488:U
484:I
480:M
476:x
469:x
453:x
441:a
423:.
419:)
416:3
409:(
404:0
396:a
392:2
385:n
372:I
368:n
353:I
335:I
323:I
315:I
258:→
254:y
248:x
225:U
215:y
212:U
209:x
205:→
201:y
195:x
172:M
163:M
159:→
155:x
152:M
129:I
125:U
116:x
112:→
108:I
105:x
60:U
56:I
52:M
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