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#1

Various solutions, all running in under one second:

RPL+

≪ 0 =sum

   1 999 FOR i

      i 3 mod 0 ==

      i 5 mod 0 ==

         or

      IF THEN i =+sum END

   NEXT

   sum

≫

or (RPL+ alt)

≪ 1000 3 5 → n a b ≪

    --n

    a b lcm =c

    n n a mod - =na

    n n b mod - =nb

    n n c mod - =nc

    '(na*((na/a)+1) + nb*((nb/b)+1) - nc*((nc/c)+1) )/2' EVAL

≫≫

or (closed-form, classic RPL)

≪ 1000 → n '((n-1)*(((n-1)/3)+1)+(n-5)*(((n-5)/5)+1)-(n-10)*(((n-10)/15)+1))/2' ≫

or (classic RPL)

≪ 0. 1. 999.

    FOR I

        IF I 3. MOD 0. ==

        THEN I +

        ELSE

            IF I 5. MOD 0. ==

            THEN I +

            END

        END

    NEXT

≫

or (JavaScript; fastest)

function () { /*as is*/

   var sum = 0;

   for (var i = 0; i<1000; i++)

      if (i%3 === 0 || i%5 === 0)

         sum += i;

   return sum;

}

#2

≪ 0 =sum

   1 1

   DO

      DUP ROT + =:val

      IF val isEven THEN val =+sum END

   UNTIL

      val>4e6

   END

   DROP DROP

   sum

≫

#3

≪ 600851475143 factors head ≫

or (GolfScript/MorphEngine)

600851475143factors head

#4

≪ -1 =r @ palindrome result

   999 11 / floor 11 * =i

   WHILE

       i 100 >

       i 999 * r > @ potential for bigger r?

      and

   REPEAT

      100 999 FOR j

         i j * →STR =s

         1 =a s size =b

         DO s[a] s[b] ++a --b

         UNTIL ≠ =:cmp a b ≥ OR

         END

         IF cmp NOT THEN r i j * max =r END

      NEXT

   i 11 - =i

   END

   r

≫

[ND1: 4s; CalcPad/Mac: 0.6s]

#5

≪ 1..20 vlcm ≫

or (using range)

≪ 1 20 range vlcm ≫

or (using fold operator)

≪ 1 20 range [lcm] fold ≫

or (using classic RPL commands)

≪ 1 20 range ≪ LCM ≫ STREAM ≫

or (GolfScript/MorphEngine)

21,tail{lcm}*

#6

≪ 1..100 =v

   v total squared

   v squared total

  -

≫

or (using classic RPL commands)

≪ 1 100 range =v

   v ≪ + ≫ STREAM SQ

   v SQ ≪ + ≫ STREAM

  -

≫

or (GolfScript/MorphEngine)

101,tail.{+}*SQSQ{+}*-

#7

≪ 1.1e5 primes 10001 at ≫

[ND1: 1.9s; CalcPad/Mac: 0.1s]

or (a bit slower)

≪ 2 1 10000 START NEXTPRIME NEXT≫

[ND1: 2.5s; CalcPad/Mac: 0.2s]

or (GolfScript/MorphEngine)

2{NEXTPRIME}10000*

#8

≪ #8_data fromString split

   5 ≪ * * * * ≫ DOSUBS

   vmax

≫

[ND1: 1.2s; CalcPad/Mac: 0.08s]

or (GolfScript/MorphEngine)

→{2{max}DOSUBS+}1-*

#8_data

"7316717653133062491922511967442657474235534919493496983520312774506326239578318016984801869478851843858615607891129494954595017379583319528532088055111254069874715852386305071569329096329522744304355766896648950445244523161731856403098711121722383113622298934233803081353362766142828064444866452387493035890729629049156044077239071381051585930796086670172427121883998797908792274921901699720888093776657273330010533678812202354218097512545405947522435258490771167055601360483958644670632441572215539753697817977846174064955149290862569321978468622482839722413756570560574902614079729686524145351004748216637048440319989000889524345065854122758866688116427171479924442928230863465674813919123162824586178664583591245665294765456828489128831426076900422421902267105562632111110937054421750694165896040807198403850962455444362981230987879927244284909188845801561660979191338754992005240636899125607176060588611646710940507754100225698315520005593572972571636269561882670428252483600823257530420752963450"

#9

function () { /*as is*/

    for (var a=1;a<500;a++)

        for (var b=1;b<500;b++)

                if (a*a+b*b == (1000-a-b)*(1000-a-b))

                    return a*b*(1000-a-b);

}

#10

≪ 2e6 primes total ≫

[ND1: 41s; CalcPad/Mac: 2.6s]

or

≪ 2e6 primes { + } STREAM ≫

[ND1: 55s; CalcPad/Mac: 3.5s]

or (JavaScript)

function () {

    var sum = 0, p = primes(2e6);

    for (var i=p.length-1; i>=0; i--)

        sum += p[i];

    return sum;

}

[ND1: 43s; CalcPad/Mac: 2.6s]

or (GolfScript/MorphEngine)

2000000primes{+}*

#11

≪ #11_data =m

   0 @ max product kept on stack

   m

   m transpose concat

   m diagonals concat

   m flip diagonals concat

   1

   ≪ 4 [ * * * MAX ] DOSUBS ≫

   DOSUBS

≫

#11_data

[[8,2,22,97,38,15,0,40,0,75,4,5,7,78,52,12,50,77,91,8],[49,49,99,40,17,81,18,57,60,87,17,40,98,43,69,48,4,56,62,0],[81,49,31,73,55,79,14,29,93,71,40,67,53,88,30,3,49,13,36,65],[52,70,95,23,4,60,11,42,69,24,68,56,1,32,56,71,37,2,36,91],[22,31,16,71,51,67,63,89,41,92,36,54,22,40,40,28,66,33,13,80],[24,47,32,60,99,3,45,2,44,75,33,53,78,36,84,20,35,17,12,50],[32,98,81,28,64,23,67,10,26,38,40,67,59,54,70,66,18,38,64,70],[67,26,20,68,2,62,12,20,95,63,94,39,63,8,40,91,66,49,94,21],[24,55,58,5,66,73,99,26,97,17,78,78,96,83,14,88,34,89,63,72],[21,36,23,9,75,0,76,44,20,45,35,14,0,61,33,97,34,31,33,95],[78,17,53,28,22,75,31,67,15,94,3,80,4,62,16,14,9,53,56,92],[16,39,5,42,96,35,31,47,55,58,88,24,0,17,54,24,36,29,85,57],[86,56,0,48,35,71,89,7,5,44,44,37,44,60,21,58,51,54,17,58],[19,80,81,68,5,94,47,69,28,73,92,13,86,52,17,77,4,89,55,40],[4,52,8,83,97,35,99,16,7,97,57,32,16,26,26,79,33,27,98,66],[88,36,68,87,57,62,20,72,3,46,33,67,46,55,12,32,63,93,53,69],[4,42,16,73,38,25,39,11,24,94,72,18,8,46,29,32,40,62,76,36],[20,69,36,41,72,30,23,88,34,62,99,69,82,67,59,85,74,4,36,16],[20,73,35,29,78,31,90,1,74,31,49,71,48,86,81,16,23,57,5,54],[1,70,54,71,83,51,54,69,16,92,33,48,61,43,52,1,89,19,67,48]]

#12

≪ 1 =counter

   0 =num

   DO

      counter =+num

      ++counter

   UNTIL

      ndivs(num)>500

   END

   num

≫

[ND1: 6.6s; CalcPad/Mac: 0.5s]

#13

≪ #13_data

   total →STR 1 10 SUB STR→

≫

#13_data

function () {         

    var data = requestURL("http://naivedesign.com/redist/PE_13_data.txt").split("");

    return data.map(function(x){return BigNum.fromString(x);});

}

#14

function () { /*as is*/

    var longest = 0, index = 0;

    var cache = []; cache[1] = 0;

    for (var i=2; i<1e6; i++) {

        var cur = i, steps = 0;

        while (cur != 1 && cur >= i) {

            cur = (cur % 2 ? (3*cur+1) : (cur/2));

            ++steps;

        }

        if (cur < i)

            steps += cache[cur];

        cache[i] = steps;

        if (steps > longest) {

            longest = steps;

            index = i;

        }

    }

    return index;

}

[ND1: 8s; CalcPad/Mac: 0.4s]

or (in RPL+)

≪ 0 =longest

   [] =cache @ length of chain at num

   0 =cache[1] @ length for num 1

   2 1e6 1 - FOR i

      i

      0 =steps

      DO

         IF DUP isEven

         THEN 2 / ELSE 3 * ++ END

         ++steps

      UNTIL

         DUP =:cur i <

      END

      DROP

      cache[cur] =+steps

      steps =cache[i]

      IF steps longest >

      THEN steps =longest i =index END

   NEXT

   index

≫

[impractically slow; CalcPad/Mac: 253s]

#15

≪ 20 → x 'round((x*2)!/(x!^2), 0)' ≫

or

≪ 20 → x ≪ x 2 * ! x ! 2 ^ / 0 RND ≫≫

#16

≪ 2 1000 →big ^ split total ≫

#18

≪ #18_data OBJ→

   2 SWAP START

      2 ≪ MAX ≫ DOSUBS

      +

   NEXT

≫

or (GolfScript/MorphEngine)

→{2{max}DOSUBS+}1-*

#18_data

[75,[95,64],[17,47,82],[18,35,87,10],[20,4,82,47,65],[19,1,23,75,3,34],[88,2,77,73,7,63,67],[99,65,4,28,6,16,70,92],[41,41,26,56,83,40,80,70,33],[41,48,72,33,47,32,37,16,94,29],[53,71,44,65,25,43,91,52,97,51,14],[70,11,33,28,77,73,17,78,39,68,17,57],[91,71,52,38,17,14,91,43,58,50,27,29,48],[63,66,4,68,89,53,67,30,73,16,69,87,40,31],[4,62,98,27,23,9,70,98,73,93,38,53,60,4,23]]

#19

≪ floor(12*100/7) ≫

#20

≪ 100 ! split total ≫

or (ignorant of split and total, using a loop)

≪ 100 ! →STR =s

   0 =sum

   1 s SIZE FOR i

      s[i] STR→ =+sum

   NEXT

   sum

≫

or (using list processing)

≪ 100 ! →STR

   split

   { STR→ }  MAP

   { + } STREAM

≫

#21

function () {     

    function isAmicable(a) {

        function d(n) {

            var divs = calculator.functions.divs(n);

            divs.pop();

            return calculator.functions.vector.total(divs);

        }

        var b = d(a);

        return (b != a && d(b) == a);

    }

    var sum = 0;

    for (var i=2; i<10000; i++)

        if (isAmicable(i))

            sum += i;

    return sum;

}

[ND1: 1.5s; CalcPad/Mac: 0.1s]

#22

≪ #22_data { →STR } MAP

   SORT

   split

   1 { NSUB SWAP

        1 { NUM 64 - } DOSUBS

        total *

   } DOSUBS

   total

≫

[ND1: 22s; CalcPad/Mac: 1.8s]

#22_data

function () { 

    var data = requestURL("http://projecteuler.net/project/names.txt");

    return eval("(" + "[" + data + "]" + ")");

}

#23

function () { /*as is*/

    var F = calculator.functions;

    var abundants = [];

    for (var i=1; i<28123; i++)

        if (F.vector.total(F.divs(i)) > i*2)

            abundants.push(i);

    var count = abundants.length;

    var arr = [];

    for (var i=0; i<count; i++)

        for (var j=0; j<=i; j++)

            arr[abundants[i]+abundants[j]] = true;

    var sum = 0;

    for (var i=0; i<28123+1000; i++)

        if (!arr[i]) sum += i;

    return sum;

}

[ND1: 21s; CalcPad/Mac: 1.7s]

#24

≪ "2" "013456789" permutate 1e6-9!*2 at + fromString ≫

[ND1: 20s; CalcPad/Mac: 1.0s]

#25

Closed-form solution

≪ ceil((1000-1+log(√5))/log((1+√5)/2)) ≫

or (using evaluation of traditional algebraic expression; note; spaces allowed in expression)

≪ 'ceil((1000-1+log(√5)) / log((1+√5)/2))' EVAL≫

or (brute-force; very slow)

≪ 1 =count

   0 1 →big

   DO

      DUP ROT +

      ++count

   UNTIL

      DUP →STR size 1000 >=

   END

   DROP DROP

   count

≫

#28

≪ 1001 → n '(4*n^3+3*n^2+8*n-9)/6'≫

#29

≪ 2 100 FOR a

      2 100 FOR b

         a b ^

      NEXT

   NEXT

   99 99 * fromElements

   removeDuplicates

   size

≫

[ND1: 11.5s; CalcPad/Mac: 1.3s]

#30

≪ 0 =sum

   10 200000 FOR i

      i split DUP SQ SQ * total

      IF i == THEN i =+sum END

   NEXT

   sum

≫

[ND1: 2.5m; CalcPad/Mac: 10.9s]

#33

function () {      

    var r = new Fraction(1);

    for (var i=12; i<=99; i++) {

        var n = split(i);

        for (var j=i+1; j<=99; j++) {

            var d = split(j);

            if (n[1]==d[0] && i/j == n[0]/d[1])

                r=r.multiply(new Fraction(i,j));

        }

    }

    return r;

}

or (in RPL+)

≪ whole =r

    12 99 FOR i

        i split =n

        i 1 + 99 FOR j

            j split =d

            IF n[2] d[1] == THEN 

               IF i j / n[1] d[2] / == THEN

                  i j →fract =*r

               END

            END

         NEXT

   NEXT 

   r

≫

[ND1: 2.3s; CalcPad/Mac: 0.15s]

#34

≪ 0 =sum

   10 50000 FOR i

      i split ! total

      IF i == THEN i =+sum END

   NEXT

   sum

≫

[ND1: 45s; CalcPad/Mac: 3.6s]

#40

function () { /*as is*/  

    var s = "", p = 1;

    for (var i=1; s.length<1e6; i++)

        s += i;

    for (var i=0; i<7; i++)

        p *= s[Math.pow(10, i)-1];

    return p;

}

#41

≪ 123456 permutate rsort =p

   1 p size FOR i

      p[i] 7000000 + =:x isPrime IFTB

   NEXT

   x

≫

#42

≪ #42_data { →STR } MAP

   split

   1 {

        { NUM 64 - } MAP

        total

        8 * incr √ decr 2 / @ inv triangle

        fract not { true } IFT

   } DOSUBS

   size

≫

[ND1: 7.5s; CalcPad/Mac: 0.7s]

#42_data

function () {  

    var data = requestURL("http://projecteuler.net/project/words.txt");

    return eval("(" + "[" + data + "]" + ")");

}

#47

≪ 1 =c false =foundIt

   DO

      c factors size

      IF 8 == THEN c 1 + factors size

      IF 8 == THEN c 2 + factors size

      IF 8 == THEN c 3 + factors size

      IF 8 == THEN true =foundIt END

      END END END

      ++c

   UNTIL foundIt

   END

   c decr

≫

[ND1: 123s; CalcPad/Mac: 9s]

#48

Classic RPL

≪ 1e10 R→I MODSTO

   0

   1 1000 FOR i

      i R→I DUP POWMOD +

   NEXT

   1e10 R→I MOD

≫

[ND1: 3.8s; CalcPad/Mac: 0.3s]

#53

≪ 0 =count

   23 100 FOR n

      3 n 3 - FOR r

         n r COMB

         IF 1e6 > THEN ++count END

      NEXT

   NEXT

   count

≫

[ND1: 5.3s; CalcPad/Mac: 0.4s]

#56

≪ 0 =m

   80 100 FOR a

      80 100 FOR b

         a b toBigNum ^ split total

         m max =m

      NEXT

   NEXT

   m

≫

[ND1: 7.4s; CalcPad/Mac: 0.6s]

#57

≪ 0 =count

    big0 =p0

    big1 =p1

    1 1000 START

        big2 p1 * p0 + =:p @ Pell number

        p1 + =:n @ numerator

        IF size p size > THEN ++count END

        p1 =p0

        p =p1

    NEXT

    count

≫

[ND1: 30s; CalcPad/Mac: 2.5s]

or (in JavaScript; note: same speed)

function () {     

    var sum = 0;

    var p0 = big0, p1 = big1;

    for (var i=0; i<1000; i++) {

        var p = BigNum["+"](BigNum["*"](big2, p1), p0); // Pell number

        var n = BigNum["+"](p, p1); // numerator

        if (String(n).length > String(p).length)

            ++sum;

        p0 = p1; p1 = p;

    }

    return sum;

}

[ND1: 30s; CalcPad/Mac: 2.5s]

#63

≪ 0 =sum

   1 9 FOR i

      'floor(1/(1-log(i)))' EVAL =+sum

   NEXT

   sum

≫

#64

≪ 2..10000

   [toSqrtCF period isEven not] map

   total

≫

[ND1: 5.5s; CalcPad/Mac: 0.4s]

#67

≪ #67_data OBJ→

   2 SWAP START

      2 ≪ MAX ≫ DOSUBS

      ADD

   NEXT

   V→

≫

#67_data

function () {       

   var data = requestURL("http://projecteuler.net/project/triangle.txt").split("");

   data.pop(); // remove empty row

   return data.map(function(x){return x.split(" ").map(function(x){return +x;});});

}

#69

function () {  /*as is*/

    var M = 1000001;

    function phi(x) {

        if(x <= 1) return 1;

        var f = fact[x];

        if(f == 1) return x-1;

        var p = 1, fp = 1;

        while((x /= f) % f == 0) { p++; fp *= f; }

        return phi(x)*(f-1)*fp;

    }

    var fact = [];

    for(var i=0; i<M; i++) fact[i]=1;

    for(var i=2; i<M; i++)

        if(fact[i]==1)

            for(var k=i+i; k<M; k+=i) if(fact[k]==1) fact[k]=i;

    var mn, mx = 0;

    for(var n=2; n<M; n++) {

        var val = n/phi(n);

        if (val > mx) { mx = val; mn = n; }

    }

    return mn;

}

[ND1: 11s; CalcPad/Mac: 0.9s]

#71

≪ 1e6 → n 'int((n-5)/7)*3+2' ≫

#72

function () {  /*as is*/

    var M = 1000001;

    function phi(x) {

        if(x <= 1) return 1;

        var f = fact[x];

        if(f == 1) return x-1;

        var p = 1, fp = 1;

        while((x /= f) % f == 0) { p++; fp *= f; }

        return phi(x)*(f-1)*fp;

    }

    var fact = [];

    for(var i=0; i<M; i++) fact[i]=1;

    for(var i=2; i<M; i++)

        if(fact[i]==1)

            for(var k=i+i; k<M; k+=i) if(fact[k]==1) fact[k]=i;

    var s = 0;

    for(var i=2; i<M; i++) s += phi(i);

    return s;

}

[ND1: 10s; CalcPad/Mac: 0.9s]

or (in RPL+)

≪ 2..1e6 phi total ≫

[ND1: 94s; CalcPad/Mac: 5.1s]

#76

(Requires the Number Theory shared folder, which defines “p”, a function to compute the number of partitions of an integer)

≪ 100 NT.p 1 - ≫

or (using immediate algebraic expression syntax)

≪ NT.p(100)-1 ≫

#77

function () {  

    function part(a, b) {

        if (a == 0)

            return 1;

        else if (a < 0 || b == 0)

            return 0;

        else return (part(a, b-1) +

            part(a-ps[b], b));

    }

    var ps = primes(5000);

    ps.unshift(0);

    var i=1;

    while (part(++i, ps.length-1) <= 5000)

        ;

    return i;

}

[ND1: 2s; CalcPad/Mac: 0.2s]

#80

≪ 1..100 squareFree 

   99 setBigFPrecision toBigF √ split

   [total] MAP

   total

≫

[ND1: 82s; CalcPad/Mac: 7.3s]

or (not using array processing, but a regular loop)

≪ 99 setBigFPrecision

   0 =sum

   1 100 FOR i

      IF i √ fract THEN

         i toBigF √

         split total =+sum

      END

   NEXT

   sum

≫

[ND1: 82s; CalcPad/Mac: 7.3s]

or (using ContinuedFraction)

≪ 99 setBigFPrecision

   0 =sum

   1 100 FOR i

      IF i √ fract THEN

         i toSqrtCF @ convert to CF

         toNumber @ get decimals

         split @ into array of digits

         total =+sum

      END

   NEXT

   sum

≫

[ND1: 82s; CalcPad/Mac: 7.3s]

#87

≪ 50e6 √ primes =p

   p SQ =p2

   p2 p * 75 resize =p3 

   p2 SQ 25 resize =p4

   p2 p3 [+] combine [ 50e6 <= ] filter

   removeDuplicates =p23

   p23 p4 [+] combine [ 50e6 <= ] filter

   removeDuplicates

   size

≫

[ND1: not practical; CalcPad/Mac: 18s]

#97

≪ 2 7830457 1e10 modpow 28433 * 1 + 1e10 mod ≫

#100

≪ 1 =:x =y

   WHILE y<1e12

   REPEAT

      x =xx

      3*x+2*y-2 =x

      4*xx+3*y-3 =y

   END

   x

≫

#102

≪ #102_data

   ≪ V→ →V2 =a →V2 =b →V2 =c

   a b cross last sign =s

   b c cross last sign s ==

   c a cross last sign s == and ≫ MAP

   total

≫

#102_data

function () {           

    var data = requestURL("http://projecteuler.net/project/triangles.txt").split("\r\n");

   data.pop(); // remove empty row

   return data.map(function(x){return x.split(",").map(function(x){return +x;});}

   );

}

#119

≪ []

   2 =a DO

      2 10 FOR b

         a b ^ =:val

         split total a == { val + } IFT

      NEXT

      ++a

   UNTIL DUP SIZE 31 ==

   END

   sort

   30 GET

≫

#120

≪ 0 =sum

   3 1000 FOR a

      a DUP isEven { 2 - } { 1 - } IFTE

      a * =+sum

   NEXT

   sum

≫

or

≪ 0 =sum

   3 1000 FOR a

      a^2-(2-mod(a,2))*a =+sum

   NEXT

   sum

≫

#123

≪ 300e3 primes DUP ≪ =p

      2*N*p>1e10

   ≫ 7038 =offset find pos ++

≫

[ND1: 7.5s; CalcPad/Mac: 0.5s]

#124

≪ 1..100000 DUP

   radical SWAP

   2 [ @tag ] DOLIST

   sort

   10000 at tag

≫

[ND1: ~31m; CalcPad/Mac: 126s]

#131

≪ 1e6 =top

   0 =sum

   1 top FOR n

      3*n*n+3*n+1 =:n top >= IFTB

      n isPrime =+sum

   NEXT

   sum

≫

#188

≪ 8 =nDigits

   1777 toBig =base

   1855 =expo

   10 toBig nDigits pow =m

   base

   1 expo-1 START

      base @swap m modpow

   NEXT

≫

[ND1: 14.5s; CalcPad/Mac: 1.1s]

Most solutions have been developed in RPL+. This guide is recommended if you want to pick up the language.

For several problems, solutions in more than one language, or taking different approaches, are shown. Run-times are reported for each, so you can see how they compare speed-wise. Note that a variety of styles are explored in these solutions. Many more variations are possible. (RPL+ can look like RPN or algebraic notation, for example.)

Run-times for CalcPad/Mac, an upcoming product that will allow WiFi folder sharing with ND1, are also included. This is to give you an idea of how MorphEngine performs on a desktop system.

The code is available for download into ND1: tap Project Euler under My Data | Downloadables.