Problem 11:Largest product in a grid
2013-08-04 00:06
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原题地址：http://projecteuler.net/problem=11
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 20
20
grid?
大意:
在以下这个20
20的网格中，四个处于同一对角线上的相邻数字用红色标了出来：
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
这四个数字的乘积是：26
63
78
14
= 1788696.
在这个20
20网格中，处于任何方向上（上，下，左，右或者对角线）的四个相邻数字的乘积的最大值是多少？
解法1：
1.遍历每一行，算出所有相邻4个数的乘积，并记录下最大乘积
2.遍历每一列，算出所有相邻4个数的乘积，并记录下最大乘积
3.遍历对角线，方向是\, 算出所有相邻4个数的乘积，并记录下最大乘积
4.遍历对角线，方向是/, 算出所有相邻4个数的乘积，并记录下最大乘积
php代码如下所示：
注：题目的中文翻译源自http://pe.spiritzhang.com
Largest product in a grid
Problem 11
In the 2020 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 20
20
grid?
大意:
在以下这个20
20的网格中，四个处于同一对角线上的相邻数字用红色标了出来：
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
这四个数字的乘积是：26
63
78
14
= 1788696.
在这个20
20网格中，处于任何方向上（上，下，左，右或者对角线）的四个相邻数字的乘积的最大值是多少？
解法1：
1.遍历每一行，算出所有相邻4个数的乘积，并记录下最大乘积
2.遍历每一列，算出所有相邻4个数的乘积，并记录下最大乘积
3.遍历对角线，方向是\, 算出所有相邻4个数的乘积，并记录下最大乘积
4.遍历对角线，方向是/, 算出所有相邻4个数的乘积，并记录下最大乘积
php代码如下所示：
$data = array(
array("08", "02", "22", "97", "38", "15", "00", "40", "00", "75", "04", "05", "07", "78", "52", "12", "50", "77", "91", "08"),
array("49", "49", "99", "40", "17", "81", "18", "57", "60", "87", "17", "40", "98", "43", "69", "48", "04", "56", "62", "00"),
array("81", "49", "31", "73", "55", "79", "14", "29", "93", "71", "40", "67", "53", "88", "30", "03", "49", "13", "36", "65"),
array("52", "70", "95", "23", "04", "60", "11", "42", "69", "24", "68", "56", "01", "32", "56", "71", "37", "02", "36", "91"),
array("22", "31", "16", "71", "51", "67", "63", "89", "41", "92", "36", "54", "22", "40", "40", "28", "66", "33", "13", "80"),
array("24", "47", "32", "60", "99", "03", "45", "02", "44", "75", "33", "53", "78", "36", "84", "20", "35", "17", "12", "50"),
array("32", "98", "81", "28", "64", "23", "67", "10", "26", "38", "40", "67", "59", "54", "70", "66", "18", "38", "64", "70"),
array("67", "26", "20", "68", "02", "62", "12", "20", "95", "63", "94", "39", "63", "08", "40", "91", "66", "49", "94", "21"),
array("24", "55", "58", "05", "66", "73", "99", "26", "97", "17", "78", "78", "96", "83", "14", "88", "34", "89", "63", "72"),
array("21", "36", "23", "09", "75", "00", "76", "44", "20", "45", "35", "14", "00", "61", "33", "97", "34", "31", "33", "95"),
array("78", "17", "53", "28", "22", "75", "31", "67", "15", "94", "03", "80", "04", "62", "16", "14", "09", "53", "56", "92"),
array("16", "39", "05", "42", "96", "35", "31", "47", "55", "58", "88", "24", "00", "17", "54", "24", "36", "29", "85", "57"),
array("86", "56", "00", "48", "35", "71", "89", "07", "05", "44", "44", "37", "44", "60", "21", "58", "51", "54", "17", "58"),
array("19", "80", "81", "68", "05", "94", "47", "69", "28", "73", "92", "13", "86", "52", "17", "77", "04", "89", "55", "40"),
array("04", "52", "08", "83", "97", "35", "99", "16", "07", "97", "57", "32", "16", "26", "26", "79", "33", "27", "98", "66"),
array("88", "36", "68", "87", "57", "62", "20", "72", "03", "46", "33", "67", "46", "55", "12", "32", "63", "93", "53", "69"),
array("04", "42", "16", "73", "38", "25", "39", "11", "24", "94", "72", "18", "08", "46", "29", "32", "40", "62", "76", "36"),
array("20", "69", "36", "41", "72", "30", "23", "88", "34", "62", "99", "69", "82", "67", "59", "85", "74", "04", "36", "16"),
array("20", "73", "35", "29", "78", "31", "90", "01", "74", "31", "49", "71", "48", "86", "81", "16", "23", "57", "05", "54"),
array("01", "70", "54", "71", "83", "51", "54", "69", "16", "92", "33", "48", "61", "43", "52", "01", "89", "19", "67", "48"),
);
$hight = 20;
$width = 20;
$number_sequence_len = 4;
$max = 0;
$max_array = array();
function set_max($arr,&$max,&$max_array){
$sum = 1;
foreach($arr as $number){
$sum *= $number;
}
if($sum>$max){
$max = $sum;
$max_array = $arr;
}
}
$groups_of_row = $width-$number_sequence_len+1;
$groups_of_col = $hight-$number_sequence_len+1;
//行
for($i=0;$i<$hight;$i++){
for($j=0;$j<$groups_of_row;$j++){
$temp = array_slice($data[$i],$j,$number_sequence_len);
set_max($temp,$max,$max_array);
}
}
//列
for($i=0;$i<$width;$i++){
for($j=0;$j<$groups_of_col;$j++){
$temp = array();
for($k=0;$k<$number_sequence_len;$k++){
$temp[] = $data[$j+$k][$i];
}
set_max($temp,$max,$max_array);
}
}
//对角：\
for($i=0;$i<$groups_of_col;$i++){
for($j=0;$j<$groups_of_row;$j++){
$temp = array();
for($k=0;$k<$number_sequence_len;$k++){
$temp[] = $data[$i+$k][$j+$k];
}
set_max($temp,$max,$max_array);
}
}
//对角：/
for($i=0;$i<$groups_of_col;$i++){
for($j=$number_sequence_len-1;$j<$width;$j++){
$temp = array();
for($k=0;$k<$number_sequence_len;$k++){
$temp[] = $data[$i+$k][$j-$k];
}
set_max($temp,$max,$max_array);
}
}
echo $max;注：题目的中文翻译源自http://pe.spiritzhang.com
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