歐拉項目問題11：Largest product in a grid
In the 20
20 grid below, four numbers along a diagonal
line have been marked in red.
08 02 22 97 38 15 00 40 00 75 04 05 07 78 52 12 50 77 91 08
49 49 99 40 17 81 18 57 60 87 17 40 98 43 69 48 04 56 62 00
81 49 31 73 55 79 14 29 93 71 40 67 53 88 30 03 49 13 36 65
52 70 95 23 04 60 11 42 69 24 68 56 01 32 56 71 37 02 36 91
22 31 16 71 51 67 63 89 41 92 36 54 22 40 40 28 66 33 13 80
24 47 32 60 99 03 45 02 44 75 33 53 78 36 84 20 35 17 12 50
32 98 81 28 64 23 67 10 26 38 40 67 59 54 70 66 18 38 64 70
67 26 20 68 02 62 12 20 95 63 94 39 63 08 40 91 66 49 94 21
24 55 58 05 66 73 99 26 97 17 78 78 96 83 14 88 34 89 63 72
21 36 23 09 75 00 76 44 20 45 35 14 00 61 33 97 34 31 33 95
78 17 53 28 22 75 31 67 15 94 03 80 04 62 16 14 09 53 56 92
16 39 05 42 96 35 31 47 55 58 88 24 00 17 54 24 36 29 85 57
86 56 00 48 35 71 89 07 05 44 44 37 44 60 21 58 51 54 17 58
19 80 81 68 05 94 47 69 28 73 92 13 86 52 17 77 04 89 55 40
04 52 08 83 97 35 99 16 07 97 57 32 16 26 26 79 33 27 98 66
88 36 68 87 57 62 20 72 03 46 33 67 46 55 12 32 63 93 53 69
04 42 16 73 38 25 39 11 24 94 72 18 08 46 29 32 40 62 76 36
20 69 36 41 72 30 23 88 34 62 99 69 82 67 59 85 74 04 36 16
20 73 35 29 78 31 90 01 74 31 49 71 48 86 81 16 23 57 05 54
01 70 54 71 83 51 54 69 16 92 33 48 61 43 52 01 89 19 67 48
The product of these numbers is 26 63 78 14
= 1788696.
What is the greatest product of four adjacent numbers in the same direction (up, down, left, right, or diagonally) in the 2020
grid?
分析
這道題目是要求找到最大的連續四個數乘積（在上，下，左，右，斜線方向），可以看出這個滿足Dynamic Programming的兩個特徵，所以可以優化查找過程而不是遍歷每一種可能。解決方案（Java）
package Problem11;
public class LargestProduct {
private static final int[] array = {8,02,22,97,38,15,00,40,00,75,04,05,07,78,52,12,50,77,91,8
,49,49,99,40,17,81,18,57,60,87,17,40,98,43,69,48,04,56,62,00
,81,49,31,73,55,79,14,29,93,71,40,67,53,88,30,03,49,13,36,65
,52,70,95,23,04,60,11,42,69,24,68,56,01,32,56,71,37,02,36,91
,22,31,16,71,51,67,63,89,41,92,36,54,22,40,40,28,66,33,13,80
,24,47,32,60,99,03,45,02,44,75,33,53,78,36,84,20,35,17,12,50
,32,98,81,28,64,23,67,10,26,38,40,67,59,54,70,66,18,38,64,70
,67,26,20,68,02,62,12,20,95,63,94,39,63,8,40,91,66,49,94,21
,24,55,58,05,66,73,99,26,97,17,78,78,96,83,14,88,34,89,63,72
,21,36,23,9,75,00,76,44,20,45,35,14,00,61,33,97,34,31,33,95
,78,17,53,28,22,75,31,67,15,94,03,80,04,62,16,14,9,53,56,92
,16,39,05,42,96,35,31,47,55,58,88,24,00,17,54,24,36,29,85,57
,86,56,00,48,35,71,89,07,05,44,44,37,44,60,21,58,51,54,17,58
,19,80,81,68,05,94,47,69,28,73,92,13,86,52,17,77,04,89,55,40
,04,52,8,83,97,35,99,16,07,97,57,32,16,26,26,79,33,27,98,66
,88,36,68,87,57,62,20,72,03,46,33,67,46,55,12,32,63,93,53,69
,04,42,16,73,38,25,39,11,24,94,72,18,8,46,29,32,40,62,76,36
,20,69,36,41,72,30,23,88,34,62,99,69,82,67,59,85,74,04,36,16
,20,73,35,29,78,31,90,01,74,31,49,71,48,86,81,16,23,57,05,54
,01,70,54,71,83,51,54,69,16,92,33,48,61,43,52,01,89,19,67,48};
private static final int level = 20;
private int maxProduct = 0;
public static void main(String[] args){
LargestProduct lp = new LargestProduct();
long startTime = System.currentTimeMillis();
lp.compute();
long endTime = System.currentTimeMillis();
System.out.println("Time: "+(endTime-startTime));
lp.print();
}
private void compute(){
for(int i=0;i<array.length;i++){
int rightProduct = 0;
int downProduct = 0;
int rightdownDiagonallyProduct = 0;
int leftdownDiagonallyProduct = 0;
int posX = i%level;
int posY = i/level;
int remainX = level - posX;
int remainY = level - posY;
int second;
int third;
int fourth;
if(remainX >3){
second = array[i+1];
third = array[i+2];
fourth = array[i+3];
rightProduct = second*third*fourth;
}
if(remainY >3){
second = array[(posY+1)*level+posX];
third = array[(posY+2)*level+posX];
fourth = array[(posY+3)*level+posX];
downProduct = second*third*fourth;
}
if(remainX >3 && remainY >3){
second = array[(posY+1)*level+posX+1];
third = array[(posY+2)*level+posX+2];
fourth = array[(posY+3)*level+posX+3];
rightdownDiagonallyProduct = second*third*fourth;
}
if(posX>=3&&remainY > 3){
second = array[(posY+1)*level+posX-1];
third = array[(posY+2)*level+posX-2];
fourth = array[(posY+3)*level+posX-3];
leftdownDiagonallyProduct = second*third*fourth;
}
int tmpProduct = rightProduct;
if(downProduct>rightProduct){
tmpProduct = downProduct;
}
if(rightdownDiagonallyProduct>tmpProduct){
tmpProduct = rightdownDiagonallyProduct;
}
if(leftdownDiagonallyProduct>tmpProduct){
tmpProduct = leftdownDiagonallyProduct;
}
if(tmpProduct>0){
tmpProduct = array[i] * tmpProduct;
if(tmpProduct > maxProduct){
maxProduct = tmpProduct;
}
}
}
}
private void print(){
System.out.println("The max product is :"+maxProduct);
}
}