I got this challenge to find the highest product of 4 consecutive numbers on a 20x20 matrix of integers.
The numbers are read line by line from a file separated by a space.
The products can be in horizontal, vertical and diagonal in both directions
My "solution" gives the wrong answer.
EDIT: I've updated the code to work without file input and added sample data; also fixed one of my mistakes that were pointed out in the comments
$data = [
[89,32,92,64,81,2,20,33,44,1,70,75,39,62,76,35,16,77,22,27],
[53,11,6,95,41,51,31,59,8,23,19,13,61,91,48,69,84,52,66,24],
[93,72,85,97,21,79,56,5,45,3,65,30,83,87,43,7,34,0,4,14],
[29,17,49,9,82,90,55,67,15,63,54,94,12,28,96,37,58,98,86,78],
[74,40,50,60,26,99,80,18,10,46,36,68,25,57,47,71,42,73,88,38],
[50,22,6,26,18,53,52,5,46,2,89,77,83,48,4,58,45,28,84,81],
[49,82,31,14,69,17,91,54,34,40,0,33,30,95,60,44,29,24,85,16],
[27,11,76,39,15,86,92,74,99,59,94,12,55,57,38,96,47,32,78,75],
[51,20,87,42,62,41,7,35,23,21,71,25,67,97,80,90,88,64,13,70],
[19,9,56,43,68,93,65,98,36,3,61,63,10,72,8,73,1,66,79,37],
[22,58,52,12,3,41,28,72,42,74,76,64,59,35,85,78,14,27,53,88],
[46,80,5,96,7,68,61,69,67,34,36,40,82,26,75,50,29,91,10,2],
[30,39,19,48,33,93,1,45,66,98,0,23,62,25,51,71,56,77,24,21],
[79,87,94,60,8,32,13,65,4,92,73,9,31,37,17,84,15,90,86,20],
[95,6,81,70,47,16,44,83,49,43,55,54,18,63,38,11,97,89,99,57],
[95,78,64,58,7,17,53,28,74,86,6,12,54,85,21,94,16,69,25,68],
[13,20,41,97,1,2,80,30,0,84,67,45,93,96,82,92,62,33,18,44],
[60,77,31,70,76,36,59,38,15,3,91,46,65,73,49,11,8,35,5,52],
[61,66,79,40,26,72,89,71,75,99,22,9,43,32,14,81,98,88,87,83],
[10,4,23,19,56,57,51,47,50,27,90,63,42,29,24,55,48,37,39,34]
];
$matrix = [];
//maximums in possible directions
$maxes = [0, 0, 0, 0];
//while ($line = trim(fgets(STDIN))) {
while ($line = current($data)) {
//the horizontal maxes can be calculated while loading
//$array = explode(" ", $line);
$array = $line;
$hMax = array_product(array_slice($array, 0, 4));
for ($i = 1; $i < (count($array)-4); $i++) {
$max = array_product(array_slice($array, $i, 4));
if($max > $hMax) {
$hMax = $max;
}
}
if ( $hMax > $maxes[0] ) {
$maxes[0] = $hMax;
}
$matrix[] = $array;
next($data);
}
// the last 3 rows can be skipped
for($i = 0; $i < (count($matrix)-4); $i++) {
for ($j = 0; $j < (count($matrix[$i])-1); $j++) {
$vMax = 1; // vertical
$dlMax = 1; // diagonal left
$drMax = 1; // diagonal rigth
for ($k = 0; $k < 5; $k++) {
$vMax *= $matrix[$i + $k][$j];
if ( $j < (count($matrix[$i]) - 4) ) {
$drMax *= $matrix[$i + $k][$j + $k];
}
if ( $j > 3 ) {
$dlMax *= $matrix[$i + $k][$j - $k];
}
}
if ( $maxes[1] < $vMax ) $maxes[1] = $vMax; // the index used to be 1 - my first mistake
if ( $maxes[2] < $dlMax ) $maxes[2] = $dlMax; // the index used to be 1 - my first mistake
if ( $maxes[3] < $drMax ) $maxes[3] = $drMax; // the index used to be 1 - my first mistake
}
}
sort($maxes);
echo end($maxes).PHP_EOL;
Where did my approach go wrong, and how can it be sped up?
Are there any math tricks that can be applied here (besides checking for zeros)?
EDIT: the solution that the code gives for the current data is 4912231320 is it correct?
I've found 2 major errors, and now the result is a plausible 67352832
I'm considering it solved for that reason, but if anyone comes up with some math trick that simplifies or makes it faster I'll give up the accepted answer.
The first mistake was
for ($k = 0; $k < 5; $k++) {
It should've been
for ($k = 0; $k < 4; $k++) {
since we are only counting 4 numbers at once, thats why the result was so large compared to 10^8
The second was
if ( $j > 3 ) {
which should've been
if ( $j > 2 ) {
which will now include one more diagonal possibility
We can consider the four directions a bottom- or right-most cell can be the last of in a sequence. If m[i][j][k][d] is the highest total for a sequence of length k coming from direction d, then:
m[i][j][1][d] = data[i][j] for all d
m[i][j][k]['E'] = data[i][j] * m[i][j - 1][k - 1]['E']
m[i][j][k]['NE'] = data[i][j] * m[i - 1][j - 1][k - 1]['NE']
m[i][j][k]['N'] = data[i][j] * m[i - 1][j][k - 1]['N']
m[i][j][k]['NW'] = data[i][j] * m[i - 1][j + 1][k - 1]['NW']
If we traverse north to south, east to west, the needed cells should have already been calculated, and, clearly, we're looking for
max(m[i][j][4][d])
for all i, j, d
i have an small issue with the way this problem is resolved.
some would say: println((0 /: ((0 until 1000).filter(x => x % 3 == 0 || x % 5 == 0))) (_+_)) is the solution witch adds to 233168
my way was to do:
$maxnumber = 1000;
for ($i = 3; $i < $maxnumber; $i += 3)
{
$t += $i;
echo $i.',';
}
echo '<br>';
for ($j = 5; $j < $maxnumber; $j += 5)
{
$d += $j;
echo $j.',';
}
echo '<br>';
echo $t;
echo '<br>';
echo $d;
echo '<br>';
echo $t+$d;
this will give me :
3,6,9,12,15,18,21,24,27,30,33,36,39,42,45,48,51,54,57,60,63,66,69,72,75,78,81,84,87,90,93,96,99,102,105,108,111,114,117,120,123,126,129,132,135,138,141,144,147,150,153,156,159,162,165,168,171,174,177,180,183,186,189,192,195,198,201,204,207,210,213,216,219,222,225,228,231,234,237,240,243,246,249,252,255,258,261,264,267,270,273,276,279,282,285,288,291,294,297,300,303,306,309,312,315,318,321,324,327,330,333,336,339,342,345,348,351,354,357,360,363,366,369,372,375,378,381,384,387,390,393,396,399,402,405,408,411,414,417,420,423,426,429,432,435,438,441,444,447,450,453,456,459,462,465,468,471,474,477,480,483,486,489,492,495,498,501,504,507,510,513,516,519,522,525,528,531,534,537,540,543,546,549,552,555,558,561,564,567,570,573,576,579,582,585,588,591,594,597,600,603,606,609,612,615,618,621,624,627,630,633,636,639,642,645,648,651,654,657,660,663,666,669,672,675,678,681,684,687,690,693,696,699,702,705,708,711,714,717,720,723,726,729,732,735,738,741,744,747,750,753,756,759,762,765,768,771,774,777,780,783,786,789,792,795,798,801,804,807,810,813,816,819,822,825,828,831,834,837,840,843,846,849,852,855,858,861,864,867,870,873,876,879,882,885,888,891,894,897,900,903,906,909,912,915,918,921,924,927,930,933,936,939,942,945,948,951,954,957,960,963,966,969,972,975,978,981,984,987,990,993,996,999
5,10,15,20,25,30,35,40,45,50,55,60,65,70,75,80,85,90,95,100,105,110,115,120,125,130,135,140,145,150,155,160,165,170,175,180,185,190,195,200,205,210,215,220,225,230,235,240,245,250,255,260,265,270,275,280,285,290,295,300,305,310,315,320,325,330,335,340,345,350,355,360,365,370,375,380,385,390,395,400,405,410,415,420,425,430,435,440,445,450,455,460,465,470,475,480,485,490,495,500,505,510,515,520,525,530,535,540,545,550,555,560,565,570,575,580,585,590,595,600,605,610,615,620,625,630,635,640,645,650,655,660,665,670,675,680,685,690,695,700,705,710,715,720,725,730,735,740,745,750,755,760,765,770,775,780,785,790,795,800,805,810,815,820,825,830,835,840,845,850,855,860,865,870,875,880,885,890,895,900,905,910,915,920,925,930,935,940,945,950,955,960,965,970,975,980,985,990,995
$t - 166833
$d - 99500
and total:
266333
why am i wrong?
Some numbers are multiples of both 3 and 5. (Your algorithm adds these numbers to the total twice.)
Because 6 * 5 == 30 and 10 * 3 == 30, you're adding the some numbers up twice.
$sum = 0;
$i = 0;
foreach(range(0, 999) as $i) {
if($i % 3 == 0 || $i % 5 == 0) $sum += $i;
}
Because you double-count numbers that are multiple of both 3 and 5, i.e. multiples of 15.
You can account for this naively by subtracting all multiples of 15.
for ($j = 15; $j < $maxnumber; $j += 15)
{
$e += $j;
echo $j.',';
}
$total = $total - $d;
In your case, if it is 15, you will add the number twice.
Try this:
$t = 0;
$d = 0;
for ($i = 0; $i <= $maxnumber; $i++){
if ($i % 3 == 0)
$t+= $i;
else if ($i % 5 == 0)
$d += $i;
}
echo $t.'<br>'.$d;
I think that in your code, if a number is a multiple of 3 and 5, it is added twice. Take 15 for example. It's in your list of multiples of 3 and in the list of multiples of 5. Is this the behaviour you want?
One of the best approach to this solution (to achieve optimum time complexity), run an Arithmetic Progression series and find the number of terms in all series by using AP formula: T=a+(n-1)d, then find sum by : S=n/2[2*a+(n-1)d]
where : a=first term ,n=no. of term , d=common deference, T=nth term
The code solution below has been implemented to suit the question above - so the values 3 and 5 are hard-coded. However, the function can modified such that values are passed in as variable parameters.
function solution($number){
$val1 = 3;
$val2 = 5;
$common_term = $val1 * $val2;
$sum_of_terms1 = calculateSumofMulitples($val1,$number);
$sum_of_terms2 = calculateSumofMulitples($val2,$number);
$sum_of_cterms = calculateSumofMulitples($common_term,$number);
$final_result = $sum_of_terms1 + $sum_of_terms2 - $sum_of_cterms;
return $final_result;
}
function calculateSumofMulitples($val, $number)
{
//first, we begin by running an aithmetic prograssion for $val up to $number say N to get the no of terms [using T=a +(n-1)d]
$no_of_terms = (int) ($number / $val);
if($number % $val == 0) $no_of_terms = (int) ( ($number-1)/$val ); //since we are computing for a no of terms below N and not up to/exactly/up to N. if N divides val with no remainder, use no of terms = (N-1)/val
//second, we compute the sum of terms [using Sn = n/2[2a + (n-1)d]
$sum_of_terms = ($no_of_terms * 0.5) * ( (2*$val) + ($no_of_terms - 1) * $val );
// sum of multiples
return $sum_of_terms;
}
You can run a single loop checking whether the number is multiple of 3 OR 5:
for ($i = 0; $i < $maxnumber; $i++)
{
if($i%3 || $i%5){
$t += $i;
echo $i.',';}
}
I think the original code is not including numbers which are multiples of both 3 and 5 in the total: if the test for multiple of 3 matches, it takes that and goes on.
If you total the multiples of 15 up to 1000, you get 33165, which is exactly the difference between your total, 266333, and the original total, 233168.
Here's my solution to the question:
<?php
$sum = 0;
$arr = [];
for($i = 1; $i < 1000; $i++){
if((int)$i % 3 === 0 || (int)$i % 5 === 0)
{
$sum += $i;
array_push($arr,$i);
}
}
echo $sum;
echo '<br>';
print_r($arr);//Displays the values meeting the criteria as an array of values