Cada peça de dominó tem dois números de 0 a 6, representados pelo número de pontos sobre sua face. Um conjunto de dominós tem exatamente uma peça de cada tipo, não havendo repetições. Pergunto: quantas peças e quantos pontos há em um conjunto de dominós?
(Each domino piece has two numbers from 0 to 6, represented by the number of points on its face. A domino set has exactly one piece of each kind, with no repetitions. I ask: how many pieces and how many points are there in a domino set?)
PCFilho
0-0
ResponderExcluir0-1
0-2
0-3
0-4
0-5
0-6
1-1
1-2
1-3
1-4
1-5
1-6
2-2
2-3
2-4
2-5
2-6
3-3
3-4
3-5
3-6
4-4
4-5
4-6
5-5
5-6
6-6
Peças : 28
Pontos : 168
O método força-bruta resolve mesmo.
ExcluirMas e se os dominós permitissem números maiores que 6? Se fossem até 10? :)
All right, let's say the highest number that appears on one half of a domino piece is n. Since the numbers go from 0 to n, that means that there are n + 1 possible numbers that can appear on one half of a domino in the set. So, the total number of dominoes in the set is (n + 1)(n + 2)/2. Each of the numbers from 0 to n appears (n + 2) times in the set, so the total number of points in the entire set is n(n + 1)(n + 2)/2, or the number of pieces times n. So, here are the stats for some common domino sets:
ExcluirDouble six: n = 6, total number of pieces = (7)(8)/2 = 28, total number of points = 28 * 6 = 168
Double nine: n = 9, total number of pieces = (10)(11)/2 = 55, total number of points = 55 * 9 = 495
Double twelve: n = 12, total number of pieces = (13)(14)/2 = 91, total number of points = 91 * 12 = 1092
Double fifteen: n = 15, total number of pieces = (16)(17)/2 = 136, total number of points = 136 * 15 = 2040
Double eighteen (yes, I have seen a set like this!): n = 18, total number of pieces = (19)(20)/2 = 190, total number of points = 190 * 18 = 3420
Those numbers get large rather quickly, don't they?
Yes! The sequence of numbers of pieces in a domino set is the following:
Excluir1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, 780, 820, 861, 903, 946, 990, 1035, 1081, 1128, 1176, 1225, 1275, 1326, 1378, 1431, ...
These numbers are known as the triangular numbers. :)
And the sequence of number of points in a domino set goes like this:
Excluir0, 3, 12, 30, 60, 105, 168, 252, 360, 495, 660, 858, 1092, 1365, 1680, 2040, 2448, 2907, 3420, 3990, 4620, 5313, 6072, 6900, 7800, 8775, 9828, 10962, 12180, 13485, 14880, 16368, 17952, 19635, 21420, 23310, 25308, 27417, 29640, 31980, 34440, ...