coloque viales en su centrífuga

12

Esta pregunta se origina en este hilo de reddit por el usuario de reddit taho_teg pero se expande a un 'rompecabezas' más general.

Tiene una centrífuga con 24 orificios para los viales distribuidos uniformemente en un círculo alrededor del eje central. Si ahora tiene varios viales y desea iniciar la centrífuga, debe asegurarse de que se coloquen de manera equilibrada. Los únicos números de viales que no puede equilibrar son 1 y 23. Obviamente, puede equilibrar 4, pero también puede equilibrar 5 haciendo un 'triángulo' con 3 viales y colocando los otros dos en dos sitios opuestos.

Objetivo

Debe escribir un programa que acepte el número de agujeros (que se distribuyen uniformemente en un círculo alrededor del eje giratorio) de su centrífuga como entrada, y que genere una lista de números de viales que no pueden equilibrarse en la centrífuga.

Tiene que hacer el cálculo y no puede simplemente codificar las soluciones precalculadas.

La entrada y la salida deben implementarse de manera que el código del programa no tenga que cambiarse para llamar al programa para diferentes entradas. También es aceptable escribir una función (o una construcción similar en su idioma) que se pueda llamar a través de una consola.

También tenga en cuenta que si tiene 6 orificios en su centrífuga, puede centrifugar 2 y 3 viales, pero no puede equilibrar 5 ya que el 'triángulo' y los dos opuestos se superpondrán en un punto. Otro ejemplo sería para n = 15, no puede equilibrar 11 viales, puede equilibrar 6 y 5 viales, pero la combinación de esas soluciones se superpondrá (por supuesto, este aún no es el criterio de que sea imposible hacerlo).

Actualizar

Parece que algunas personas no entendieron el ejemplo dado, así que hice un gráfico aquí. POR FAVOR, escriba una breve descripción de cómo funciona su algoritmo, así como algunos resultados de ejemplo para verificación. Por favor incluya los siguientes ejemplos:

n = 1, 6, 10, 24, 63, 100 = 10^2, 163 (prime), 40320 = 8!, 65536=2^2^2^2^2, 105953 (prime)

Tenga en cuenta que 40320 y 65536 producirán grandes listas, tal vez sea una buena idea indicar solo la longitud de esas listas.

Si conoce algunos números interesantes para agregar a esa lista, ¡hágamelo saber! El algoritmo debería funcionar al menos hasta n = 1'000'000. 5 viales colocados equilibrados en una centrífuga de 24 agujeros

Salidas de ejemplo:

Estos son algunos resultados de ejemplo, pero quizás defectuosos porque los calculé manualmente.

1: 1
2: 1
3: 1,2
4: 1,3
5: 1,2,3,4
6: 1,5
7: 1,2,3,4,5,6
8: 1,3,5,7
9: 1,2,4,5,7,8
10:1,3,7,9
11:1,2,3,4,5,6,7,8,9,10
12:1,11
13:1,2,3,4,5,6,7,8,9,10,11,12
14:1,3,5,9,11,13
15:1,2,4,7,8,11,13,14

Insinuación

Si tiene una centrífuga con n agujeros, y no puede equilibrar, por ejemplo, 6 viales, tampoco podrá equilibrar n-6 viales: es básicamente la misma tarea equilibrar m viales en una centrífuga vacía o equilibrar una centrífuga llena quitando m viales. Entonces, si tiene el número m en su lista, también deberá incluir nm .

falla
fuente
3
¿No deberíamos necesitar tener los viales espaciados uniformemente para estar equilibrados? No veo cómo el ejemplo de 5 viales en 24 agujeros satisface esto. Una "mitad" de la centrífuga tendrá más viales que la otra mitad. Eso suena como la definición de desequilibrado para mí.
Thorn
66
Creo que por "equilibrado" se entiende que el centro de masa de los viales está verticalmente por encima o por debajo del centro de masa de la centrífuga.
Peter Taylor
2
@Thorn necesitas pensar en dos dimensiones, no en una. Las coordenadas de los primeros 3 viales son (0,1), (-sqrt (3) / 2, -1 / 2) y (+ sqrt (3) / 2, -1 / 2.) La disposición de 5 viales no es simétrico (aparte de posiblemente un plano espejo) pero está equilibrado. Es bastante común que las ruedas de los automóviles tengan diferentes números de radios y tuercas de rueda (de nuevo, no simétricos, pero totalmente equilibrados porque los radios forman un conjunto equilibrado y las tuercas forman un conjunto equilibrado). Google 7 spoke wheely eche un vistazo.
Level River St
2
"El algoritmo debería funcionar al menos hasta n = 1'000'000". Solo para estar seguro: ¿Realmente te referías al algoritmo o al programa? Mi algoritmo funciona tan bien para n = 1.000.000 como lo hace para n = 10. Sin embargo, el programa tiene algunos problemas.
Wrzlprmft
1
@ edc65. equilibrado! = simétrico ... siempre que divida sus viales en subgrupos, donde cada subgrupo esté en un estado simétrico, entonces la suma de la fuerza externa de todos los subgrupos estará en un estado equilibrado.
Eoin Campbell

Respuestas:

5

Salvia - 102 104/115

¿Por qué usar la teoría de números, cuando hay fuerza bruta?

v=lambda n:[j for j in range(n+1)if all(sum(e^(i*2*I*pi/n)for i in c)for c in Combinations(range(n),j))]

Para un número determinado de viales, esto abarca todas las formas de colocar los viales y calcula su centro de masa mediante el uso de aritmética compleja. Si el centro de masa es cero para ninguna de estas formas, se devuelve el número.

Desafortunadamente, esto no funciona en ciertos casos (10,14), porque Sage no puede simplificar algunas expresiones a cero (lo que puede estar relacionado con este error ). Uno podría considerar esto como una falla del intérprete y no del programa y aún así decir que el algoritmo y el programa están bien.

La siguiente alternativa de 113 caracteres se basa en flotantes en lugar de símbolos y no sufre estos problemas:

v=lambda n:[j for j in range(n+1)if all(abs(sum(exp(i*2j*pi/n)for i in c))>1e-9for c in Combinations(range(n),j))]

Salida de prueba de la versión de 113 caracteres ( for n in range(14): print n,v(n)):

0 []
1 [1]
2 [1]
3 [1, 2]
4 [1, 3]
5 [1, 2, 3, 4]
6 [1, 5]
7 [1, 2, 3, 4, 5, 6]
8 [1, 3, 5, 7]
9 [1, 2, 4, 5, 7, 8]
10 [1, 3, 7, 9]
11 [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
12 [1, 11]
13 [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]
14 [1, 3, 5, 9, 11, 13]

No quería esperar el tiempo de ejecución por más n.


Esto se origina en la siguiente solución de Python. La aritmética exacta y no tener que importar algunos módulos es algo bastante.

Python - 173 154 156

from itertools import*
from cmath import*
v=lambda n:[j for j in range(n+1)if all(abs(sum(exp(i*2j*pi/n)for i in c))>1e-9for c in combinations(range(n),j))]

Salida de prueba de esta variante ( for n in range(24): print n,v(n)):

0 []
1 [1]
2 [1]
3 [1, 2]
4 [1, 3]
5 [1, 2, 3, 4]
6 [1, 5]
7 [1, 2, 3, 4, 5, 6]
8 [1, 3, 5, 7]
9 [1, 2, 4, 5, 7, 8]
10 [1, 3, 7, 9]
11 [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
12 [1, 11]
13 [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]
14 [1, 3, 5, 9, 11, 13]
15 [1, 2, 4, 7, 8, 11, 13, 14]
16 [1, 3, 5, 7, 9, 11, 13, 15]
17 [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]
18 [1, 17]
19 [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]
20 [1, 3, 17, 19]
21 [1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 19, 20]
22 [1, 3, 5, 7, 9, 13, 15, 17, 19, 21]
23 [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]
24 [1, 23]

No quería esperar el tiempo de ejecución por más n.

Wrzlprmft
fuente
¡Me gusta la idea de usar las raíces complejas de la unidad! ¿Podría mostrar algunos resultados de ejemplo solo para verificación? Publiqué algunas sugerencias.
flawr
4

Lua - 197

Un método de fuerza no bruta, crea una lista de factores y los descarta. También descarta los números que se pueden obtener con la suma de esos factores, siempre que el factor más grande utilizado sea menor que la cantidad de agujeros sin llenar. Uno siempre se imprime y no se usa en el algoritmo.

i=io.read("*n")f={}print(1)for z=2,i do
x=z
if i%x<1 then
table.insert(f,1,x)end
for q=1,#f do
y=f[q]x=x*math.min(1,z%y)while x>=y and x-1~=y and y<=i-z do x=x-y end
end
if x>0 then print(z)end
end

Ejemplo de salida: (algunos se ponen como rangos para que no exceda el límite de caracteres)

1: 1

6: 1,5

10:1,3,7,9

24:1,23

63:1,2,4,5,8,11,13,17,20,22,23,25,26,29,32,34,38,41,44,47,50,53,58,59,61,62

100:1,3,13,23,33,43,53,63,73,83,93,97,99

163:1-162

40320:1,11,13,17,19,29,31,37,41,43,61,71,73,97,113,121,127,139,157,169,179,181,191,193,209,211,221,223,241,251,253,263,269,271,277,281,289,299,307,313,331,337,347,349,353,361,373,377,379,397,401,403,409,421,431,433,437,439,449,461,467,473,479,481,491,493,499,517,521,523,529,533,541,547,571,577,587,589,593,601,607,613,617,619,631,641,653,659,671,673,683,689,691,697,701,703,709,713,731,733,737,739,751,757,761,769,781,793,811,817,841,851,853,857,859,869,871,877,881,883,907,913,929,937,953,961,971,977,979,989,991,997,1003,1009,1019,1021,1027,1033,1037,1039,1049,1051,1069,1073,1079,1081,1093,1121,1133,1139,1151,1153,1163,1171,1177,1181,1189,1193,1201,1213,1217,1223,1237,1243,1249,1261,1271,1273,1277,1279,1289,1291,1297,1301,1303,1321,1331,1333,1357,1361,1363,1369,1373,1381,1387,1409,1417,1429,1441,1451,1453,1457,1459,1469,1471,1481,1483,1489,1501,1511,1513,1531,1537,1553,1567,1579,1597,1601,1609,1619,1621,1633,1643,1649,1651,1661,1663,1681,1691,1693,1697,1699,1709,1711,1717,1721,1723,1741,1751,1753,1777,1793,1801,1807,1819,1837,1849,1859,1861,1871,1873,1889,1891,1901,1903,1921,1931,1933,1937,1949,1951,1957,1961,1963,1969,1991,1993,2011,2017,2027,2029,2033,2041,2047,2053,2057,2059,2077,2081,2087,2089,2101,2113,2129,2137,2143,2161,2171,2173,2197,2207,2209,2221,2227,2237,2239,2251,2257,2269,2273,2281,2297,2311,2321,2353,2369,2381,2419,2431,2449,2461,2477,2491,2503,2509,2521,2531,2533,2537,2539,2549,2551,2557,2561,2563,2581,2591,2593,2617,2633,2641,2647,2659,2677,2689,2699,2701,2707,2713,2717,2719,2729,2731,2749,2753,2759,2761,2773,2801,2809,2827,2833,2843,2857,2867,2869,2879,2881,2893,2897,2899,2909,2911,2917,2921,2923,2929,2941,2951,2953,2971,2977,2993,3001,3007,3019,3037,3041,3049,3061,3071,3083,3089,3091,3103,3121,3131,3133,3149,3151,3161,3169,3179,3181,3187,3193,3211,3217,3229,3233,3251,3253,3257,3259,3277,3281,3289,3293,3301,3313,3317,3319,3329,3341,3347,3359,3361,3371,3373,3377,3379,3389,3391,3397,3401,3403,3421,3431,3433,3457,3473,3481,3487,3499,3517,3529,3539,3541,3551,3553,3569,3571,3581,3583,3601,3611,3613,3623,3629,3631,3637,3641,3649,3659,3667,3673,3691,3697,3707,3709,3713,3721,3733,3737,3739,3757,3761,3763,3769,3781,3791,3793,3797,3799,3809,3821,3827,3833,3839,3841,3851,3853,3859,3877,3881,3883,3889,3893,3901,3907,3931,3937,3947,3949,3953,3961,3967,3973,3977,3979,3991,4001,4013,4019,4031,4033,4043,4049,4051,4057,4061,4063,4069,4073,4093,4097,4103,4117,4129,4153,4159,4171,4177,4187,4189,4201,4211,4213,4223,4237,4241,4243,4253,4267,4273,4283,4297,4301,4303,4309,4313,4321,4331,4337,4339,4363,4369,4379,4381,4387,4393,4409,4411,4427,4429,4433,4441,4447,4453,4463,4469,4471,4477,4481,4493,4499,4511,4513,4517,4523,4537,4541,4553,4561,4577,4601,4607,4609,4619,4621,4637,4649,4661,4673,4691,4703,4717,4721,4733,4751,4757,4769,4787,4793,4801,4811,4813,4817,4829,4841,4853,4859,4877,4883,4889,4897,4901,4913,4919,4939,4957,4961,4973,4979,4997,5003,5009,5017,5021,5027,5041,5051,5053,5057,5059,5069,5071,5077,5081,5083,5101,5111,5113,5137,5153,5161,5167,5179,5197,5209,5219,5221,5231,5233,5249,5251,5261,5263,5281,5291,5293,5303,5309,5311,5317,5321,5329,5339,5347,5353,5371,5377,5387,5389,5393,5401,5413,5417,5419,5437,5441,5443,5449,5461,5471,5473,5477,5479,5489,5501,5507,5513,5519,5521,5531,5533,5539,5557,5561,5563,5569,5573,5581,5587,5611,5617,5627,5629,5633,5641,5647,5653,5657,5659,5671,5681,5693,5699,5711,5713,5723,5729,5731,5737,5741,5743,5749,5753,5771,5773,5777,5779,5791,5797,5801,5809,5821,5833,5851,5857,5881,5899,5917,5921,5939,5941,5951,5953,5963,5969,5981,5983,6001,6011,6023,6029,6031,6037,6049,6059,6061,6067,6073,6091,6107,6109,6113,6121,6131,6133,6137,6157,6161,6163,6169,6173,6191,6193,6197,6199,6221,6227,6233,6239,6241,6253,6257,6259,6277,6281,6283,6289,6301,6313,6331,6337,6347,6353,6361,6367,6373,6379,6401,6413,6431,6443,6449,6451,6457,6463,6469,6473,6481,6491,6493,6497,6499,6509,6511,6521,6523,6529,6541,6551,6553,6571,6577,6593,6611,6613,6617,6619,6631,6637,6641,6649,6667,6673,6689,6697,6721,6731,6733,6737,6739,6749,6751,6757,6761,6763,6781,6791,6793,6817,6833,6841,6847,6859,6877,6889,6899,6901,6911,6913,6929,6931,6941,6943,6961,6971,6973,6983,6989,6991,6997,7001,7009,7019,7027,7033,7051,7057,7067,7069,7073,7081,7093,7097,7099,7117,7121,7123,7129,7141,7151,7153,7157,7159,7169,7181,7187,7193,7199,7201,7211,7213,7219,7237,7241,7243,7249,7253,7261,7267,7291,7297,7307,7309,7313,7321,7327,7333,7337,7339,7351,7361,7373,7379,7391,7393,7403,7409,7411,7417,7421,7423,7429,7433,7451,7453,7457,7459,7471,7477,7481,7489,7501,7513,7531,7537,7561,7571,7573,7577,7579,7589,7591,7597,7601,7603,7627,7633,7649,7657,7673,7681,7691,7697,7699,7709,7711,7717,7723,7729,7739,7741,7747,7753,7757,7759,7769,7771,7789,7793,7799,7801,7813,7841,7853,7859,7871,7873,7883,7891,7897,7901,7909,7913,7921,7933,7937,7943,7957,7963,7969,7981,7991,7993,7997,7999,8009,8011,8017,8021,8023,8041,8051,8053,8077,8081,8083,8089,8093,8101,8107,8129,8137,8149,8161,8177,8191,8203,8209,8219,8221,8233,8243,8257,8269,8273,8287,8299,8317,8327,8329,8333,8341,8353,8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65536: This makes my answer go over the character limit (32768 numbers)

105953: 1-105952
Nexo
fuente
Creo que algo está mal aquí, no puede equilibrar uno en una centrífuga de 100 agujeros, pero puede equilibrar 8,18,28,48,58,68,78,88 (esta lista tal vez no esté completa).
flawr
Debo haber cortado accidentalmente el uno, en cuanto a los otros números, tengo que investigar eso
Nexus
¡Muchas gracias por actualizar tu lista! Si es posible, podría contar la cantidad de resultados para los casos en los que obtiene listas tan enormes, no pensé que realmente =)
fallara el
@flawr Actualicé la lista
Nexus
0

Pyth - 39 37 bytes

Una traducción directa de la respuesta de Python de @ Wrzlprmft.

f.Am>.asm^.n1c*.jZyk.nZQd^10_9.cUQTSQ

Explicación y probablemente más golf próximamente.

Pruébelo en línea aquí .

Maltysen
fuente