985:
938:
891:
844:
797:
708:
682:
621:
567:
1020:
1013:
1006:
999:
992:
973:
966:
959:
952:
945:
926:
919:
912:
905:
898:
879:
872:
865:
858:
851:
832:
825:
818:
811:
804:
743:
736:
729:
722:
715:
696:
689:
675:
668:
661:
649:
642:
635:
628:
614:
602:
595:
588:
581:
574:
93:
25:
108:
A right-handed practitioner holds the cards from above in the left hand and from below in the right hand. The deck is separated into two preferably equal parts by simply lifting up half the cards with the right thumb slightly and pushing the left hand's packet forward away from the right hand. The
158:
An out-shuffle has the same result as removing the top and bottom cards, doing an in-shuffle on the remaining cards, and then replacing the top and bottom cards in their original positions. Repeated out-shuffles cannot reverse the order of the entire deck, only the middle n−2 cards. Mathematical
1883:
1274:
1382:
109:
two packets are often crossed and tapped against each other to align them. They are then pushed together on the short sides and bent either up or down. The cards will then alternately fall onto each other, ideally alternating one by one from each half, much like a
162:
An in-shuffle has the same result as adding one extraneous card at the top and one extraneous card at the bottom, doing an out-shuffle on the enlarged deck, and then removing the extraneous cards. Repeated in-shuffles can reverse the order of the deck.
124:, the above method was used, and he calls it the "faro dealer's shuffle". Maskelyne was the first to give clear instructions, but the shuffle was used and associated with faro earlier, as discovered mostly by the mathematician and magician
1712:
1701:
1149:
139:
A perfect faro shuffle, where the cards are perfectly alternated, requires the shuffler to cut the deck into two equal stacks and apply just the right pressure when pushing the half decks into each other.
85:, in which half of the deck is held in each hand with the thumbs inward, then cards are released by the thumbs so that they fall to the table interleaved. Diaconis, Graham, and Kantor also call this
1285:
1041:
discovered that a controlled series of in- and out-shuffles can be used to move the top card of the deck down into any desired position. The trick is to express the card's desired position as a
409:
311:
262:
458:
1052:). Shuffle in, out, in, out. Deal ten cards off the top of the deck; the eleventh will be your original card. Notice that it doesn't matter whether you express the number ten as 1010
2006:
1967:
1940:
1913:
1598:
1513:
1551:
1447:
1144:
1107:
1486:
1417:
1571:
354:
334:
207:
187:
100:
Mathematicians use the term "faro shuffle" to describe a precise rearrangement of a deck into two equal piles of 26 cards which are then interleaved perfectly.
46:
1878:{\displaystyle {\mbox{sgn}}(\rho _{n})=(-1)^{\lfloor n/2\rfloor }={\begin{cases}+1&n\equiv 0,1{\pmod {4}}\\-1&n\equiv 2,3{\pmod {4}}\end{cases}}}
1606:
411:. For example, if one manages to perform eight out-shuffles in a row, then the deck of 52 cards will be restored to its original order, because
1269:{\displaystyle {\begin{pmatrix}1&2&3&4&\cdots &2n-1&2n\\1&n+1&2&n+2&\cdots &n&2n\end{pmatrix}}}
3337:
755:
The following shows the order of the deck after each out-shuffle. A deck of this size returns to its original order after 4 out-shuffles.
166:
If one can do perfect in-shuffles, then 26 shuffles will reverse the order of the deck and 26 more will restore it to its original order.
525:
The following shows the order of the deck after each in-shuffle. A deck of this size returns to its original order after 3 in-shuffles.
3332:
3327:
33:
493:
3376:
500:
3622:
2530:
2330:
3275:
1377:{\displaystyle k\mapsto {\begin{cases}{\frac {k+1}{2}}&k\ {\text{odd}}\\n+{\frac {k}{2}}&k\ {\text{even}}\end{cases}}}
120:
ends with the cards in two equal piles that the dealer must combine to deal them for the next game. According to the magician
3735:
147:, while one that moves the original top card to second and the original bottom card to second from the bottom is known as an
3753:
3717:
2878:
3657:
2469:
3313:
3173:
3001:
143:
A faro shuffle that leaves the original top card at the top and the original bottom card at the bottom is known as an
96:
Comparison of a perfect faro out-shuffle and in-shuffle, the numbers denoting each card's positions before the shuffle
3093:
2236:
3726:
3298:
3293:
1048:
For example, to move the top card down so that there are ten cards above it, express the number ten in binary (1010
2971:
2087:
486:=2, 4, 6, 8, 10, 12 ..., the number of in-shuffles needed are: 2, 4, 3, 6, 10, 12, 4, 8, 18, 6, 11, ... (sequence
3615:
3208:
2821:
2622:
2501:
2496:
2484:
2865:
359:
267:
212:
3784:
3708:
3391:
2718:
414:
113:. A flourish can be added by springing the packets together by applying pressure and bending them from above.
3516:
3303:
2091:
1060:; preliminary out-shuffles will not affect the outcome because out-shuffles always keep the top card on top.
3629:
3461:
2817:
2813:
3799:
3198:
3168:
2750:
2516:
2362:
2323:
3308:
2991:
2768:
2129:
984:
937:
890:
843:
796:
707:
681:
620:
566:
3401:
3053:
1783:
1300:
3270:
3260:
3138:
3118:
2936:
2733:
1972:
1019:
1012:
1005:
998:
991:
972:
965:
958:
951:
944:
925:
918:
911:
904:
897:
878:
871:
864:
857:
850:
831:
824:
817:
810:
803:
742:
735:
728:
721:
714:
695:
688:
674:
667:
660:
648:
641:
634:
627:
613:
601:
594:
587:
580:
573:
3794:
3265:
3113:
3063:
3043:
38:
3650:
2474:
2316:
2207:
Sharps and Flats: A Complete
Revelation of the Secrets of Cheating at Games of Chance and Skill
1945:
1918:
1891:
1576:
1491:
3587:
2956:
2205:
3762:
3689:
3406:
3381:
3143:
3017:
2677:
2662:
2627:
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2463:
2347:
2226:
1518:
1422:
1119:
1082:
3681:
3472:
2713:
2699:
2687:
2667:
2617:
2479:
2424:
2264:
2201:
2165:
1459:
1390:
468:
121:
463:
In other words, the number of in-shuffles required to return a deck of cards of even size
8:
3789:
3608:
3366:
2873:
2682:
2436:
2419:
511:
460:. However, only 6 faro out-shuffles are required to restore the order of a 64-card deck.
2268:
3643:
3503:
3498:
2996:
2295:
1556:
472:
339:
319:
264:. For example, 52 consecutive in-shuffles restore the order of a 52-card deck, because
192:
172:
3549:
3537:
3493:
3386:
3347:
3233:
3193:
3022:
2961:
2441:
2358:
2280:
2232:
2149:
503:, it follows that there are infinitely many deck sizes which require the full set of
3601:
3594:
3559:
3438:
3319:
3252:
3183:
3123:
2632:
2272:
2180:
2144:
117:
2490:
3223:
3218:
2916:
2552:
2429:
2381:
2276:
2161:
1073:
136:
The faro shuffle is a controlled shuffle that does not fully randomize a deck.
3744:
3636:
3542:
3522:
3477:
3418:
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2125:
2117:
125:
3158:
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2845:
2840:
2797:
2758:
2728:
2723:
2412:
2407:
2376:
2371:
2121:
1042:
1696:{\displaystyle {\mbox{sgn}}(\rho _{n})=(-1)^{n+1}{\mbox{sgn}}(\rho _{n-1}).}
3482:
3423:
3213:
3108:
3103:
3038:
2986:
2835:
2802:
2781:
2386:
2339:
2284:
1038:
152:
82:
3048:
2255:
Kolata, Gina (April 1982). "Perfect
Shuffles and Their Relation to Math".
1113:
is the permutation that splits the set into 2 piles and interleaves them:
510:
The analogous operation to an out-shuffle for an infinite sequence is the
3674:
3508:
3455:
3450:
3396:
3178:
3068:
2946:
2830:
2506:
2294:
Jain, Peiyush (May 2008). "A simple in-place algorithm for in-shuffles".
1069:
3564:
3554:
3428:
3228:
3098:
2921:
2853:
2602:
2184:
1045:, and then do an in-shuffle for each 1 and an out-shuffle for each 0.
3527:
3352:
3148:
3088:
2931:
2672:
2446:
79:
3163:
2966:
2611:
2595:
2581:
2300:
151:. These names were coined by the magician and computer programmer
3487:
3058:
2981:
2565:
2558:
2308:
159:
theorems regarding faro shuffles tend to refer to out-shuffles.
92:
24:
2951:
2883:
2645:
2638:
2574:
2541:
110:
2926:
2705:
2228:
Magic Tricks, Card
Shuffling, and Dynamic Computer Memories
1871:
1370:
488:
2402:
2166:"The Cycles of the Multiway Perfect Shuffle Permutation"
1072:, a perfect shuffle can be considered an element of the
2173:
1717:
1662:
1611:
1158:
1975:
1948:
1921:
1894:
1715:
1609:
1579:
1559:
1521:
1494:
1462:
1425:
1393:
1288:
1152:
1122:
1085:
417:
362:
342:
322:
270:
215:
195:
175:
2116:
2000:
1961:
1934:
1907:
1877:
1695:
1592:
1565:
1545:
1507:
1480:
1441:
1411:
1376:
1268:
1138:
1101:
452:
403:
348:
336:perfect out-shuffles will restore the order of an
328:
305:
256:
201:
181:
522:For simplicity, we will use a deck of six cards.
189:perfect in-shuffles will restore the order of an
3776:
2159:
1419:-perfect shuffle permutation is the element of
3338:Worshipful Company of Makers of Playing Cards
2324:
1770:
1756:
2231:. The Mathematical Association of America.
3333:Saga Arashiyama Museum of Arts and Culture
2331:
2317:
404:{\displaystyle 2^{k}\equiv 1{\pmod {n-1}}}
306:{\displaystyle 2^{52}\equiv 1{\pmod {53}}}
257:{\displaystyle 2^{k}\equiv 1{\pmod {n+1}}}
16:Perfectly interleaved playing card shuffle
2299:
2200:
2148:
453:{\displaystyle 2^{8}\equiv 1{\pmod {51}}}
91:
49:of all important aspects of the article.
3623:The Card Sharp with the Ace of Diamonds
1063:
1032:
3777:
3276:International Skat Players Association
2254:
2224:
2078:Diaconis, Graham, and Kantor 1983, 193
2024:Diaconis, Graham, and Kantor 1983, 188
45:Please consider expanding the lead to
3377:Blackstone's Card Trick Without Cards
3360:
2312:
2130:"The mathematics of perfect shuffles"
501:Artin's conjecture on primitive roots
2293:
1888:The first few perfect shuffles are:
467:, to original order is given by the
18:
3658:Violin and Playing Cards on a Table
3251:
1860:
1853:
1817:
1810:
442:
387:
295:
240:
131:
13:
3328:Musée Français de la Carte à Jouer
3314:International Playing-Card Society
3002:United States Playing Card Company
14:
3811:
2338:
3299:Fournier Museum of Playing Cards
3294:Cary Collection of Playing Cards
1018:
1011:
1004:
997:
990:
983:
971:
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659:
647:
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633:
626:
619:
612:
600:
593:
586:
579:
572:
565:
482:For example, for a deck size of
23:
3616:Card Players in a Rich Interior
2137:Advances in Applied Mathematics
2088:Real v recreational mathematics
435:
380:
288:
233:
37:may be too short to adequately
2097:
2081:
2072:
2063:
2054:
2045:
2036:
2027:
2018:
1982:
1976:
1864:
1854:
1821:
1811:
1752:
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1723:
1687:
1668:
1646:
1636:
1630:
1617:
1540:
1522:
1475:
1463:
1406:
1394:
1292:
1279:In other words, it is the map
446:
436:
397:
381:
299:
289:
250:
234:
103:
47:provide an accessible overview
1:
3517:The Phantom of the Card Table
3304:German Doppelkopf Association
2972:Nürnberger-Spielkarten-Verlag
2210:. Longmans, Green and Company
2110:
2001:{\displaystyle (23)\in S_{4}}
1706:The sign is thus 4-periodic:
3736:Patience and solitaire games
3630:Cardplayers in a Sunlit Room
3462:The Expert at the Card Table
2277:10.1126/science.216.4545.505
2150:10.1016/0196-8858(83)90009-X
2103:Ellis, Fan, and Shallit 2002
1515:, is the composition of the
1453:piles and interleaves them.
7:
3754:Tarot and Tarock card games
3718:Non trick-taking card games
3199:Richard Valentine Pitchford
3169:Master of the Playing Cards
10:
3816:
3309:German Playing Card Museum
1488:-perfect shuffle, denoted
517:
3703:
3666:
3580:
3573:
3346:
3242:
3081:
3031:
3010:
2909:
2900:
2864:
2811:
2790:
2767:
2749:
2742:
2655:
2529:
2455:
2395:
2357:
2346:
2225:Morris, S. Brent (1998).
1962:{\displaystyle \rho _{2}}
1935:{\displaystyle \rho _{1}}
1908:{\displaystyle \rho _{0}}
1593:{\displaystyle \rho _{n}}
1553:-perfect shuffle with an
1508:{\displaystyle \rho _{n}}
1449:that splits the set into
3271:International Skat Court
3261:British Skat Association
2011:
3727:Trick-taking card games
3266:German Skat Association
3044:Bielefelder Spielkarten
1573:-cycle, so the sign of
1546:{\displaystyle (2,n-1)}
3651:Still Life with a Poem
2854:Guru (circular whirls)
2002:
1963:
1936:
1909:
1879:
1697:
1594:
1567:
1547:
1509:
1482:
1443:
1442:{\displaystyle S_{kn}}
1413:
1378:
1270:
1140:
1139:{\displaystyle S_{2n}}
1103:
1102:{\displaystyle S_{2n}}
454:
405:
350:
330:
307:
258:
203:
183:
97:
89:, when used in magic.
3785:Card game terminology
3709:Historical card games
3690:Play Your Cards Right
3407:Twenty-One Card Trick
3382:The Circus Card Trick
3144:Johann Kaspar Hechtel
2470:Archaeology awareness
2464:Ambraser Hofjagdspiel
2003:
1969:is the transposition
1964:
1937:
1910:
1880:
1698:
1595:
1568:
1548:
1510:
1483:
1481:{\displaystyle (2,n)}
1444:
1414:
1412:{\displaystyle (k,n)}
1379:
1271:
1141:
1104:
455:
406:
351:
331:
308:
259:
204:
184:
95:
3473:History of cardistry
2957:J.O. Öberg & Son
2480:Flemish Hunting Deck
2160:Ellis, J.; Fan, H.;
1973:
1946:
1919:
1892:
1713:
1607:
1577:
1557:
1519:
1492:
1460:
1423:
1391:
1286:
1150:
1120:
1083:
1064:Group theory aspects
1033:As deck manipulation
469:multiplicative order
415:
360:
340:
320:
268:
213:
193:
173:
3667:Film and television
3609:The Card Players II
3367:The Acme of Control
2269:1982Sci...216..505K
2051:Maskelyne 1894, 204
1079:More generally, in
512:interleave sequence
3800:Permutation groups
3644:Dogs Playing Poker
3209:Johann Georg Rauch
2997:U.S. Games Systems
2359:Playing card suits
2185:10.46298/dmtcs.308
2069:Morris 1998, 11–12
1998:
1959:
1932:
1905:
1875:
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1721:
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1666:
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1563:
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1099:
450:
401:
346:
326:
303:
254:
199:
179:
98:
3772:
3771:
3699:
3698:
3550:Si Stebbins stack
3414:
3413:
3392:Out of This World
3387:The Four Burglars
3348:Card manipulation
3284:
3283:
3234:Fusajiro Yamauchi
3194:Ferdinand Piatnik
3077:
3076:
2896:
2895:
2623:Portuguese-suited
2525:
2524:
2502:Most-wanted Iraqi
2485:Hamas most wanted
2442:Curse of Scotland
2263:(4545): 505–506.
2094:, April 10, 2014.
1942:are trivial, and
1720:
1665:
1614:
1566:{\displaystyle n}
1387:Analogously, the
1365:
1361:
1352:
1332:
1328:
1319:
1026:
1025:
749:
748:
479: + 1).
349:{\displaystyle n}
329:{\displaystyle k}
202:{\displaystyle n}
182:{\displaystyle k}
64:
63:
3807:
3767:
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3758:
3752:
3749:
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3707:
3602:The Card Players
3595:The Bezique Game
3578:
3577:
3560:Three-card monte
3439:Cards in the hat
3358:
3357:
3320:The Playing-Card
3249:
3248:
3184:Samuel J. Murray
3124:Thierry Depaulis
2907:
2906:
2791:German and Swiss
2747:
2746:
2553:Four-colour pack
2355:
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2043:
2042:Morris 1998, 111
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188:
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180:
132:Perfect shuffles
76:dovetail shuffle
59:
56:
50:
27:
19:
3815:
3814:
3810:
3809:
3808:
3806:
3805:
3804:
3775:
3774:
3773:
3768:
3765:
3759:
3756:
3750:
3747:
3741:
3738:
3732:
3729:
3723:
3720:
3714:
3711:
3705:
3695:
3662:
3569:
3410:
3350:
3342:
3280:
3244:
3238:
3224:Charles Troedel
3219:Howard Thurston
3139:Richard Harding
3073:
3054:Charles Goodall
3027:
3006:
2917:ASS Altenburger
2902:
2892:
2889:Tens of Myriads
2879:Strings of Cash
2866:Chinese (Money)
2860:
2807:
2786:
2763:
2738:
2651:
2532:
2521:
2451:
2430:Queen of spades
2391:
2349:
2342:
2337:
2243:
2241:
2239:
2213:
2211:
2202:Maskelyne, John
2189:
2187:
2168:
2132:
2118:Diaconis, Persi
2113:
2108:
2107:
2102:
2098:
2086:
2082:
2077:
2073:
2068:
2064:
2059:
2055:
2050:
2046:
2041:
2037:
2033:Morris 1998, 13
2032:
2028:
2023:
2019:
2014:
1992:
1988:
1974:
1971:
1970:
1953:
1949:
1947:
1944:
1943:
1926:
1922:
1920:
1917:
1916:
1899:
1895:
1893:
1890:
1889:
1869:
1868:
1852:
1835:
1826:
1825:
1809:
1792:
1779:
1778:
1762:
1755:
1751:
1730:
1726:
1716:
1714:
1711:
1710:
1675:
1671:
1661:
1649:
1645:
1624:
1620:
1610:
1608:
1605:
1604:
1584:
1580:
1578:
1575:
1574:
1558:
1555:
1554:
1520:
1517:
1516:
1499:
1495:
1493:
1490:
1489:
1461:
1458:
1457:
1430:
1426:
1424:
1421:
1420:
1392:
1389:
1388:
1368:
1367:
1362:
1354:
1344:
1335:
1334:
1329:
1321:
1305:
1303:
1296:
1295:
1287:
1284:
1283:
1259:
1258:
1250:
1245:
1240:
1229:
1224:
1213:
1207:
1206:
1198:
1184:
1179:
1174:
1169:
1164:
1154:
1153:
1151:
1148:
1147:
1127:
1123:
1121:
1118:
1117:
1111:perfect shuffle
1090:
1086:
1084:
1081:
1080:
1074:symmetric group
1066:
1059:
1055:
1051:
1035:
786:
769:
556:
539:
520:
487:
434:
422:
418:
416:
413:
412:
379:
367:
363:
361:
358:
357:
341:
338:
337:
321:
318:
317:
287:
275:
271:
269:
266:
265:
232:
220:
216:
214:
211:
210:
194:
191:
190:
174:
171:
170:
134:
106:
78:is a method of
60:
54:
51:
44:
32:This article's
28:
17:
12:
11:
5:
3813:
3803:
3802:
3797:
3795:Card shuffling
3792:
3787:
3770:
3769:
3704:
3701:
3700:
3697:
3696:
3694:
3693:
3686:
3678:
3670:
3668:
3664:
3663:
3661:
3654:
3647:
3640:
3637:The Cardsharps
3633:
3626:
3619:
3612:
3605:
3598:
3591:
3584:
3582:
3575:
3571:
3570:
3568:
3567:
3562:
3557:
3552:
3547:
3546:
3545:
3540:
3535:
3525:
3523:Second dealing
3520:
3513:
3512:
3511:
3506:
3501:
3496:
3485:
3480:
3478:House of cards
3475:
3470:
3465:
3458:
3453:
3448:
3443:
3442:
3441:
3431:
3426:
3421:
3419:Bottom dealing
3415:
3412:
3411:
3409:
3404:
3399:
3394:
3389:
3384:
3379:
3374:
3372:Ambitious Card
3369:
3364:
3362:
3355:
3344:
3343:
3341:
3340:
3335:
3330:
3325:
3324:
3323:
3311:
3306:
3301:
3296:
3291:
3285:
3282:
3281:
3279:
3278:
3273:
3268:
3263:
3257:
3255:
3246:
3240:
3239:
3237:
3236:
3231:
3226:
3221:
3216:
3211:
3206:
3204:Franco Pratesi
3201:
3196:
3191:
3186:
3181:
3176:
3171:
3166:
3161:
3156:
3154:Sekiryo Kaneda
3151:
3146:
3141:
3136:
3131:
3126:
3121:
3116:
3111:
3106:
3101:
3096:
3091:
3085:
3083:
3082:Notable people
3079:
3078:
3075:
3074:
3072:
3071:
3066:
3061:
3056:
3051:
3046:
3041:
3035:
3033:
3029:
3028:
3026:
3025:
3020:
3014:
3012:
3008:
3007:
3005:
3004:
2999:
2994:
2989:
2984:
2979:
2974:
2969:
2964:
2959:
2954:
2949:
2944:
2939:
2934:
2929:
2924:
2919:
2913:
2911:
2904:
2898:
2897:
2894:
2893:
2891:
2890:
2887:
2881:
2876:
2870:
2868:
2862:
2861:
2859:
2858:
2857:
2856:
2848:
2843:
2838:
2833:
2827:
2825:
2809:
2808:
2806:
2805:
2800:
2794:
2792:
2788:
2787:
2785:
2784:
2779:
2773:
2771:
2765:
2764:
2762:
2761:
2755:
2753:
2744:
2740:
2739:
2737:
2736:
2731:
2726:
2721:
2716:
2711:
2710:
2709:
2702:
2692:
2691:
2690:
2685:
2680:
2670:
2665:
2659:
2657:
2653:
2652:
2650:
2649:
2642:
2635:
2630:
2625:
2620:
2615:
2608:
2607:
2606:
2599:
2592:
2589:Hyakunin Isshu
2585:
2571:
2570:
2569:
2555:
2550:
2545:
2537:
2535:
2527:
2526:
2523:
2522:
2520:
2519:
2517:Transformation
2514:
2512:Stuttgart pack
2509:
2504:
2499:
2497:Jerry's Nugget
2494:
2487:
2482:
2477:
2472:
2467:
2459:
2457:
2456:Specific decks
2453:
2452:
2450:
2449:
2444:
2439:
2434:
2433:
2432:
2422:
2417:
2416:
2415:
2410:
2399:
2397:
2393:
2392:
2390:
2389:
2384:
2379:
2374:
2368:
2366:
2352:
2344:
2343:
2336:
2335:
2328:
2321:
2313:
2307:
2306:
2290:
2289:
2251:
2250:
2237:
2221:
2220:
2197:
2196:
2156:
2155:
2143:(2): 175–196.
2112:
2109:
2106:
2105:
2096:
2080:
2071:
2062:
2060:Morris 1998, 8
2053:
2044:
2035:
2026:
2016:
2015:
2013:
2010:
1995:
1991:
1987:
1984:
1981:
1978:
1956:
1952:
1929:
1925:
1902:
1898:
1886:
1885:
1872:
1866:
1863:
1859:
1856:
1851:
1848:
1845:
1842:
1839:
1836:
1834:
1831:
1828:
1827:
1823:
1820:
1816:
1813:
1808:
1805:
1802:
1799:
1796:
1793:
1791:
1788:
1785:
1784:
1782:
1777:
1772:
1769:
1765:
1761:
1758:
1754:
1750:
1747:
1744:
1741:
1738:
1733:
1729:
1725:
1704:
1703:
1692:
1689:
1684:
1681:
1678:
1674:
1670:
1658:
1655:
1652:
1648:
1644:
1641:
1638:
1635:
1632:
1627:
1623:
1619:
1587:
1583:
1562:
1542:
1539:
1536:
1533:
1530:
1527:
1524:
1502:
1498:
1477:
1474:
1471:
1468:
1465:
1436:
1433:
1429:
1408:
1405:
1402:
1399:
1396:
1385:
1384:
1371:
1358:
1355:
1351:
1348:
1343:
1340:
1337:
1336:
1325:
1322:
1318:
1314:
1311:
1308:
1302:
1301:
1299:
1294:
1291:
1277:
1276:
1263:
1257:
1254:
1251:
1249:
1246:
1244:
1241:
1239:
1236:
1233:
1230:
1228:
1225:
1223:
1220:
1217:
1214:
1212:
1209:
1208:
1205:
1202:
1199:
1197:
1194:
1191:
1188:
1185:
1183:
1180:
1178:
1175:
1173:
1170:
1168:
1165:
1163:
1160:
1159:
1157:
1133:
1130:
1126:
1096:
1093:
1089:
1065:
1062:
1057:
1053:
1049:
1034:
1031:
1030:
1029:
1028:
1027:
1024:
1023:
1016:
1009:
1002:
995:
988:
981:
977:
976:
969:
962:
955:
948:
941:
934:
930:
929:
922:
915:
908:
901:
894:
887:
883:
882:
875:
868:
861:
854:
847:
840:
836:
835:
828:
821:
814:
807:
800:
793:
789:
788:
783:
780:
777:
774:
771:
766:
753:
752:
751:
750:
747:
746:
739:
732:
725:
718:
711:
704:
700:
699:
692:
685:
678:
671:
664:
657:
653:
652:
645:
638:
631:
624:
617:
610:
606:
605:
598:
591:
584:
577:
570:
563:
559:
558:
553:
550:
547:
544:
541:
536:
519:
516:
448:
445:
441:
438:
433:
430:
425:
421:
399:
396:
393:
390:
386:
383:
378:
375:
370:
366:
356:-card deck if
345:
325:
301:
298:
294:
291:
286:
283:
278:
274:
252:
249:
246:
243:
239:
236:
231:
228:
223:
219:
209:-card deck if
198:
178:
133:
130:
126:Persi Diaconis
122:John Maskelyne
105:
102:
74:(British), or
62:
61:
41:the key points
31:
29:
22:
15:
9:
6:
4:
3:
2:
3812:
3801:
3798:
3796:
3793:
3791:
3788:
3786:
3783:
3782:
3780:
3764:
3755:
3746:
3737:
3728:
3719:
3710:
3702:
3692:
3691:
3687:
3684:
3683:
3682:Playing Cards
3679:
3677:
3676:
3672:
3671:
3669:
3665:
3660:
3659:
3655:
3653:
3652:
3648:
3646:
3645:
3641:
3639:
3638:
3634:
3632:
3631:
3627:
3625:
3624:
3620:
3618:
3617:
3613:
3611:
3610:
3606:
3604:
3603:
3599:
3597:
3596:
3592:
3590:
3589:
3585:
3583:
3579:
3576:
3574:Art and media
3572:
3566:
3563:
3561:
3558:
3556:
3553:
3551:
3548:
3544:
3541:
3539:
3536:
3534:
3531:
3530:
3529:
3526:
3524:
3521:
3519:
3518:
3514:
3510:
3507:
3505:
3502:
3500:
3497:
3495:
3492:
3491:
3489:
3486:
3484:
3481:
3479:
3476:
3474:
3471:
3469:
3468:Herrmann pass
3466:
3464:
3463:
3459:
3457:
3454:
3452:
3449:
3447:
3446:Cardistry-Con
3444:
3440:
3437:
3436:
3435:
3434:Card throwing
3432:
3430:
3427:
3425:
3422:
3420:
3417:
3416:
3408:
3405:
3403:
3400:
3398:
3395:
3393:
3390:
3388:
3385:
3383:
3380:
3378:
3375:
3373:
3370:
3368:
3365:
3363:
3359:
3356:
3354:
3349:
3345:
3339:
3336:
3334:
3331:
3329:
3326:
3322:
3321:
3317:
3316:
3315:
3312:
3310:
3307:
3305:
3302:
3300:
3297:
3295:
3292:
3290:
3289:52 Plus Joker
3287:
3286:
3277:
3274:
3272:
3269:
3267:
3264:
3262:
3259:
3258:
3256:
3254:
3250:
3247:
3245:organisations
3241:
3235:
3232:
3230:
3227:
3225:
3222:
3220:
3217:
3215:
3212:
3210:
3207:
3205:
3202:
3200:
3197:
3195:
3192:
3190:
3189:David Parlett
3187:
3185:
3182:
3180:
3177:
3175:
3172:
3170:
3167:
3165:
3162:
3160:
3157:
3155:
3152:
3150:
3147:
3145:
3142:
3140:
3137:
3135:
3134:Lennart Green
3132:
3130:
3129:S. W. Erdnase
3127:
3125:
3122:
3120:
3117:
3115:
3112:
3110:
3107:
3105:
3102:
3100:
3097:
3095:
3092:
3090:
3087:
3086:
3084:
3080:
3070:
3067:
3065:
3062:
3060:
3057:
3055:
3052:
3050:
3047:
3045:
3042:
3040:
3037:
3036:
3034:
3030:
3024:
3021:
3019:
3016:
3015:
3013:
3009:
3003:
3000:
2998:
2995:
2993:
2990:
2988:
2985:
2983:
2980:
2978:
2977:Oishi Tengudo
2975:
2973:
2970:
2968:
2965:
2963:
2960:
2958:
2955:
2953:
2950:
2948:
2945:
2943:
2942:France Cartes
2940:
2938:
2935:
2933:
2930:
2928:
2925:
2923:
2920:
2918:
2915:
2914:
2912:
2910:Manufacturers
2908:
2905:
2901:Manufacturers
2899:
2888:
2885:
2882:
2880:
2877:
2875:
2872:
2871:
2869:
2867:
2863:
2855:
2852:
2851:
2849:
2847:
2844:
2842:
2839:
2837:
2834:
2832:
2829:
2828:
2826:
2823:
2819:
2815:
2810:
2804:
2801:
2799:
2796:
2795:
2793:
2789:
2783:
2780:
2778:
2775:
2774:
2772:
2770:
2766:
2760:
2757:
2756:
2754:
2752:
2748:
2745:
2741:
2735:
2732:
2730:
2727:
2725:
2724:Stripped deck
2722:
2720:
2717:
2715:
2712:
2708:
2707:
2703:
2701:
2698:
2697:
2696:
2693:
2689:
2686:
2684:
2681:
2679:
2676:
2675:
2674:
2671:
2669:
2666:
2664:
2661:
2660:
2658:
2654:
2648:
2647:
2643:
2641:
2640:
2636:
2634:
2631:
2629:
2626:
2624:
2621:
2619:
2616:
2614:
2613:
2609:
2605:
2604:
2600:
2598:
2597:
2593:
2591:
2590:
2586:
2584:
2583:
2579:
2578:
2577:
2576:
2572:
2568:
2567:
2563:
2562:
2561:
2560:
2556:
2554:
2551:
2549:
2546:
2544:
2543:
2539:
2538:
2536:
2534:
2528:
2518:
2515:
2513:
2510:
2508:
2505:
2503:
2500:
2498:
2495:
2493:
2492:
2491:Hofämterspiel
2488:
2486:
2483:
2481:
2478:
2476:
2473:
2471:
2468:
2466:
2465:
2461:
2460:
2458:
2454:
2448:
2445:
2443:
2440:
2438:
2435:
2431:
2428:
2427:
2426:
2423:
2421:
2418:
2414:
2413:Ace of hearts
2411:
2409:
2408:Ace of spades
2406:
2405:
2404:
2401:
2400:
2398:
2394:
2388:
2385:
2383:
2380:
2378:
2375:
2373:
2370:
2369:
2367:
2364:
2360:
2356:
2353:
2351:
2345:
2341:
2340:Playing cards
2334:
2329:
2327:
2322:
2320:
2315:
2314:
2311:
2302:
2297:
2292:
2291:
2286:
2282:
2278:
2274:
2270:
2266:
2262:
2258:
2253:
2252:
2240:
2238:0-883-85527-5
2234:
2230:
2229:
2223:
2222:
2209:
2208:
2203:
2199:
2198:
2186:
2182:
2178:
2174:
2167:
2163:
2158:
2157:
2151:
2146:
2142:
2138:
2131:
2127:
2126:Kantor, W. M.
2123:
2122:Graham, R. L.
2119:
2115:
2114:
2100:
2093:
2092:Peter Cameron
2089:
2084:
2075:
2066:
2057:
2048:
2039:
2030:
2021:
2017:
2009:
1993:
1989:
1985:
1979:
1954:
1950:
1927:
1923:
1900:
1896:
1861:
1857:
1849:
1846:
1843:
1840:
1837:
1832:
1829:
1818:
1814:
1806:
1803:
1800:
1797:
1794:
1789:
1786:
1780:
1775:
1767:
1763:
1759:
1748:
1745:
1739:
1731:
1727:
1709:
1708:
1707:
1690:
1682:
1679:
1676:
1672:
1656:
1653:
1650:
1642:
1639:
1633:
1625:
1621:
1603:
1602:
1601:
1585:
1581:
1560:
1537:
1534:
1531:
1528:
1525:
1500:
1496:
1472:
1469:
1466:
1454:
1452:
1434:
1431:
1427:
1403:
1400:
1397:
1356:
1349:
1346:
1341:
1338:
1323:
1316:
1312:
1309:
1306:
1297:
1289:
1282:
1281:
1280:
1261:
1255:
1252:
1247:
1242:
1237:
1234:
1231:
1226:
1221:
1218:
1215:
1210:
1203:
1200:
1195:
1192:
1189:
1186:
1181:
1176:
1171:
1166:
1161:
1155:
1131:
1128:
1124:
1116:
1115:
1114:
1112:
1094:
1091:
1087:
1077:
1075:
1071:
1061:
1046:
1044:
1043:binary number
1040:
1021:
1017:
1014:
1010:
1007:
1003:
1000:
996:
993:
989:
986:
982:
979:
978:
974:
970:
967:
963:
960:
956:
953:
949:
946:
942:
939:
935:
932:
931:
927:
923:
920:
916:
913:
909:
906:
902:
899:
895:
892:
888:
885:
884:
880:
876:
873:
869:
866:
862:
859:
855:
852:
848:
845:
841:
838:
837:
833:
829:
826:
822:
819:
815:
812:
808:
805:
801:
798:
794:
791:
790:
784:
781:
778:
775:
772:
767:
764:
763:
760:
759:
758:
757:
756:
744:
740:
737:
733:
730:
726:
723:
719:
716:
712:
709:
705:
702:
701:
697:
693:
690:
686:
683:
679:
676:
672:
669:
665:
662:
658:
655:
654:
650:
646:
643:
639:
636:
632:
629:
625:
622:
618:
615:
611:
608:
607:
603:
599:
596:
592:
589:
585:
582:
578:
575:
571:
568:
564:
561:
560:
554:
551:
548:
545:
542:
537:
534:
533:
530:
529:
528:
527:
526:
523:
515:
513:
508:
506:
502:
499:According to
497:
495:
490:
485:
480:
478:
474:
470:
466:
461:
443:
439:
431:
428:
423:
419:
394:
391:
388:
384:
376:
373:
368:
364:
343:
323:
314:
296:
292:
284:
281:
276:
272:
247:
244:
241:
237:
229:
226:
221:
217:
196:
176:
167:
164:
160:
156:
154:
150:
146:
141:
137:
129:
127:
123:
119:
114:
112:
101:
94:
90:
88:
87:the technique
84:
83:playing cards
81:
77:
73:
72:weave shuffle
69:
58:
48:
42:
40:
35:
30:
26:
21:
20:
3766:}}
3763:Occult tarot
3760:{{
3757:}}
3751:{{
3748:}}
3742:{{
3739:}}
3733:{{
3730:}}
3724:{{
3721:}}
3715:{{
3712:}}
3706:{{
3688:
3680:
3673:
3656:
3649:
3642:
3635:
3628:
3621:
3614:
3607:
3600:
3593:
3588:The Acrobats
3586:
3532:
3515:
3483:One-way deck
3460:
3424:Card marking
3402:Spelling Bee
3318:
3214:Juan Tamariz
3119:Dan and Dave
3109:Derren Brown
3104:David Blaine
3039:Aspioti-ELKA
2987:Ravensburger
2704:
2644:
2637:
2610:
2601:
2594:
2587:
2580:
2573:
2564:
2557:
2540:
2489:
2462:
2350:52-card deck
2260:
2256:
2242:. Retrieved
2227:
2212:. Retrieved
2206:
2188:. Retrieved
2176:
2172:
2140:
2136:
2099:
2083:
2074:
2065:
2056:
2047:
2038:
2029:
2020:
1887:
1705:
1455:
1450:
1386:
1278:
1110:
1078:
1067:
1047:
1039:Alex Elmsley
1036:
754:
524:
521:
509:
504:
498:
483:
481:
476:
464:
462:
316:In general,
315:
169:In general,
168:
165:
161:
157:
153:Alex Elmsley
148:
144:
142:
138:
135:
115:
107:
99:
86:
75:
71:
70:(American),
68:faro shuffle
67:
65:
55:October 2018
52:
36:
34:lead section
3675:Card Sharks
3565:Trick decks
3456:Double lift
3397:Rising Card
3179:Anton Moser
3174:John McLeod
3159:René Lavand
3114:Lewis Cohen
3094:Hubert Auer
3069:Waddingtons
3064:Lewis Cohen
2947:F.X. Schmid
2743:Other suits
2531:Other packs
2507:Politicards
2179:: 169–180.
2162:Shallit, J.
1070:mathematics
1056:or 00001010
145:out-shuffle
104:Description
3790:Card magic
3779:Categories
3555:Spoke card
3504:mechanic's
3499:magician's
3490:and grips
3429:Card sharp
3243:Groups and
3229:Dai Vernon
3099:Bryan Berg
2922:Cartamundi
2903:and brands
2886:of Strings
2822:Portuguese
2673:Court card
2633:Tarot pack
2603:uta-garuta
2111:References
507:shuffles.
149:in-shuffle
116:A game of
3581:Paintings
3538:Gilbreath
3528:Shuffling
3494:gambler's
3353:cardistry
3149:Ricky Jay
3089:Lee Asher
3049:C.L. Wüst
2932:Dal Negro
2850:Japanese
2719:Nicknames
2533:and decks
2447:Beer card
2348:Standard
2301:0805.1598
1986:∈
1951:ρ
1924:ρ
1897:ρ
1841:≡
1830:−
1798:≡
1771:⌋
1757:⌊
1746:−
1728:ρ
1680:−
1673:ρ
1640:−
1622:ρ
1582:ρ
1535:−
1497:ρ
1293:↦
1243:⋯
1193:−
1182:⋯
1037:Magician
429:≡
392:−
374:≡
282:≡
227:≡
80:shuffling
39:summarize
3164:Ed Marlo
2967:Nintendo
2937:Fournier
2695:Pip card
2612:Kvitlech
2596:kabufuda
2582:hanafuda
2475:Charruan
2382:Diamonds
2285:17735734
2204:(1894).
2164:(2002).
2128:(1983).
3451:Cutting
3059:Dondorf
3032:Defunct
3023:Grimaud
3018:Bicycle
2982:Piatnik
2962:Modiano
2884:Myriads
2818:Spanish
2814:Italian
2812:Latin (
2777:Shields
2734:Unicode
2628:Russian
2566:ganjapa
2559:Ganjifa
2548:Chinese
2265:Bibcode
2257:Science
518:Example
492:in the
489:A002326
3685:(1896)
3543:Zarrow
3509:Tenkai
3361:Tricks
3011:Brands
2952:Gemaco
2846:Swords
2841:Batons
2798:Acorns
2759:Leaves
2751:German
2678:knight
2663:Banner
2646:Zi pai
2639:Tujeon
2618:Polish
2575:Karuta
2542:As-Nas
2377:Hearts
2372:Spades
2363:French
2283:
2244:26 Dec
2235:
2214:26 Dec
2190:26 Dec
1360:
1327:
1109:, the
792:Start
785:Bottom
562:Start
555:Bottom
473:modulo
111:zipper
3745:Poker
3488:Palms
2992:Trefl
2927:Copag
2836:Coins
2803:Bells
2782:Roses
2769:Swiss
2729:Talon
2714:Joker
2700:deuce
2688:Unter
2668:Blank
2656:Cards
2425:Queen
2396:Ranks
2387:Clubs
2296:arXiv
2169:(PDF)
2133:(PDF)
2012:Notes
787:Card
770:Card
765:Step
557:Card
540:Card
535:Step
471:of 2
3533:faro
3351:and
3253:Skat
2874:Cash
2831:Cups
2706:Weli
2683:Ober
2437:Jack
2420:King
2281:PMID
2246:2013
2233:ISBN
2216:2013
2192:2013
1915:and
1600:is:
1456:The
1364:even
494:OEIS
118:Faro
66:The
2403:Ace
2273:doi
2261:216
2181:doi
2145:doi
1858:mod
1815:mod
1719:sgn
1664:sgn
1613:sgn
1331:odd
1068:In
768:Top
538:Top
496:).
440:mod
385:mod
293:mod
238:mod
3781::
2279:.
2271:.
2259:.
2175:.
2171:.
2139:.
2135:.
2124:;
2120:;
2090:,
2008:.
1980:23
1076:.
980:4
933:3
886:2
839:1
782:5
779:4
776:3
773:2
703:3
656:2
609:1
552:5
549:4
546:3
543:2
514:.
444:51
313:.
297:53
277:52
155:.
128:.
2824:)
2820:/
2816:/
2365:)
2361:(
2332:e
2325:t
2318:v
2304:.
2298::
2287:.
2275::
2267::
2248:.
2218:.
2194:.
2183::
2177:5
2153:.
2147::
2141:4
1994:4
1990:S
1983:)
1977:(
1955:2
1928:1
1901:0
1865:)
1862:4
1855:(
1850:3
1847:,
1844:2
1838:n
1833:1
1822:)
1819:4
1812:(
1807:1
1804:,
1801:0
1795:n
1790:1
1787:+
1781:{
1776:=
1768:2
1764:/
1760:n
1753:)
1749:1
1743:(
1740:=
1737:)
1732:n
1724:(
1691:.
1688:)
1683:1
1677:n
1669:(
1657:1
1654:+
1651:n
1647:)
1643:1
1637:(
1634:=
1631:)
1626:n
1618:(
1586:n
1561:n
1541:)
1538:1
1532:n
1529:,
1526:2
1523:(
1501:n
1476:)
1473:n
1470:,
1467:2
1464:(
1451:k
1435:n
1432:k
1428:S
1407:)
1404:n
1401:,
1398:k
1395:(
1357:k
1350:2
1347:k
1342:+
1339:n
1324:k
1317:2
1313:1
1310:+
1307:k
1298:{
1290:k
1262:)
1256:n
1253:2
1248:n
1238:2
1235:+
1232:n
1227:2
1222:1
1219:+
1216:n
1211:1
1204:n
1201:2
1196:1
1190:n
1187:2
1177:4
1172:3
1167:2
1162:1
1156:(
1146:=
1132:n
1129:2
1125:S
1095:n
1092:2
1088:S
1058:2
1054:2
1050:2
505:n
484:n
477:n
475:(
465:n
447:)
437:(
432:1
424:8
420:2
398:)
395:1
389:n
382:(
377:1
369:k
365:2
344:n
324:k
300:)
290:(
285:1
273:2
251:)
248:1
245:+
242:n
235:(
230:1
222:k
218:2
197:n
177:k
57:)
53:(
43:.
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