원심 분리기에 유리 병을 놓으십시오


12

이 질문은 reddit 사용자 taho_teg에 의해이 reddit 스레드 에서 시작 되지만 더 일반적인 '퍼즐'로 확장됩니다.

바이알을위한 24 개의 구멍이있는 원심 분리기가 중심 축 주위에 원으로 균일하게 분포되어 있습니다. 이제 여러 개의 바이알이 있고 원심 분리기를 시작하려는 경우 바이알을 균형 잡힌 방식으로 배치해야합니다. 균형을 맞출 수없는 바이알의 수는 1과 23입니다. 예를 들어 4의 균형을 잡을 수는 있지만, 3 개의 바이알로 '삼각형'을 만들고 두 개의 반대쪽에 두 개의 바이알을 배치하여 5의 균형을 맞출 수도 있습니다.

원심 분리기의 구멍 수 (회전 축 주위에 원으로 균일하게 분포되어 있음)를 입력으로 받아들이고 원심 분리기에서 균형을 맞출 수없는 바이알 수 목록을 출력하는 프로그램을 작성해야합니다.

계산을 수행해야하며 사전 계산 된 솔루션을 하드 코딩 할 수 없습니다.

다른 입력을 위해 프로그램을 호출하기 위해 프로그램 코드를 변경할 필요가 없도록 입력 및 출력을 구현해야합니다. 콘솔을 통해 호출 할 수있는 함수 (또는 해당 언어의 유사한 구문)를 작성하는 것도 허용됩니다.

원심 분리기에 6 개의 구멍이있는 경우 2 및 3 바이알을 원심 분리 할 수 ​​있지만 '삼각형'과 반대쪽 2 개가 한 지점에서 겹치므로 5의 균형을 맞출 수 없습니다. 또 다른 예로는 n = 15의 경우 11 바이알의 균형을 맞출 수없고 6 및 5 바이알의 균형을 맞출 수 있지만 이러한 솔루션의 조합이 겹칩니다 (물론 아직 그렇게 할 수없는 기준은 아닙니다).

최신 정보

일부 사람들은 주어진 예를 이해하지 못하는 것 같습니다. 그래서 여기에 그래픽을 만들었습니다. 제발 당신의 알고리즘이 작동하는 방법에 대한 간단한 설명뿐만 아니라 검증을위한 몇 가지 예를 들어 출력을 작성합니다. 다음 예를 포함하십시오 :

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

40320 및 65536은 거대한 목록을 생성하므로 해당 목록의 길이 만 나타내는 것이 좋습니다.

이 목록에 추가 할 흥미로운 숫자를 알고 있다면 알려주십시오. 알고리즘은 최소한 n = 1'000'000까지 작동해야합니다. 24 구 원심 분리기에 균형을 둔 5 개의 바이알

출력 예 :

이것들은 예제 출력이지만 수동으로 계산했기 때문에 아마도 결함이 있습니다.

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

힌트

n 개의 구멍 이있는 원심 분리기가 있고 6 개의 바이알과 균형을 맞출 수없는 경우 n-6 개의 바이알 을 뜯을 수 없습니다. 기본적으로 빈 원심 분리기에서 m 바이알의 균형을 맞추거나 채워진 원심 분리기의 균형 을 맞추는 것은 동일한 작업입니다 m 바이알 을 빼앗아 서 따라서 리스트에 m 이 있다면 nm 도 포함해야합니다 .


3
유리 병을 골고루 균등하게 배치해야합니까? 24 홀 예제의 5 개 바이알이 어떻게이를 충족시키는 지 알 수 없습니다. 원심 분리기의 한 "반"은 다른 절반보다 더 많은 약병을 갖습니다. 그것은 저에게 불균형의 정의처럼 들립니다.
Thorn

6
"균형"이 의미하는 것은 유리 병의 질량 중심이 원심 분리기의 질량 중심 위나 아래에 있다는 것입니다.
피터 테일러

2
@Thorn 당신은 하나가 아니라 2 차원으로 생각해야합니다. 처음 3 개의 바이알의 좌표는 (0,1), (-sqrt (3) / 2, -1 / 2) 및 (+ sqrt (3) / 2, -1 / 2)입니다. 5 개의 바이알 배열이 아닙니다. 대칭 (거울 평면 제외)이지만 균형이 맞습니다. 자동차 바퀴에는 다른 수의 스포크와 휠 너트 (대칭은 아니지만 스포크가 균형 잡힌 세트를 형성하고 너트가 균형 잡힌 세트를 형성하기 때문에 완전히 균형 잡힌) 7 spoke wheel가있는 것이 일반적입니다.
Level River St

2
"알고리즘은 적어도 n = 1'000'000까지 작동해야합니다." 확실하게 : 정말로 알고리즘을 의미 했습니까 , 아니면 프로그램 을 의미 했습니까 ? 내 알고리즘은 n = 10의 경우와 마찬가지로 n = 1.000.000에서도 잘 작동합니다. 그러나 프로그램에는 몇 가지 문제가 있습니다.
Wrzlprmft

1
@ edc65. 밸런스드! = 대칭 ... 바이알을 하위 그룹으로 나누는 한 각 하위 그룹이 대칭 상태 인 경우 모든 하위 그룹의 총 외 향력은 균형 상태가됩니다.
Eoin Campbell

답변:


5

세이지 - 102 115분의 104

무차별 대입이있을 때 왜 수 이론을 사용합니까?

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))]

주어진 수의 바이알에 대해, 이것은 복잡한 산술을 사용하여 바이알을 배치하고 질량 중심을 계산하는 모든 방법을 거칩니다. 이러한 방법으로 질량 중심이 0이 아니면 숫자가 반환됩니다.

불행히도 Sage는 일부 표현식을 0으로 단순화하지 못하기 때문에 특정 경우 (10,14)에서는 작동하지 않습니다 ( 이 버그 와 관련 있을 수 있음 ). 이것을 프로그램이 아니라 인터프리터의 결함으로 간주하고 알고리즘과 프로그램이 훌륭하다고 말할 수 있습니다.

다음 113 문자 대안은 기호 대신 부동 소수점을 사용하며 이러한 문제가 발생하지 않습니다.

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))]

113 자 버전 ( for n in range(14): print n,v(n)) 의 테스트 출력 :

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]

나는 런타임을 더 이상 기다리기를 원하지 않았다 n.


이것은 다음 Python 솔루션에서 시작됩니다. 정확한 산술과 일부 모듈을 가져올 필요는 없습니다.

파이썬 – 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))]

이 변형의 테스트 출력 ( 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]

나는 런타임을 더 이상 기다리기를 원하지 않았다 n.


나는 복잡한 단위 뿌리를 사용하는 아이디어를 좋아합니다! 확인을 위해 출력 예를 보여 주시겠습니까? 몇 가지 제안을 게시했습니다.
flawr

4

루아-197

무차별 강제 방법으로 요인 목록을 작성하고 제외시킵니다. 또한 사용되는 가장 큰 요소가 채워지지 않은 구멍의 양보다 작은 한 이러한 요소를 추가하여 얻을 수있는 숫자를 배제합니다. 하나는 항상 인쇄되며 알고리즘에서 사용되지 않습니다.

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

출력 예 : (일부는 범위로 넣어 문자 제한을 초과하지 않음)

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

나는 여기에 문제가 있다고 생각합니다 .100 구멍 원심 분리기에서 균형을 맞출 수는 없지만 8,18,28,48,58,68,78,88의 균형을 잡을 수 있습니다 (이 목록은 완전하지 않을 수 있습니다).
flawr

실수로 다른 숫자를 잘라 내야합니다. 다른 숫자도 마찬가지입니다.
Nexus

목록을 업데이트 해 주셔서 감사합니다! 가능하다면, 당신은 단지 당신이 그런 거대한 목록을 얻을 경우에 출력의 수를 셀 수 - 내가 생각하지 않았다 throu 정말 =)
flawr

@flawr 나는 목록을 업데이트했다
Nexus

당사 사이트를 사용함과 동시에 당사의 쿠키 정책개인정보 보호정책을 읽고 이해하였음을 인정하는 것으로 간주합니다.
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