小学奥数题Java编程（2）8．把0—9这十个数字填到右图的圆圈内，使得五条线上的数字和构成一个等差数列，而且这
小学奥数题Java编程（2）
8．把0—9这十个数字填到右图的圆圈内，使得五条线上的数字和构成一个等差数列，而且这个等差数列的各项之和为55，那么这个等差数列的公差有2种可能的取值。?
?解：设顶点分别为ABCDE，有45+A+B+C+D+E=55，?
所以A+B+C+D+E=10，则除之外的另外5个数（即边上的）为45-10=35，?利用求和公式?5（Ａ１＋A1+４Ｄ）/2=55，?得A1+2d=11，A1大于等于6，且为奇数，?只能取7或9???这样d只能为1或者2????经试验能填出来??
两种情况??公差d等于1或者2。
package com.ssll.collect.pc;import java.util.Arrays;import java.util.List;public class PNCDemo { public static void main(String[] args) { System.out.println("===== demo permutation :"); //for(List<Integer> list : Permutation.of(Arrays.asList(0,1,2,3,4,5,6,7,8,9))) //System.out.println(list); for(List<Integer> list : Permutation.of(Arrays.asList(0,1,2,3,4))){ for(List<Integer> list2 : Permutation.of(Arrays.asList(5,6,7,8,9))){ isArithmeticProgression(list, list2); } } } public static int sum(int i, List<Integer> list1, List<Integer> list2){ return i<4? list1.get(i)+list1.get(i+1)+list2.get(i) :list1.get(i)+list1.get(0)+list2.get(i); } public static boolean isArithmeticProgression(List<Integer> list1, List<Integer> list2){ int presum =0; int predef = 0; int thissum =0; presum= sum(0, list1,list2); thissum= sum(1, list1,list2); predef = thissum-presum; for(int i=2; i<5; i++){ presum = thissum; thissum= sum(i, list1,list2); if(predef!=thissum-presum) return false; } System.out.println("def="+predef+" list1="+list1+" list2="+list2); return true; }}输出结果为：
def=1 list1=[0, 1, 2, 3, 4] list2=[8, 7, 6, 5, 9]
def=0 list1=[0, 2, 4, 1, 3] list2=[9, 5, 6, 7, 8]
def=0 list1=[0, 3, 1, 4, 2] list2=[8, 7, 6, 5, 9]
def=1 list1=[0, 3, 2, 1, 4] list2=[6, 5, 8, 7, 9]
def=-1 list1=[0, 4, 1, 2, 3] list2=[9, 7, 8, 5, 6]
def=-1 list1=[0, 4, 3, 2, 1] list2=[9, 5, 6, 7, 8]
def=1 list1=[1, 0, 4, 2, 3] list2=[8, 6, 5, 7, 9]
def=1 list1=[1, 2, 0, 4, 3] list2=[6, 8, 7, 5, 9]
def=1 list1=[1, 3, 0, 2, 4] list2=[5, 7, 9, 6, 8]
def=0 list1=[1, 3, 0, 2, 4] list2=[7, 8, 9, 5, 6]
def=-1 list1=[1, 3, 2, 4, 0] list2=[9, 7, 5, 6, 8]
def=-1 list1=[1, 3, 4, 0, 2] list2=[9, 5, 7, 8, 6]
def=0 list1=[1, 4, 2, 0, 3] list2=[6, 5, 9, 8, 7]
def=-1 list1=[1, 4, 2, 0, 3] list2=[8, 6, 9, 7, 5]
def=2 list1=[2, 0, 1, 3, 4] list2=[5, 8, 7, 6, 9]
def=1 list1=[2, 0, 1, 4, 3] list2=[7, 9, 6, 5, 8]
def=2 list1=[2, 0, 3, 1, 4] list2=[5, 6, 7, 8, 9]
def=0 list1=[2, 0, 3, 1, 4] list2=[9, 8, 7, 6, 5]
def=1 list1=[2, 1, 0, 3, 4] list2=[6, 9, 8, 5, 7]
def=1 list1=[2, 1, 0, 4, 3] list2=[6, 9, 7, 5, 8]
def=1 list1=[2, 1, 4, 0, 3] list2=[6, 5, 7, 9, 8]
def=-1 list1=[2, 3, 0, 4, 1] list2=[8, 9, 7, 5, 6]
def=-1 list1=[2, 3, 4, 0, 1] list2=[8, 5, 7, 9, 6]
def=-1 list1=[2, 3, 4, 1, 0] list2=[8, 5, 6, 9, 7]
def=0 list1=[2, 4, 1, 3, 0] list2=[5, 6, 7, 8, 9]
def=-2 list1=[2, 4, 1, 3, 0] list2=[9, 8, 7, 6, 5]
def=-1 list1=[2, 4, 3, 0, 1] list2=[7, 5, 8, 9, 6]
def=-2 list1=[2, 4, 3, 1, 0] list2=[9, 6, 7, 8, 5]
def=1 list1=[3, 0, 2, 4, 1] list2=[6, 8, 5, 7, 9]
def=0 list1=[3, 0, 2, 4, 1] list2=[8, 9, 5, 6, 7]
def=1 list1=[3, 1, 0, 4, 2] list2=[5, 9, 7, 6, 8]
def=1 list1=[3, 1, 2, 0, 4] list2=[5, 7, 9, 8, 6]
def=0 list1=[3, 1, 4, 2, 0] list2=[7, 6, 5, 9, 8]
def=-1 list1=[3, 1, 4, 2, 0] list2=[9, 7, 5, 8, 6]
def=-1 list1=[3, 2, 4, 0, 1] list2=[8, 6, 7, 9, 5]
def=-1 list1=[3, 4, 0, 2, 1] list2=[6, 8, 9, 7, 5]
def=1 list1=[4, 0, 1, 2, 3] list2=[5, 9, 8, 7, 6]
def=1 list1=[4, 0, 3, 2, 1] list2=[5, 7, 6, 9, 8]
def=-1 list1=[4, 1, 2, 3, 0] list2=[8, 9, 6, 7, 5]
def=0 list1=[4, 1, 3, 0, 2] list2=[6, 7, 8, 9, 5]
def=0 list1=[4, 2, 0, 3, 1] list2=[5, 9, 8, 7, 6]
def=-1 list1=[4, 3, 2, 1, 0] list2=[6, 7, 8, 9, 5]
这里面用了来自于 http://blog.csdn.net/raistlic/article/details/7844812
的排列组合类，使得计算大大简化，输出结果中list1为五个顶点上的数，list2为五条边上的数，从结果上可以看出有五个差，0，+1，－1， ＋2，－2，很多解都仅提到了1和2，这也是这道题的争议之处。
