Euler 001

arethmeticUnion_general.js

/*Generate a list of all Combinations*/
function getCombinations(list){
    var combinations = []; //All combinations
    var combination = [];  //Single combination
    var quantity = (1 << list.length);
    for (var i = 0; i < quantity ; i++){
        combination = [];
        for (var j=0;j<list.length;j++) {
            if ((i & (1 << j))){ 
                combination.push(list[j]);
            }
        }
        if (combination.length !== 0) {
            combinations.push(combination);
        }
    }
    return combinations;
}

/*Generate a product from a list of numbers*/
function listProduct(list){
    var product=1;
    for (var i=0;i<list.length;i++){
        product *=list[i];
    }
    return product;
}

/*Return the arithmetic sum*/
function arithmeticSum (a, n){
    return (1/2)*n*(a+a*n);
}

/*Generate the arithmetic sum of a series based on a multiple*/
function arithmeticSumFromSeriesMultiple (series, multiple){
    var a = multiple;
    var n = Math.floor(series/multiple);
    return arithmeticSum(a, n);
}

    
function sumMultiples(range, multiples){
    var sum= 0;
    var subsetSums = [];
    
    var multiplesCombination=getCombinations(multiples);
    for (var i=0;i<multiplesCombination.length;i++){
        
        //Generate product from combinations of multiples
        //and
        //Find individual sums of all combinations.
        subsetSums.push(
            arithmeticSumFromSeriesMultiple(
                range, 
                listProduct(multiplesCombination[i])
            )
        );
    }

    for (var i=1; i< subsetSums.length + 1;i++){
        //Check if i is an even base 2.
        if ((i & (i - 1)) == 0){
            sum += subsetSums[i-1];
        } else {
            sum -= subsetSums[i-1];
        }
    }
    return sum;
}

console.log(sumMultiples(999, [3,5]));

arethmeticUnion_specific.js

function sumMultiplesOfTwoOrThree(n){
    var sum= 0;
    var arithmeticSum = function(a, n){
        return (1/2)*n*(a+a*n);
    };
    var arithmeticSumFromSeriesMultiple = function(series, multiple){
        series -=1; //Euler problem #1 wants numbers less than 1000.
        var a = multiple;
        var n = Math.floor(series/multiple);
        return arithmeticSum(a, n);
    }
    sum += arithmeticSumFromSeriesMultiple(1000, 3);
    sum += arithmeticSumFromSeriesMultiple(1000, 5)
    sum -= arithmeticSumFromSeriesMultiple(1000, 15); 
    return sum;
}
console.log(sumMultiplesOfTwoAndThree(1000))

bruteforce_general.js

function sumMultiples(n, multiples){
    var sum= 0;
    var isMultiple = function(n, multiple){
        return ((n % multiple) === 0);
    };
    
    var commonMultiples = function (n,multiples){
        var isCommon = false;
        for (var i=0;i<multiples.length;i++){
            isCommon = (isCommon || isMultiple(n, multiples[i]));
        }
        return isCommon;
    };
    for (var i=1;i<n;i++) {
        if (commonMultiples (i,multiples)) {
            sum += i;
        }
    }
    return sum;
}
console.log(sumMultiples(1000, [3,5]))

bruteforce_specific.js

function sumMultiplesOfTwoAndThree(n){
    var sum= 0;
    for (var i=1;i<n;i++) {
        if (((i % 3) === 0) || ((i % 5) === 0)) {
            sum += i;
        }
    }
    return sum;
}
console.log(sumMultiplesOfTwoAndThree(1000))

combinatorics

function getCombinations(list){
    var combinations = []; //All combinations
    var combination = [];  //Single combination
    var quantity = (1 << list.length);
    for (var i = 0; i < quantity ; i++){
        combination = [];
        for (var j=0;j<list.length;j++) {
            if ((i & (1 << j))){ 
                combination.push(list[j]);
            }
        }
        if (combination.length !== 0) {
            combinations.push(combination);
        }
    }
    return combinations;
}

function listProduct(list){
    var product=1;
    for (var i=0;i<list.length;i++){
        product *=list[i];
    }
    return product;
}

var data=getCombinations([3,5,15,100]);
var output=[];
for (var i=0;i<data.length;i++){
    output.push(listProduct(data[i]));
}
console.log(data);
console.log(output);

Inclusion-exclusion.tex

$ | \bigcup\limits_{i=1}^n A_i | = $

$ + ( \sum\limits_{i=0}^n | A_i | ) $

$ - ( \sum\limits_{i,j:1 \le i<j \le n} | A_i \cap A_j | ) $

$ + ( \sum\limits_{i,j,k:1 \le i < j < k \le n} | A_i \cap A_j \cap A_k | ) $

$ - ( \sum\limits_{i,j,k,l:1 \le i < j < k < l \le n} | A_i \cap A_j \cap A_k \cap A_l | ) $

$ \vdots $

$ + ( (-1)^{n-1} | A_1 \cap A_2  \cap \ldots  \cap A_{n-1}  \cap A_n |) $