Fraction reduction
You are encouraged to solve this task according to the task description, using any language you may know.
There is a fine line between numerator and denominator. ─── anonymous
A method to "reduce" some reducible fractions is to cross out a digit from the numerator and the denominator. An example is:
16 16──── and then (simply) cross─out the sixes: ──── 6464
resulting in:
1 ─── 4
Naturally, this "method" of reduction must reduce to the proper value (shown as a fraction).
This "method" is also known as anomalous cancellation and also accidental cancellation.
(Of course, this "method" shouldn't be taught to impressionable or gullible minds.) 😇
 Task
Find and show some fractions that can be reduced by the above "method".
 show 2digit fractions found (like the example shown above)
 show 3digit fractions
 show 4digit fractions
 show 5digit fractions (and higher) (optional)
 show each (above) ndigit fractions separately from other different nsized fractions, don't mix different "sizes" together
 for each "size" fraction, only show a dozen examples (the 1^{st} twelve found)
 (it's recognized that not every programming solution will have the same generation algorithm)
 for each "size" fraction:
 show a count of how many reducible fractions were found. The example (above) is size 2
 show a count of which digits were crossed out (one line for each different digit)
 for each "size" fraction, show a count of how many were found. The example (above) is size 2
 show each ndigit example (to be shown on one line):
 show each ndigit fraction
 show each reduced ndigit fraction
 show what digit was crossed out for the numerator and the denominator
 Task requirements/restrictions

 only proper fractions and their reductions (the result) are to be used (no vulgar fractions)
 only positive fractions are to be used (no negative signs anywhere)
 only base ten integers are to be used for the numerator and denominator
 no zeros (decimal digit) can be used within the numerator or the denominator
 the numerator and denominator should be composed of the same number of digits
 no digit can be repeated in the numerator
 no digit can be repeated in the denominator
 (naturally) there should be a shared decimal digit in the numerator and the denominator
 fractions can be shown as 16/64 (for example)
Show all output here, on this page.
 Somewhat related task

 Farey sequence (It concerns fractions.)
 References

 Wikipedia entry: proper and improper fractions.
 Wikipedia entry: anomalous cancellation and/or accidental cancellation.
Contents
Ada[edit]
with Ada.Integer_Text_IO; use Ada.Integer_Text_IO;
with Ada.Text_IO; use Ada.Text_IO;
procedure Fraction_Reduction is
type Int_Array is array (Natural range <>) of Integer;
function indexOf(haystack : Int_Array; needle : Integer) return Integer is
idx : Integer := 0;
begin
for straw of haystack loop
if straw = needle then
return idx;
else
idx := idx + 1;
end if;
end loop;
return 1;
end IndexOf;
function getDigits(n, le : in Integer;
digit_array : in out Int_Array) return Boolean is
n_local : Integer := n;
le_local : Integer := le;
r : Integer;
begin
while n_local > 0 loop
r := n_local mod 10;
if r = 0 or indexOf(digit_array, r) >= 0 then
return False;
end if;
le_local := le_local  1;
digit_array(le_local) := r;
n_local := n_local / 10;
end loop;
return True;
end getDigits;
function removeDigit(digit_array : Int_Array;
le, idx : Integer) return Integer is
sum : Integer := 0;
pow : Integer := 10 ** (le  2);
begin
for i in 0 .. le  1 loop
if i /= idx then
sum := sum + digit_array(i) * pow;
pow := pow / 10;
end if;
end loop;
return sum;
end removeDigit;
lims : constant array (0 .. 3) of Int_Array (0 .. 1) :=
((12, 97), (123, 986), (1234, 9875), (12345, 98764));
count : Int_Array (0 .. 4) := (others => 0);
omitted : array (0 .. 4) of Int_Array (0 .. 9) :=
(others => (others => 0));
begin
Ada.Integer_Text_IO.Default_Width := 0;
for i in lims'Range loop
declare
nDigits, dDigits : Int_Array (0 .. i + 1);
digit, dix, rn, rd : Integer;
begin
for n in lims(i)(0) .. lims(i)(1) loop
nDigits := (others => 0);
if getDigits(n, i + 2, nDigits) then
for d in n + 1 .. lims(i)(1) + 1 loop
dDigits := (others => 0);
if getDigits(d, i + 2, dDigits) then
for nix in nDigits'Range loop
digit := nDigits(nix);
dix := indexOf(dDigits, digit);
if dix >= 0 then
rn := removeDigit(nDigits, i + 2, nix);
rd := removeDigit(dDigits, i + 2, dix);
 'n/d = rn/rd' is same as 'n*rd = rn*d'
if n*rd = rn*d then
count(i) := count(i) + 1;
omitted(i)(digit) :=
omitted(i)(digit) + 1;
if count(i) <= 12 then
Put (n);
Put ("/");
Put (d);
Put (" = ");
Put (rn);
Put ("/");
Put (rd);
Put (" by omitting ");
Put (digit);
Put_Line ("'s");
end if;
end if;
end if;
end loop;
end if;
end loop;
end if;
end loop;
end;
New_Line;
end loop;
for i in 2 .. 5 loop
Put ("There are ");
Put (count(i  2));
Put (" ");
Put (i);
Put_Line ("digit fractions of which:");
for j in 1 .. 9 loop
if omitted(i  2)(j) /= 0 then
Put (omitted(i  2)(j), Width => 6);
Put (" have ");
Put (j);
Put_Line ("'s omitted");
end if;
end loop;
New_Line;
end loop;
end Fraction_Reduction;
C[edit]
#include <stdbool.h>
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
typedef struct IntArray_t {
int *ptr;
size_t length;
} IntArray;
IntArray make(size_t size) {
IntArray temp;
temp.ptr = calloc(size, sizeof(int));
temp.length = size;
return temp;
}
void destroy(IntArray *ia) {
if (ia>ptr != NULL) {
free(ia>ptr);
ia>ptr = NULL;
ia>length = 0;
}
}
void zeroFill(IntArray dst) {
memset(dst.ptr, 0, dst.length * sizeof(int));
}
int indexOf(const int n, const IntArray ia) {
size_t i;
for (i = 0; i < ia.length; i++) {
if (ia.ptr[i] == n) {
return i;
}
}
return 1;
}
bool getDigits(int n, int le, IntArray digits) {
while (n > 0) {
int r = n % 10;
if (r == 0  indexOf(r, digits) >= 0) {
return false;
}
le;
digits.ptr[le] = r;
n /= 10;
}
return true;
}
int removeDigit(IntArray digits, size_t le, size_t idx) {
static const int POWS[] = { 1, 10, 100, 1000, 10000 };
int sum = 0;
int pow = POWS[le  2];
size_t i;
for (i = 0; i < le; i++) {
if (i == idx) continue;
sum += digits.ptr[i] * pow;
pow /= 10;
}
return sum;
}
int main() {
int lims[4][2] = { { 12, 97 }, { 123, 986 }, { 1234, 9875 }, { 12345, 98764 } };
int count[5] = { 0 };
int omitted[5][10] = { {0} };
size_t upperBound = sizeof(lims) / sizeof(lims[0]);
size_t i;
for (i = 0; i < upperBound; i++) {
IntArray nDigits = make(i + 2);
IntArray dDigits = make(i + 2);
int n;
for (n = lims[i][0]; n <= lims[i][1]; n++) {
int d;
bool nOk;
zeroFill(nDigits);
nOk = getDigits(n, i + 2, nDigits);
if (!nOk) {
continue;
}
for (d = n + 1; d <= lims[i][1] + 1; d++) {
size_t nix;
bool dOk;
zeroFill(dDigits);
dOk = getDigits(d, i + 2, dDigits);
if (!dOk) {
continue;
}
for (nix = 0; nix < nDigits.length; nix++) {
int digit = nDigits.ptr[nix];
int dix = indexOf(digit, dDigits);
if (dix >= 0) {
int rn = removeDigit(nDigits, i + 2, nix);
int rd = removeDigit(dDigits, i + 2, dix);
if ((double)n / d == (double)rn / rd) {
count[i]++;
omitted[i][digit]++;
if (count[i] <= 12) {
printf("%d/%d = %d/%d by omitting %d's\n", n, d, rn, rd, digit);
}
}
}
}
}
}
printf("\n");
destroy(&nDigits);
destroy(&dDigits);
}
for (i = 2; i <= 5; i++) {
int j;
printf("There are %d %ddigit fractions of which:\n", count[i  2], i);
for (j = 1; j <= 9; j++) {
if (omitted[i  2][j] == 0) {
continue;
}
printf("%6d have %d's omitted\n", omitted[i  2][j], j);
}
printf("\n");
}
return 0;
}
 Output:
16/64 = 1/4 by omitting 6's 19/95 = 1/5 by omitting 9's 26/65 = 2/5 by omitting 6's 49/98 = 4/8 by omitting 9's 132/231 = 12/21 by omitting 3's 134/536 = 14/56 by omitting 3's 134/938 = 14/98 by omitting 3's 136/238 = 16/28 by omitting 3's 138/345 = 18/45 by omitting 3's 139/695 = 13/65 by omitting 9's 143/341 = 13/31 by omitting 4's 146/365 = 14/35 by omitting 6's 149/298 = 14/28 by omitting 9's 149/596 = 14/56 by omitting 9's 149/894 = 14/84 by omitting 9's 154/253 = 14/23 by omitting 5's 1234/4936 = 124/496 by omitting 3's 1239/6195 = 123/615 by omitting 9's 1246/3649 = 126/369 by omitting 4's 1249/2498 = 124/248 by omitting 9's 1259/6295 = 125/625 by omitting 9's 1279/6395 = 127/635 by omitting 9's 1283/5132 = 128/512 by omitting 3's 1297/2594 = 127/254 by omitting 9's 1297/3891 = 127/381 by omitting 9's 1298/2596 = 128/256 by omitting 9's 1298/3894 = 128/384 by omitting 9's 1298/5192 = 128/512 by omitting 9's 12349/24698 = 1234/2468 by omitting 9's 12356/67958 = 1236/6798 by omitting 5's 12358/14362 = 1258/1462 by omitting 3's 12358/15364 = 1258/1564 by omitting 3's 12358/17368 = 1258/1768 by omitting 3's 12358/19372 = 1258/1972 by omitting 3's 12358/21376 = 1258/2176 by omitting 3's 12358/25384 = 1258/2584 by omitting 3's 12359/61795 = 1235/6175 by omitting 9's 12364/32596 = 1364/3596 by omitting 2's 12379/61895 = 1237/6185 by omitting 9's 12386/32654 = 1386/3654 by omitting 2's There are 4 2digit fractions of which: 2 have 6's omitted 2 have 9's omitted There are 122 3digit fractions of which: 9 have 3's omitted 1 have 4's omitted 6 have 5's omitted 15 have 6's omitted 16 have 7's omitted 15 have 8's omitted 60 have 9's omitted There are 660 4digit fractions of which: 14 have 1's omitted 25 have 2's omitted 92 have 3's omitted 14 have 4's omitted 29 have 5's omitted 63 have 6's omitted 16 have 7's omitted 17 have 8's omitted 390 have 9's omitted There are 5087 5digit fractions of which: 75 have 1's omitted 40 have 2's omitted 376 have 3's omitted 78 have 4's omitted 209 have 5's omitted 379 have 6's omitted 591 have 7's omitted 351 have 8's omitted 2988 have 9's omitted
C#[edit]
using System;
namespace FractionReduction {
class Program {
static int IndexOf(int n, int[] s) {
for (int i = 0; i < s.Length; i++) {
if (s[i] == n) {
return i;
}
}
return 1;
}
static bool GetDigits(int n, int le, int[] digits) {
while (n > 0) {
var r = n % 10;
if (r == 0  IndexOf(r, digits) >= 0) {
return false;
}
le;
digits[le] = r;
n /= 10;
}
return true;
}
static int RemoveDigit(int[] digits, int le, int idx) {
int[] pows = { 1, 10, 100, 1000, 10000 };
var sum = 0;
var pow = pows[le  2];
for (int i = 0; i < le; i++) {
if (i == idx) continue;
sum += digits[i] * pow;
pow /= 10;
}
return sum;
}
static void Main() {
var lims = new int[,] { { 12, 97 }, { 123, 986 }, { 1234, 9875 }, { 12345, 98764 } };
var count = new int[5];
var omitted = new int[5, 10];
var upperBound = lims.GetLength(0);
for (int i = 0; i < upperBound; i++) {
var nDigits = new int[i + 2];
var dDigits = new int[i + 2];
var blank = new int[i + 2];
for (int n = lims[i, 0]; n <= lims[i, 1]; n++) {
blank.CopyTo(nDigits, 0);
var nOk = GetDigits(n, i + 2, nDigits);
if (!nOk) {
continue;
}
for (int d = n + 1; d <= lims[i, 1] + 1; d++) {
blank.CopyTo(dDigits, 0);
var dOk = GetDigits(d, i + 2, dDigits);
if (!dOk) {
continue;
}
for (int nix = 0; nix < nDigits.Length; nix++) {
var digit = nDigits[nix];
var dix = IndexOf(digit, dDigits);
if (dix >= 0) {
var rn = RemoveDigit(nDigits, i + 2, nix);
var rd = RemoveDigit(dDigits, i + 2, dix);
if ((double)n / d == (double)rn / rd) {
count[i]++;
omitted[i, digit]++;
if (count[i] <= 12) {
Console.WriteLine("{0}/{1} = {2}/{3} by omitting {4}'s", n, d, rn, rd, digit);
}
}
}
}
}
}
Console.WriteLine();
}
for (int i = 2; i <= 5; i++) {
Console.WriteLine("There are {0} {1}digit fractions of which:", count[i  2], i);
for (int j = 1; j <= 9; j++) {
if (omitted[i  2, j] == 0) {
continue;
}
Console.WriteLine("{0,6} have {1}'s omitted", omitted[i  2, j], j);
}
Console.WriteLine();
}
}
}
}
 Output:
16/64 = 1/4 by omitting 6's 19/95 = 1/5 by omitting 9's 26/65 = 2/5 by omitting 6's 49/98 = 4/8 by omitting 9's 132/231 = 12/21 by omitting 3's 134/536 = 14/56 by omitting 3's 134/938 = 14/98 by omitting 3's 136/238 = 16/28 by omitting 3's 138/345 = 18/45 by omitting 3's 139/695 = 13/65 by omitting 9's 143/341 = 13/31 by omitting 4's 146/365 = 14/35 by omitting 6's 149/298 = 14/28 by omitting 9's 149/596 = 14/56 by omitting 9's 149/894 = 14/84 by omitting 9's 154/253 = 14/23 by omitting 5's 1234/4936 = 124/496 by omitting 3's 1239/6195 = 123/615 by omitting 9's 1246/3649 = 126/369 by omitting 4's 1249/2498 = 124/248 by omitting 9's 1259/6295 = 125/625 by omitting 9's 1279/6395 = 127/635 by omitting 9's 1283/5132 = 128/512 by omitting 3's 1297/2594 = 127/254 by omitting 9's 1297/3891 = 127/381 by omitting 9's 1298/2596 = 128/256 by omitting 9's 1298/3894 = 128/384 by omitting 9's 1298/5192 = 128/512 by omitting 9's 12349/24698 = 1234/2468 by omitting 9's 12356/67958 = 1236/6798 by omitting 5's 12358/14362 = 1258/1462 by omitting 3's 12358/15364 = 1258/1564 by omitting 3's 12358/17368 = 1258/1768 by omitting 3's 12358/19372 = 1258/1972 by omitting 3's 12358/21376 = 1258/2176 by omitting 3's 12358/25384 = 1258/2584 by omitting 3's 12359/61795 = 1235/6175 by omitting 9's 12364/32596 = 1364/3596 by omitting 2's 12379/61895 = 1237/6185 by omitting 9's 12386/32654 = 1386/3654 by omitting 2's There are 4 2digit fractions of which: 2 have 6's omitted 2 have 9's omitted There are 122 3digit fractions of which: 9 have 3's omitted 1 have 4's omitted 6 have 5's omitted 15 have 6's omitted 16 have 7's omitted 15 have 8's omitted 60 have 9's omitted There are 660 4digit fractions of which: 14 have 1's omitted 25 have 2's omitted 92 have 3's omitted 14 have 4's omitted 29 have 5's omitted 63 have 6's omitted 16 have 7's omitted 17 have 8's omitted 390 have 9's omitted There are 5087 5digit fractions of which: 75 have 1's omitted 40 have 2's omitted 376 have 3's omitted 78 have 4's omitted 209 have 5's omitted 379 have 6's omitted 591 have 7's omitted 351 have 8's omitted 2988 have 9's omitted
C++[edit]
#include <array>
#include <iomanip>
#include <iostream>
#include <vector>
int indexOf(const std::vector<int> &haystack, int needle) {
auto it = haystack.cbegin();
auto end = haystack.cend();
int idx = 0;
for (; it != end; it = std::next(it)) {
if (*it == needle) {
return idx;
}
idx++;
}
return 1;
}
bool getDigits(int n, int le, std::vector<int> &digits) {
while (n > 0) {
auto r = n % 10;
if (r == 0  indexOf(digits, r) >= 0) {
return false;
}
le;
digits[le] = r;
n /= 10;
}
return true;
}
int removeDigit(const std::vector<int> &digits, int le, int idx) {
static std::array<int, 5> pows = { 1, 10, 100, 1000, 10000 };
int sum = 0;
auto pow = pows[le  2];
for (int i = 0; i < le; i++) {
if (i == idx) continue;
sum += digits[i] * pow;
pow /= 10;
}
return sum;
}
int main() {
std::vector<std::pair<int, int>> lims = { {12, 97}, {123, 986}, {1234, 9875}, {12345, 98764} };
std::array<int, 5> count;
std::array<std::array<int, 10>, 5> omitted;
std::fill(count.begin(), count.end(), 0);
std::for_each(omitted.begin(), omitted.end(),
[](auto &a) {
std::fill(a.begin(), a.end(), 0);
}
);
for (size_t i = 0; i < lims.size(); i++) {
std::vector<int> nDigits(i + 2);
std::vector<int> dDigits(i + 2);
for (int n = lims[i].first; n <= lims[i].second; n++) {
std::fill(nDigits.begin(), nDigits.end(), 0);
bool nOk = getDigits(n, i + 2, nDigits);
if (!nOk) {
continue;
}
for (int d = n + 1; d <= lims[i].second + 1; d++) {
std::fill(dDigits.begin(), dDigits.end(), 0);
bool dOk = getDigits(d, i + 2, dDigits);
if (!dOk) {
continue;
}
for (size_t nix = 0; nix < nDigits.size(); nix++) {
auto digit = nDigits[nix];
auto dix = indexOf(dDigits, digit);
if (dix >= 0) {
auto rn = removeDigit(nDigits, i + 2, nix);
auto rd = removeDigit(dDigits, i + 2, dix);
if ((double)n / d == (double)rn / rd) {
count[i]++;
omitted[i][digit]++;
if (count[i] <= 12) {
std::cout << n << '/' << d << " = " << rn << '/' << rd << " by omitting " << digit << "'s\n";
}
}
}
}
}
}
std::cout << '\n';
}
for (int i = 2; i <= 5; i++) {
std::cout << "There are " << count[i  2] << ' ' << i << "digit fractions of which:\n";
for (int j = 1; j <= 9; j++) {
if (omitted[i  2][j] == 0) {
continue;
}
std::cout << std::setw(6) << omitted[i  2][j] << " have " << j << "'s omitted\n";
}
std::cout << '\n';
}
return 0;
}
 Output:
16/64 = 1/4 by omitting 6's 19/95 = 1/5 by omitting 9's 26/65 = 2/5 by omitting 6's 49/98 = 4/8 by omitting 9's 132/231 = 12/21 by omitting 3's 134/536 = 14/56 by omitting 3's 134/938 = 14/98 by omitting 3's 136/238 = 16/28 by omitting 3's 138/345 = 18/45 by omitting 3's 139/695 = 13/65 by omitting 9's 143/341 = 13/31 by omitting 4's 146/365 = 14/35 by omitting 6's 149/298 = 14/28 by omitting 9's 149/596 = 14/56 by omitting 9's 149/894 = 14/84 by omitting 9's 154/253 = 14/23 by omitting 5's 1234/4936 = 124/496 by omitting 3's 1239/6195 = 123/615 by omitting 9's 1246/3649 = 126/369 by omitting 4's 1249/2498 = 124/248 by omitting 9's 1259/6295 = 125/625 by omitting 9's 1279/6395 = 127/635 by omitting 9's 1283/5132 = 128/512 by omitting 3's 1297/2594 = 127/254 by omitting 9's 1297/3891 = 127/381 by omitting 9's 1298/2596 = 128/256 by omitting 9's 1298/3894 = 128/384 by omitting 9's 1298/5192 = 128/512 by omitting 9's 12349/24698 = 1234/2468 by omitting 9's 12356/67958 = 1236/6798 by omitting 5's 12358/14362 = 1258/1462 by omitting 3's 12358/15364 = 1258/1564 by omitting 3's 12358/17368 = 1258/1768 by omitting 3's 12358/19372 = 1258/1972 by omitting 3's 12358/21376 = 1258/2176 by omitting 3's 12358/25384 = 1258/2584 by omitting 3's 12359/61795 = 1235/6175 by omitting 9's 12364/32596 = 1364/3596 by omitting 2's 12379/61895 = 1237/6185 by omitting 9's 12386/32654 = 1386/3654 by omitting 2's There are 4 2digit fractions of which: 2 have 6's omitted 2 have 9's omitted There are 122 3digit fractions of which: 9 have 3's omitted 1 have 4's omitted 6 have 5's omitted 15 have 6's omitted 16 have 7's omitted 15 have 8's omitted 60 have 9's omitted There are 660 4digit fractions of which: 14 have 1's omitted 25 have 2's omitted 92 have 3's omitted 14 have 4's omitted 29 have 5's omitted 63 have 6's omitted 16 have 7's omitted 17 have 8's omitted 390 have 9's omitted There are 5087 5digit fractions of which: 75 have 1's omitted 40 have 2's omitted 376 have 3's omitted 78 have 4's omitted 209 have 5's omitted 379 have 6's omitted 591 have 7's omitted 351 have 8's omitted 2988 have 9's omitted
D[edit]
import std.range;
import std.stdio;
int indexOf(Range, Element)(Range haystack, scope Element needle)
if (isInputRange!Range) {
int idx;
foreach (straw; haystack) {
if (straw == needle) {
return idx;
}
idx++;
}
return 1;
}
bool getDigits(int n, int le, int[] digits) {
while (n > 0) {
auto r = n % 10;
if (r == 0  indexOf(digits, r) >= 0) {
return false;
}
le;
digits[le] = r;
n /= 10;
}
return true;
}
int removeDigit(int[] digits, int le, int idx) {
enum pows = [ 1, 10, 100, 1_000, 10_000 ];
int sum = 0;
auto pow = pows[le  2];
for (int i = 0; i < le; i++) {
if (i == idx) continue;
sum += digits[i] * pow;
pow /= 10;
}
return sum;
}
void main() {
auto lims = [ [ 12, 97 ], [ 123, 986 ], [ 1234, 9875 ], [ 12345, 98764 ] ];
int[5] count;
int[10][5] omitted;
for (int i = 0; i < lims.length; i++) {
auto nDigits = new int[i + 2];
auto dDigits = new int[i + 2];
for (int n = lims[i][0]; n <= lims[i][1]; n++) {
nDigits[] = 0;
bool nOk = getDigits(n, i + 2, nDigits);
if (!nOk) {
continue;
}
for (int d = n + 1; d <= lims[i][1] + 1; d++) {
dDigits[] = 0;
bool dOk = getDigits(d, i + 2, dDigits);
if (!dOk) {
continue;
}
for (int nix = 0; nix < nDigits.length; nix++) {
auto digit = nDigits[nix];
auto dix = indexOf(dDigits, digit);
if (dix >= 0) {
auto rn = removeDigit(nDigits, i + 2, nix);
auto rd = removeDigit(dDigits, i + 2, dix);
if (cast(double)n / d == cast(double)rn / rd) {
count[i]++;
omitted[i][digit]++;
if (count[i] <= 12) {
writefln("%d/%d = %d/%d by omitting %d's", n, d, rn, rd, digit);
}
}
}
}
}
}
writeln;
}
for (int i = 2; i <= 5; i++) {
writefln("There are %d %ddigit fractions of which:", count[i  2], i);
for (int j = 1; j <= 9; j++) {
if (omitted[i  2][j] == 0) {
continue;
}
writefln("%6s have %d's omitted", omitted[i  2][j], j);
}
writeln;
}
}
 Output:
16/64 = 1/4 by omitting 6's 19/95 = 1/5 by omitting 9's 26/65 = 2/5 by omitting 6's 49/98 = 4/8 by omitting 9's 132/231 = 12/21 by omitting 3's 134/536 = 14/56 by omitting 3's 134/938 = 14/98 by omitting 3's 136/238 = 16/28 by omitting 3's 138/345 = 18/45 by omitting 3's 139/695 = 13/65 by omitting 9's 143/341 = 13/31 by omitting 4's 146/365 = 14/35 by omitting 6's 149/298 = 14/28 by omitting 9's 149/596 = 14/56 by omitting 9's 149/894 = 14/84 by omitting 9's 154/253 = 14/23 by omitting 5's 1234/4936 = 124/496 by omitting 3's 1239/6195 = 123/615 by omitting 9's 1246/3649 = 126/369 by omitting 4's 1249/2498 = 124/248 by omitting 9's 1259/6295 = 125/625 by omitting 9's 1279/6395 = 127/635 by omitting 9's 1283/5132 = 128/512 by omitting 3's 1297/2594 = 127/254 by omitting 9's 1297/3891 = 127/381 by omitting 9's 1298/2596 = 128/256 by omitting 9's 1298/3894 = 128/384 by omitting 9's 1298/5192 = 128/512 by omitting 9's 12349/24698 = 1234/2468 by omitting 9's 12356/67958 = 1236/6798 by omitting 5's 12358/14362 = 1258/1462 by omitting 3's 12358/15364 = 1258/1564 by omitting 3's 12358/17368 = 1258/1768 by omitting 3's 12358/19372 = 1258/1972 by omitting 3's 12358/21376 = 1258/2176 by omitting 3's 12358/25384 = 1258/2584 by omitting 3's 12359/61795 = 1235/6175 by omitting 9's 12364/32596 = 1364/3596 by omitting 2's 12379/61895 = 1237/6185 by omitting 9's 12386/32654 = 1386/3654 by omitting 2's There are 4 2digit fractions of which: 2 have 6's omitted 2 have 9's omitted There are 122 3digit fractions of which: 9 have 3's omitted 1 have 4's omitted 6 have 5's omitted 15 have 6's omitted 16 have 7's omitted 15 have 8's omitted 60 have 9's omitted There are 660 4digit fractions of which: 14 have 1's omitted 25 have 2's omitted 92 have 3's omitted 14 have 4's omitted 29 have 5's omitted 63 have 6's omitted 16 have 7's omitted 17 have 8's omitted 390 have 9's omitted There are 5087 5digit fractions of which: 75 have 1's omitted 40 have 2's omitted 376 have 3's omitted 78 have 4's omitted 209 have 5's omitted 379 have 6's omitted 591 have 7's omitted 351 have 8's omitted 2988 have 9's omitted
Go[edit]
Version 1[edit]
This produces the stats for 5digit fractions in less than 25 seconds but takes a much longer 15.5 minutes to process the 6digit case. Timings are for an Intel Core i78565U machine.
package main
import (
"fmt"
"time"
)
func indexOf(n int, s []int) int {
for i, j := range s {
if n == j {
return i
}
}
return 1
}
func getDigits(n, le int, digits []int) bool {
for n > 0 {
r := n % 10
if r == 0  indexOf(r, digits) >= 0 {
return false
}
le
digits[le] = r
n /= 10
}
return true
}
var pows = [5]int{1, 10, 100, 1000, 10000}
func removeDigit(digits []int, le, idx int) int {
sum := 0
pow := pows[le2]
for i := 0; i < le; i++ {
if i == idx {
continue
}
sum += digits[i] * pow
pow /= 10
}
return sum
}
func main() {
start := time.Now()
lims := [5][2]int{
{12, 97},
{123, 986},
{1234, 9875},
{12345, 98764},
{123456, 987653},
}
var count [5]int
var omitted [5][10]int
for i, lim := range lims {
nDigits := make([]int, i+2)
dDigits := make([]int, i+2)
blank := make([]int, i+2)
for n := lim[0]; n <= lim[1]; n++ {
copy(nDigits, blank)
nOk := getDigits(n, i+2, nDigits)
if !nOk {
continue
}
for d := n + 1; d <= lim[1]+1; d++ {
copy(dDigits, blank)
dOk := getDigits(d, i+2, dDigits)
if !dOk {
continue
}
for nix, digit := range nDigits {
if dix := indexOf(digit, dDigits); dix >= 0 {
rn := removeDigit(nDigits, i+2, nix)
rd := removeDigit(dDigits, i+2, dix)
if float64(n)/float64(d) == float64(rn)/float64(rd) {
count[i]++
omitted[i][digit]++
if count[i] <= 12 {
fmt.Printf("%d/%d = %d/%d by omitting %d's\n", n, d, rn, rd, digit)
}
}
}
}
}
}
fmt.Println()
}
for i := 2; i <= 6; i++ {
fmt.Printf("There are %d %ddigit fractions of which:\n", count[i2], i)
for j := 1; j <= 9; j++ {
if omitted[i2][j] == 0 {
continue
}
fmt.Printf("%6d have %d's omitted\n", omitted[i2][j], j)
}
fmt.Println()
}
fmt.Printf("Took %s\n", time.Since(start))
}
 Output:
16/64 = 1/4 by omitting 6's 19/95 = 1/5 by omitting 9's 26/65 = 2/5 by omitting 6's 49/98 = 4/8 by omitting 9's 132/231 = 12/21 by omitting 3's 134/536 = 14/56 by omitting 3's 134/938 = 14/98 by omitting 3's 136/238 = 16/28 by omitting 3's 138/345 = 18/45 by omitting 3's 139/695 = 13/65 by omitting 9's 143/341 = 13/31 by omitting 4's 146/365 = 14/35 by omitting 6's 149/298 = 14/28 by omitting 9's 149/596 = 14/56 by omitting 9's 149/894 = 14/84 by omitting 9's 154/253 = 14/23 by omitting 5's 1234/4936 = 124/496 by omitting 3's 1239/6195 = 123/615 by omitting 9's 1246/3649 = 126/369 by omitting 4's 1249/2498 = 124/248 by omitting 9's 1259/6295 = 125/625 by omitting 9's 1279/6395 = 127/635 by omitting 9's 1283/5132 = 128/512 by omitting 3's 1297/2594 = 127/254 by omitting 9's 1297/3891 = 127/381 by omitting 9's 1298/2596 = 128/256 by omitting 9's 1298/3894 = 128/384 by omitting 9's 1298/5192 = 128/512 by omitting 9's 12349/24698 = 1234/2468 by omitting 9's 12356/67958 = 1236/6798 by omitting 5's 12358/14362 = 1258/1462 by omitting 3's 12358/15364 = 1258/1564 by omitting 3's 12358/17368 = 1258/1768 by omitting 3's 12358/19372 = 1258/1972 by omitting 3's 12358/21376 = 1258/2176 by omitting 3's 12358/25384 = 1258/2584 by omitting 3's 12359/61795 = 1235/6175 by omitting 9's 12364/32596 = 1364/3596 by omitting 2's 12379/61895 = 1237/6185 by omitting 9's 12386/32654 = 1386/3654 by omitting 2's 123459/617295 = 12345/61725 by omitting 9's 123468/493872 = 12468/49872 by omitting 3's 123469/173524 = 12469/17524 by omitting 3's 123469/193546 = 12469/19546 by omitting 3's 123469/213568 = 12469/21568 by omitting 3's 123469/283645 = 12469/28645 by omitting 3's 123469/493876 = 12469/49876 by omitting 3's 123469/573964 = 12469/57964 by omitting 3's 123479/617395 = 12347/61735 by omitting 9's 123495/172893 = 12345/17283 by omitting 9's 123548/679514 = 12348/67914 by omitting 5's 123574/325786 = 13574/35786 by omitting 2's There are 4 2digit fractions of which: 2 have 6's omitted 2 have 9's omitted There are 122 3digit fractions of which: 9 have 3's omitted 1 have 4's omitted 6 have 5's omitted 15 have 6's omitted 16 have 7's omitted 15 have 8's omitted 60 have 9's omitted There are 660 4digit fractions of which: 14 have 1's omitted 25 have 2's omitted 92 have 3's omitted 14 have 4's omitted 29 have 5's omitted 63 have 6's omitted 16 have 7's omitted 17 have 8's omitted 390 have 9's omitted There are 5087 5digit fractions of which: 75 have 1's omitted 40 have 2's omitted 376 have 3's omitted 78 have 4's omitted 209 have 5's omitted 379 have 6's omitted 591 have 7's omitted 351 have 8's omitted 2988 have 9's omitted There are 9778 6digit fractions of which: 230 have 1's omitted 256 have 2's omitted 921 have 3's omitted 186 have 4's omitted 317 have 5's omitted 751 have 6's omitted 262 have 7's omitted 205 have 8's omitted 6650 have 9's omitted Took 15m38.231915709s
Version 2[edit]
Rather than iterate through all numbers in the ndigit range and check if they contain unique nonzero digits, this generates all such numbers to start with which turns out to be a much more efficient approach  more than 20 times faster than before.
package main
import (
"fmt"
"time"
)
type result struct {
n int
nine [9]int
}
func indexOf(n int, s []int) int {
for i, j := range s {
if n == j {
return i
}
}
return 1
}
func bIndexOf(b bool, s []bool) int {
for i, j := range s {
if b == j {
return i
}
}
return 1
}
func toNumber(digits []int, removeDigit int) int {
digits2 := digits
if removeDigit != 0 {
digits2 = make([]int, len(digits))
copy(digits2, digits)
d := indexOf(removeDigit, digits2)
copy(digits2[d:], digits2[d+1:])
digits2[len(digits2)1] = 0
digits2 = digits2[:len(digits2)1]
}
res := digits2[0]
for i := 1; i < len(digits2); i++ {
res = res*10 + digits2[i]
}
return res
}
func nDigits(n int) []result {
var res []result
digits := make([]int, n)
var used [9]bool
for i := 0; i < n; i++ {
digits[i] = i + 1
used[i] = true
}
for {
var nine [9]int
for i := 0; i < len(used); i++ {
if used[i] {
nine[i] = toNumber(digits, i+1)
}
}
res = append(res, result{toNumber(digits, 0), nine})
found := false
for i := n  1; i >= 0; i {
d := digits[i]
if !used[d1] {
panic("something went wrong with 'used' array")
}
used[d1] = false
for j := d; j < 9; j++ {
if !used[j] {
used[j] = true
digits[i] = j + 1
for k := i + 1; k < n; k++ {
digits[k] = bIndexOf(false, used[:]) + 1
used[digits[k]1] = true
}
found = true
break
}
}
if found {
break
}
}
if !found {
break
}
}
return res
}
func main() {
start := time.Now()
for n := 2; n <= 5; n++ {
rs := nDigits(n)
count := 0
var omitted [9]int
for i := 0; i < len(rs)1; i++ {
xn, rn := rs[i].n, rs[i].nine
for j := i + 1; j < len(rs); j++ {
xd, rd := rs[j].n, rs[j].nine
for k := 0; k < 9; k++ {
yn, yd := rn[k], rd[k]
if yn != 0 && yd != 0 &&
float64(xn)/float64(xd) == float64(yn)/float64(yd) {
count++
omitted[k]++
if count <= 12 {
fmt.Printf("%d/%d => %d/%d (removed %d)\n", xn, xd, yn, yd, k+1)
}
}
}
}
}
fmt.Printf("%ddigit fractions found:%d, omitted %v\n\n", n, count, omitted)
}
fmt.Printf("Took %s\n", time.Since(start))
}
 Output:
16/64 => 1/4 (removed 6) 19/95 => 1/5 (removed 9) 26/65 => 2/5 (removed 6) 49/98 => 4/8 (removed 9) 2digit fractions found:4, omitted [0 0 0 0 0 2 0 0 2] 132/231 => 12/21 (removed 3) 134/536 => 14/56 (removed 3) 134/938 => 14/98 (removed 3) 136/238 => 16/28 (removed 3) 138/345 => 18/45 (removed 3) 139/695 => 13/65 (removed 9) 143/341 => 13/31 (removed 4) 146/365 => 14/35 (removed 6) 149/298 => 14/28 (removed 9) 149/596 => 14/56 (removed 9) 149/894 => 14/84 (removed 9) 154/253 => 14/23 (removed 5) 3digit fractions found:122, omitted [0 0 9 1 6 15 16 15 60] 1234/4936 => 124/496 (removed 3) 1239/6195 => 123/615 (removed 9) 1246/3649 => 126/369 (removed 4) 1249/2498 => 124/248 (removed 9) 1259/6295 => 125/625 (removed 9) 1279/6395 => 127/635 (removed 9) 1283/5132 => 128/512 (removed 3) 1297/2594 => 127/254 (removed 9) 1297/3891 => 127/381 (removed 9) 1298/2596 => 128/256 (removed 9) 1298/3894 => 128/384 (removed 9) 1298/5192 => 128/512 (removed 9) 4digit fractions found:660, omitted [14 25 92 14 29 63 16 17 390] 12349/24698 => 1234/2468 (removed 9) 12356/67958 => 1236/6798 (removed 5) 12358/14362 => 1258/1462 (removed 3) 12358/15364 => 1258/1564 (removed 3) 12358/17368 => 1258/1768 (removed 3) 12358/19372 => 1258/1972 (removed 3) 12358/21376 => 1258/2176 (removed 3) 12358/25384 => 1258/2584 (removed 3) 12359/61795 => 1235/6175 (removed 9) 12364/32596 => 1364/3596 (removed 2) 12379/61895 => 1237/6185 (removed 9) 12386/32654 => 1386/3654 (removed 2) 5digit fractions found:5087, omitted [75 40 376 78 209 379 591 351 2988] 123459/617295 => 12345/61725 (removed 9) 123468/493872 => 12468/49872 (removed 3) 123469/173524 => 12469/17524 (removed 3) 123469/193546 => 12469/19546 (removed 3) 123469/213568 => 12469/21568 (removed 3) 123469/283645 => 12469/28645 (removed 3) 123469/493876 => 12469/49876 (removed 3) 123469/573964 => 12469/57964 (removed 3) 123479/617395 => 12347/61735 (removed 9) 123495/172893 => 12345/17283 (removed 9) 123548/679514 => 12348/67914 (removed 5) 123574/325786 => 13574/35786 (removed 2) 6digit fractions found:9778, omitted [230 256 921 186 317 751 262 205 6650] Took 42.251172302s
J[edit]
The algorithm generates all potential rational fractions of given size in base 10 and successively applies conditions to restrict the candidates. By avoiding boxing and rational numbers this version is much quicker than that which may be found in the page history.
Filter=: (#~`)(`:6)
assert 'ac' : 1 0 1"_ Filter 'abc'
intersect=:.^:2
assert 'ab' : 'abc'intersect'razb'
odometer=: (4$.$.)@:($&1)
Note 'odometer 2 3'
0 0
0 1
0 2
1 0
1 1
1 2
)
common=: 0 e. ~:
assert common 1 2 1
assert . common 1 2 3
o=: '123456789' {~ [: [email protected]:common"1 Filter [email protected]:(#&9) NB. o is y unique digits, all of them
f=: ,:"1/&g~ NB. f computes a table of all numerators and denominators pairs
mask=: [: </~&i. # NB. the lower triangle will become proper fractions
av=: (([: , mask) # ,/)@:f NB. antivulgarization
c=: [: [email protected]:,/"2 Filter av NB. ensure common digit(s)
fac=: [: ([: common ,&:~.&:q:&:"./)"2 Filter c NB. assure a common factor
NB. This common factor filter might be useful in a future fully tacit version of the program.
cancellation=: monad define
NDL =. c y NB. vector of literal numerator and denominator
NB. retain reducible fractions
ND =. ". NDL NB. integral version of NDL
MASK=. ([: common ,&:~.&:q:/)"1 ND NB. assure a common factor
FRAC=. _2 x: MASK # ND NB. division
CANDIDATES=. MASK # NDL
rat=. , 'r'&,
result=. 0 3 $ a:
for_i. i. # CANDIDATES do.
fraction =. i { FRAC
pair=. i { CANDIDATES
for_d. intersect/ pair do.
trial=. pair ."1 d
if. fraction = _2 x: ". trial do.
result =. result , (rat/pair) ; (rat/trial) ; d
end.
end.
end.
result
)
A=: cancellation&.>2 3 4 5 report=:[: (/:_2&{"1)(((4 ": #) , ' ' , 's' ,~ _1&({::)@:{.)/.~ {:"1) summary=: ' reducibles' ,~ ":@# dozen=: ({.~ (12 <. #))L:_1 boxdraw_j_ 0 NB. pretty boxes 9!:17]0 1 NB. width centering within displayed box (report&.> , summary&.> ,: dozen) A ┌─────────────┬─────────────────┬─────────────────────┬─────────────────────────┐ │ 2 6s │ 9 3s │ 14 1s │ 75 1s │ │ 2 9s │ 1 4s │ 25 2s │ 40 2s │ │ │ 6 5s │ 92 3s │ 376 3s │ │ │ 15 6s │ 14 4s │ 78 4s │ │ │ 16 7s │ 29 5s │ 209 5s │ │ │ 15 8s │ 63 6s │ 379 6s │ │ │ 60 9s │ 16 7s │ 591 7s │ │ │ │ 17 8s │ 351 8s │ │ │ │ 390 9s │ 2988 9s │ ├─────────────┼─────────────────┼─────────────────────┼─────────────────────────┤ │4 reducibles │ 122 reducibles │ 660 reducibles │ 5087 reducibles │ ├─────────────┼─────────────────┼─────────────────────┼─────────────────────────┤ │┌─────┬───┬─┐│┌───────┬─────┬─┐│┌─────────┬───────┬─┐│┌───────────┬─────────┬─┐│ ││16r64│1r4│6│││132r231│12r21│3│││1234r4936│124r496│3│││12349r24698│1234r2468│9││ │├─────┼───┼─┤│├───────┼─────┼─┤│├─────────┼───────┼─┤│├───────────┼─────────┼─┤│ ││19r95│1r5│9│││134r536│14r56│3│││1239r6195│123r615│9│││12356r67958│1236r6798│5││ │├─────┼───┼─┤│├───────┼─────┼─┤│├─────────┼───────┼─┤│├───────────┼─────────┼─┤│ ││26r65│2r5│6│││134r938│14r98│3│││1246r3649│126r369│4│││12358r14362│1258r1462│3││ │├─────┼───┼─┤│├───────┼─────┼─┤│├─────────┼───────┼─┤│├───────────┼─────────┼─┤│ ││49r98│4r8│9│││136r238│16r28│3│││1249r2498│124r248│9│││12358r15364│1258r1564│3││ │└─────┴───┴─┘│├───────┼─────┼─┤│├─────────┼───────┼─┤│├───────────┼─────────┼─┤│ │ ││138r345│18r45│3│││1259r6295│125r625│9│││12358r17368│1258r1768│3││ │ │├───────┼─────┼─┤│├─────────┼───────┼─┤│├───────────┼─────────┼─┤│ │ ││139r695│13r65│9│││1279r6395│127r635│9│││12358r19372│1258r1972│3││ │ │├───────┼─────┼─┤│├─────────┼───────┼─┤│├───────────┼─────────┼─┤│ │ ││143r341│13r31│4│││1283r5132│128r512│3│││12358r21376│1258r2176│3││ │ │├───────┼─────┼─┤│├─────────┼───────┼─┤│├───────────┼─────────┼─┤│ │ ││146r365│14r35│6│││1297r2594│127r254│9│││12358r25384│1258r2584│3││ │ │├───────┼─────┼─┤│├─────────┼───────┼─┤│├───────────┼─────────┼─┤│ │ ││149r298│14r28│9│││1297r3891│127r381│9│││12359r61795│1235r6175│9││ │ │├───────┼─────┼─┤│├─────────┼───────┼─┤│├───────────┼─────────┼─┤│ │ ││149r596│14r56│9│││1298r2596│128r256│9│││12364r32596│1364r3596│2││ │ │├───────┼─────┼─┤│├─────────┼───────┼─┤│├───────────┼─────────┼─┤│ │ ││149r894│14r84│9│││1298r3894│128r384│9│││12379r61895│1237r6185│9││ │ │├───────┼─────┼─┤│├─────────┼───────┼─┤│├───────────┼─────────┼─┤│ │ ││154r253│14r23│5│││1298r5192│128r512│9│││12386r32654│1386r3654│2││ │ │└───────┴─────┴─┘│└─────────┴───────┴─┘│└───────────┴─────────┴─┘│ └─────────────┴─────────────────┴─────────────────────┴─────────────────────────┘
Java[edit]
import java.util.ArrayList;
import java.util.Collections;
import java.util.HashMap;
import java.util.List;
import java.util.Map;
public class FractionReduction {
public static void main(String[] args) {
for ( int size = 2 ; size <= 5 ; size++ ) {
reduce(size);
}
}
private static void reduce(int numDigits) {
System.out.printf("Fractions with digits of length %d where cancellation is valid. Examples:%n", numDigits);
// Generate allowed numerator's and denominator's
int min = (int) Math.pow(10, numDigits1);
int max = (int) Math.pow(10, numDigits)  1;
List<Integer> values = new ArrayList<>();
for ( int number = min ; number <= max ; number++ ) {
if ( isValid(number) ) {
values.add(number);
}
}
Map<Integer,Integer> cancelCount = new HashMap<>();
int size = values.size();
int solutions = 0;
for ( int nIndex = 0 ; nIndex < size  1 ; nIndex++ ) {
int numerator = values.get(nIndex);
// Must be proper fraction
for ( int dIndex = nIndex + 1 ; dIndex < size ; dIndex++ ) {
int denominator = values.get(dIndex);
for ( int commonDigit : digitsInCommon(numerator, denominator) ) {
int numRemoved = removeDigit(numerator, commonDigit);
int denRemoved = removeDigit(denominator, commonDigit);
if ( numerator * denRemoved == denominator * numRemoved ) {
solutions++;
cancelCount.merge(commonDigit, 1, (v1, v2) > v1 + v2);
if ( solutions <= 12 ) {
System.out.printf(" When %d is removed, %d/%d = %d/%d%n", commonDigit, numerator, denominator, numRemoved, denRemoved);
}
}
}
}
}
System.out.printf("Number of fractions where cancellation is valid = %d.%n", solutions);
List<Integer> sorted = new ArrayList<>(cancelCount.keySet());
Collections.sort(sorted);
for ( int removed : sorted ) {
System.out.printf(" The digit %d was removed %d times.%n", removed, cancelCount.get(removed));
}
System.out.println();
}
private static int[] powers = new int[] {1, 10, 100, 1000, 10000, 100000};
// Remove the specified digit.
private static int removeDigit(int n, int removed) {
int m = 0;
int pow = 0;
while ( n > 0 ) {
int r = n % 10;
if ( r != removed ) {
m = m + r*powers[pow];
pow++;
}
n /= 10;
}
return m;
}
// Assumes no duplicate digits individually in n1 or n2  part of task
private static List<Integer> digitsInCommon(int n1, int n2) {
int[] count = new int[10];
List<Integer> common = new ArrayList<>();
while ( n1 > 0 ) {
int r = n1 % 10;
count[r] += 1;
n1 /= 10;
}
while ( n2 > 0 ) {
int r = n2 % 10;
if ( count[r] > 0 ) {
common.add(r);
}
n2 /= 10;
}
return common;
}
// No repeating digits, no digit is zero.
private static boolean isValid(int num) {
int[] count = new int[10];
while ( num > 0 ) {
int r = num % 10;
if ( r == 0  count[r] == 1 ) {
return false;
}
count[r] = 1;
num /= 10;
}
return true;
}
}
 Output:
Fractions with digits of length 2 where cancellation is valid. Examples: When 6 is removed, 16/64 = 1/4 When 9 is removed, 19/95 = 1/5 When 6 is removed, 26/65 = 2/5 When 9 is removed, 49/98 = 4/8 Number of fractions where cancellation is valid = 4. The digit 6 was removed 2 times. The digit 9 was removed 2 times. Fractions with digits of length 3 where cancellation is valid. Examples: When 3 is removed, 132/231 = 12/21 When 3 is removed, 134/536 = 14/56 When 3 is removed, 134/938 = 14/98 When 3 is removed, 136/238 = 16/28 When 3 is removed, 138/345 = 18/45 When 9 is removed, 139/695 = 13/65 When 4 is removed, 143/341 = 13/31 When 6 is removed, 146/365 = 14/35 When 9 is removed, 149/298 = 14/28 When 9 is removed, 149/596 = 14/56 When 9 is removed, 149/894 = 14/84 When 5 is removed, 154/253 = 14/23 Number of fractions where cancellation is valid = 122. The digit 3 was removed 9 times. The digit 4 was removed 1 times. The digit 5 was removed 6 times. The digit 6 was removed 15 times. The digit 7 was removed 16 times. The digit 8 was removed 15 times. The digit 9 was removed 60 times. Fractions with digits of length 4 where cancellation is valid. Examples: When 3 is removed, 1234/4936 = 124/496 When 9 is removed, 1239/6195 = 123/615 When 4 is removed, 1246/3649 = 126/369 When 9 is removed, 1249/2498 = 124/248 When 9 is removed, 1259/6295 = 125/625 When 9 is removed, 1279/6395 = 127/635 When 3 is removed, 1283/5132 = 128/512 When 9 is removed, 1297/2594 = 127/254 When 9 is removed, 1297/3891 = 127/381 When 9 is removed, 1298/2596 = 128/256 When 9 is removed, 1298/3894 = 128/384 When 9 is removed, 1298/5192 = 128/512 Number of fractions where cancellation is valid = 660. The digit 1 was removed 14 times. The digit 2 was removed 25 times. The digit 3 was removed 92 times. The digit 4 was removed 14 times. The digit 5 was removed 29 times. The digit 6 was removed 63 times. The digit 7 was removed 16 times. The digit 8 was removed 17 times. The digit 9 was removed 390 times. Fractions with digits of length 5 where cancellation is valid. Examples: When 9 is removed, 12349/24698 = 1234/2468 When 5 is removed, 12356/67958 = 1236/6798 When 3 is removed, 12358/14362 = 1258/1462 When 3 is removed, 12358/15364 = 1258/1564 When 3 is removed, 12358/17368 = 1258/1768 When 3 is removed, 12358/19372 = 1258/1972 When 3 is removed, 12358/21376 = 1258/2176 When 3 is removed, 12358/25384 = 1258/2584 When 9 is removed, 12359/61795 = 1235/6175 When 2 is removed, 12364/32596 = 1364/3596 When 9 is removed, 12379/61895 = 1237/6185 When 2 is removed, 12386/32654 = 1386/3654 Number of fractions where cancellation is valid = 5087. The digit 1 was removed 75 times. The digit 2 was removed 40 times. The digit 3 was removed 376 times. The digit 4 was removed 78 times. The digit 5 was removed 209 times. The digit 6 was removed 379 times. The digit 7 was removed 591 times. The digit 8 was removed 351 times. The digit 9 was removed 2988 times.
Julia[edit]
using Combinatorics
toi(set) = parse(Int, join(set, ""))
drop1(c, set) = toi(filter(x > x != c, set))
function anomalouscancellingfractions(numdigits)
ret = Vector{Tuple{Int, Int, Int, Int, Int}}()
for nset in permutations(1:9, numdigits), dset in permutations(1:9, numdigits)
if nset < dset # only proper fractions
for c in nset
if c in dset # a common digit exists
n, d, nn, dd = toi(nset), toi(dset), drop1(c, nset), drop1(c, dset)
if n // d == nn // dd # anomalous cancellation
push!(ret, (n, d, nn, dd, c))
end
end
end
end
end
ret
end
function testfractionreduction(maxdigits=5)
for i in 2:maxdigits
results = anomalouscancellingfractions(i)
println("\nFor $i digits, there were ", length(results),
" fractions with anomalous cancellation.")
numcounts = zeros(Int, 9)
for r in results
numcounts[r[5]] += 1
end
for (j, count) in enumerate(numcounts)
count > 0 && println("The digit $j was crossed out $count times.")
end
println("Examples:")
for j in 1:min(length(results), 12)
r = results[j]
println(r[1], "/", r[2], " = ", r[3], "/", r[4], " ($(r[5]) crossed out)")
end
end
end
testfractionreduction()
 Output:
For 2 digits, there were 4 fractions with anomalous cancellation. The digit 6 was crossed out 2 times. The digit 9 was crossed out 2 times. Examples: 16/64 = 1/4 (6 crossed out) 19/95 = 1/5 (9 crossed out) 26/65 = 2/5 (6 crossed out) 49/98 = 4/8 (9 crossed out) For 3 digits, there were 122 fractions with anomalous cancellation. The digit 3 was crossed out 9 times. The digit 4 was crossed out 1 times. The digit 5 was crossed out 6 times. The digit 6 was crossed out 15 times. The digit 7 was crossed out 16 times. The digit 8 was crossed out 15 times. The digit 9 was crossed out 60 times. Examples: 132/231 = 12/21 (3 crossed out) 134/536 = 14/56 (3 crossed out) 134/938 = 14/98 (3 crossed out) 136/238 = 16/28 (3 crossed out) 138/345 = 18/45 (3 crossed out) 139/695 = 13/65 (9 crossed out) 143/341 = 13/31 (4 crossed out) 146/365 = 14/35 (6 crossed out) 149/298 = 14/28 (9 crossed out) 149/596 = 14/56 (9 crossed out) 149/894 = 14/84 (9 crossed out) 154/253 = 14/23 (5 crossed out) For 4 digits, there were 660 fractions with anomalous cancellation. The digit 1 was crossed out 14 times. The digit 2 was crossed out 25 times. The digit 3 was crossed out 92 times. The digit 4 was crossed out 14 times. The digit 5 was crossed out 29 times. The digit 6 was crossed out 63 times. The digit 7 was crossed out 16 times. The digit 8 was crossed out 17 times. The digit 9 was crossed out 390 times. Examples: 1234/4936 = 124/496 (3 crossed out) 1239/6195 = 123/615 (9 crossed out) 1246/3649 = 126/369 (4 crossed out) 1249/2498 = 124/248 (9 crossed out) 1259/6295 = 125/625 (9 crossed out) 1279/6395 = 127/635 (9 crossed out) 1283/5132 = 128/512 (3 crossed out) 1297/2594 = 127/254 (9 crossed out) 1297/3891 = 127/381 (9 crossed out) 1298/2596 = 128/256 (9 crossed out) 1298/3894 = 128/384 (9 crossed out) 1298/5192 = 128/512 (9 crossed out) For 5 digits, there were 5087 fractions with anomalous cancellation. The digit 1 was crossed out 75 times. The digit 2 was crossed out 40 times. The digit 3 was crossed out 376 times. The digit 4 was crossed out 78 times. The digit 5 was crossed out 209 times. The digit 6 was crossed out 379 times. The digit 7 was crossed out 591 times. The digit 8 was crossed out 351 times. The digit 9 was crossed out 2988 times. Examples: 12349/24698 = 1234/2468 (9 crossed out) 12356/67958 = 1236/6798 (5 crossed out) 12358/14362 = 1258/1462 (3 crossed out) 12358/15364 = 1258/1564 (3 crossed out) 12358/17368 = 1258/1768 (3 crossed out) 12358/19372 = 1258/1972 (3 crossed out) 12358/21376 = 1258/2176 (3 crossed out) 12358/25384 = 1258/2584 (3 crossed out) 12359/61795 = 1235/6175 (9 crossed out) 12364/32596 = 1364/3596 (2 crossed out) 12379/61895 = 1237/6185 (9 crossed out) 12386/32654 = 1386/3654 (2 crossed out)
Kotlin[edit]
fun indexOf(n: Int, s: IntArray): Int {
for (i_j in s.withIndex()) {
if (n == i_j.value) {
return i_j.index
}
}
return 1
}
fun getDigits(n: Int, le: Int, digits: IntArray): Boolean {
var mn = n
var mle = le
while (mn > 0) {
val r = mn % 10
if (r == 0  indexOf(r, digits) >= 0) {
return false
}
mle
digits[mle] = r
mn /= 10
}
return true
}
val pows = intArrayOf(1, 10, 100, 1_000, 10_000)
fun removeDigit(digits: IntArray, le: Int, idx: Int): Int {
var sum = 0
var pow = pows[le  2]
for (i in 0 until le) {
if (i == idx) {
continue
}
sum += digits[i] * pow
pow /= 10
}
return sum
}
fun main() {
val lims = listOf(
Pair(12, 97),
Pair(123, 986),
Pair(1234, 9875),
Pair(12345, 98764)
)
val count = IntArray(5)
var omitted = arrayOf<Array<Int>>()
for (i in 0 until 5) {
var array = arrayOf<Int>()
for (j in 0 until 10) {
array += 0
}
omitted += array
}
for (i_lim in lims.withIndex()) {
val i = i_lim.index
val lim = i_lim.value
val nDigits = IntArray(i + 2)
val dDigits = IntArray(i + 2)
val blank = IntArray(i + 2) { 0 }
for (n in lim.first..lim.second) {
blank.copyInto(nDigits)
val nOk = getDigits(n, i + 2, nDigits)
if (!nOk) {
continue
}
for (d in n + 1..lim.second + 1) {
blank.copyInto(dDigits)
val dOk = getDigits(d, i + 2, dDigits)
if (!dOk) {
continue
}
for (nix_digit in nDigits.withIndex()) {
val dix = indexOf(nix_digit.value, dDigits)
if (dix >= 0) {
val rn = removeDigit(nDigits, i + 2, nix_digit.index)
val rd = removeDigit(dDigits, i + 2, dix)
if (n.toDouble() / d.toDouble() == rn.toDouble() / rd.toDouble()) {
count[i]++
omitted[i][nix_digit.value]++
if (count[i] <= 12) {
println("$n/$d = $rn/$rd by omitting ${nix_digit.value}'s")
}
}
}
}
}
}
println()
}
for (i in 2..5) {
println("There are ${count[i  2]} $idigit fractions of which:")
for (j in 1..9) {
if (omitted[i  2][j] == 0) {
continue
}
println("%6d have %d's omitted".format(omitted[i  2][j], j))
}
println()
}
}
 Output:
16/64 = 1/4 by omitting 6's 19/95 = 1/5 by omitting 9's 26/65 = 2/5 by omitting 6's 49/98 = 4/8 by omitting 9's 132/231 = 12/21 by omitting 3's 134/536 = 14/56 by omitting 3's 134/938 = 14/98 by omitting 3's 136/238 = 16/28 by omitting 3's 138/345 = 18/45 by omitting 3's 139/695 = 13/65 by omitting 9's 143/341 = 13/31 by omitting 4's 146/365 = 14/35 by omitting 6's 149/298 = 14/28 by omitting 9's 149/596 = 14/56 by omitting 9's 149/894 = 14/84 by omitting 9's 154/253 = 14/23 by omitting 5's 1234/4936 = 124/496 by omitting 3's 1239/6195 = 123/615 by omitting 9's 1246/3649 = 126/369 by omitting 4's 1249/2498 = 124/248 by omitting 9's 1259/6295 = 125/625 by omitting 9's 1279/6395 = 127/635 by omitting 9's 1283/5132 = 128/512 by omitting 3's 1297/2594 = 127/254 by omitting 9's 1297/3891 = 127/381 by omitting 9's 1298/2596 = 128/256 by omitting 9's 1298/3894 = 128/384 by omitting 9's 1298/5192 = 128/512 by omitting 9's 12349/24698 = 1234/2468 by omitting 9's 12356/67958 = 1236/6798 by omitting 5's 12358/14362 = 1258/1462 by omitting 3's 12358/15364 = 1258/1564 by omitting 3's 12358/17368 = 1258/1768 by omitting 3's 12358/19372 = 1258/1972 by omitting 3's 12358/21376 = 1258/2176 by omitting 3's 12358/25384 = 1258/2584 by omitting 3's 12359/61795 = 1235/6175 by omitting 9's 12364/32596 = 1364/3596 by omitting 2's 12379/61895 = 1237/6185 by omitting 9's 12386/32654 = 1386/3654 by omitting 2's There are 4 2digit fractions of which: 2 have 6's omitted 2 have 9's omitted There are 122 3digit fractions of which: 9 have 3's omitted 1 have 4's omitted 6 have 5's omitted 15 have 6's omitted 16 have 7's omitted 15 have 8's omitted 60 have 9's omitted There are 660 4digit fractions of which: 14 have 1's omitted 25 have 2's omitted 92 have 3's omitted 14 have 4's omitted 29 have 5's omitted 63 have 6's omitted 16 have 7's omitted 17 have 8's omitted 390 have 9's omitted There are 5087 5digit fractions of which: 75 have 1's omitted 40 have 2's omitted 376 have 3's omitted 78 have 4's omitted 209 have 5's omitted 379 have 6's omitted 591 have 7's omitted 351 have 8's omitted 2988 have 9's omitted
Lua[edit]
function indexOf(haystack, needle)
for idx,straw in pairs(haystack) do
if straw == needle then
return idx
end
end
return 1
end
function getDigits(n, le, digits)
while n > 0 do
local r = n % 10
if r == 0 or indexOf(digits, r) > 0 then
return false
end
le = le  1
digits[le + 1] = r
n = math.floor(n / 10)
end
return true
end
function removeDigit(digits, le, idx)
local pows = { 1, 10, 100, 1000, 10000 }
local sum = 0
local pow = pows[le  2 + 1]
for i = 1, le do
if i ~= idx then
sum = sum + digits[i] * pow
pow = math.floor(pow / 10)
end
end
return sum
end
function main()
local lims = { {12, 97}, {123, 986}, {1234, 9875}, {12345, 98764} }
local count = { 0, 0, 0, 0, 0 }
local omitted = {
{ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 },
{ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 },
{ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 },
{ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 },
{ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 },
}
for i,_ in pairs(lims) do
local nDigits = {}
local dDigits = {}
for j = 1, i + 2  1 do
nDigits[j] = 1
dDigits[j] = 1
end
for n = lims[i][1], lims[i][2] do
for j,_ in pairs(nDigits) do
nDigits[j] = 0
end
local nOk = getDigits(n, i + 2  1, nDigits)
if nOk then
for d = n + 1, lims[i][2] + 1 do
for j,_ in pairs(dDigits) do
dDigits[j] = 0
end
local dOk = getDigits(d, i + 2  1, dDigits)
if dOk then
for nix,_ in pairs(nDigits) do
local digit = nDigits[nix]
local dix = indexOf(dDigits, digit)
if dix >= 0 then
local rn = removeDigit(nDigits, i + 2  1, nix)
local rd = removeDigit(dDigits, i + 2  1, dix)
if (n / d) == (rn / rd) then
count[i] = count[i] + 1
omitted[i][digit + 1] = omitted[i][digit + 1] + 1
if count[i] <= 12 then
print(string.format("%d/%d = %d/%d by omitting %d's", n, d, rn, rd, digit))
end
end
end
end
end
end
end
end
print()
end
for i = 2, 5 do
print("There are "..count[i  2 + 1].." "..i.."digit fractions of which:")
for j = 1, 9 do
if omitted[i  2 + 1][j + 1] > 0 then
print(string.format("%6d have %d's omitted", omitted[i  2 + 1][j + 1], j))
end
end
print()
end
end
main()
 Output:
16/64 = 1/4 by omitting 6's 19/95 = 1/5 by omitting 9's 26/65 = 2/5 by omitting 6's 49/98 = 4/8 by omitting 9's 132/231 = 12/21 by omitting 3's 134/536 = 14/56 by omitting 3's 134/938 = 14/98 by omitting 3's 136/238 = 16/28 by omitting 3's 138/345 = 18/45 by omitting 3's 139/695 = 13/65 by omitting 9's 143/341 = 13/31 by omitting 4's 146/365 = 14/35 by omitting 6's 149/298 = 14/28 by omitting 9's 149/596 = 14/56 by omitting 9's 149/894 = 14/84 by omitting 9's 154/253 = 14/23 by omitting 5's 1234/4936 = 124/496 by omitting 3's 1239/6195 = 123/615 by omitting 9's 1246/3649 = 126/369 by omitting 4's 1249/2498 = 124/248 by omitting 9's 1259/6295 = 125/625 by omitting 9's 1279/6395 = 127/635 by omitting 9's 1283/5132 = 128/512 by omitting 3's 1297/2594 = 127/254 by omitting 9's 1297/3891 = 127/381 by omitting 9's 1298/2596 = 128/256 by omitting 9's 1298/3894 = 128/384 by omitting 9's 1298/5192 = 128/512 by omitting 9's 12349/24698 = 1234/2468 by omitting 9's 12356/67958 = 1236/6798 by omitting 5's 12358/14362 = 1258/1462 by omitting 3's 12358/15364 = 1258/1564 by omitting 3's 12358/17368 = 1258/1768 by omitting 3's 12358/19372 = 1258/1972 by omitting 3's 12358/21376 = 1258/2176 by omitting 3's 12358/25384 = 1258/2584 by omitting 3's 12359/61795 = 1235/6175 by omitting 9's 12364/32596 = 1364/3596 by omitting 2's 12379/61895 = 1237/6185 by omitting 9's 12386/32654 = 1386/3654 by omitting 2's There are 4 2digit fractions of which: 2 have 6's omitted 2 have 9's omitted There are 122 3digit fractions of which: 9 have 3's omitted 1 have 4's omitted 6 have 5's omitted 15 have 6's omitted 16 have 7's omitted 15 have 8's omitted 60 have 9's omitted There are 660 4digit fractions of which: 14 have 1's omitted 25 have 2's omitted 92 have 3's omitted 14 have 4's omitted 29 have 5's omitted 63 have 6's omitted 16 have 7's omitted 17 have 8's omitted 390 have 9's omitted There are 5087 5digit fractions of which: 75 have 1's omitted 40 have 2's omitted 376 have 3's omitted 78 have 4's omitted 209 have 5's omitted 379 have 6's omitted 591 have 7's omitted 351 have 8's omitted 2988 have 9's omitted
MiniZinc[edit]
The Model[edit]
%Fraction Reduction. Nigel Galloway, September 5th., 2019
include "alldifferent.mzn"; include "member.mzn";
int: S;
array [1..9] of int: Pn=[1,10,100,1000,10000,100000,1000000,10000000,100000000];
array [1..S] of var 1..9: Nz; constraint alldifferent(Nz);
array [1..S] of var 1..9: Gz; constraint alldifferent(Gz);
var int: n; constraint n=sum(n in 1..S)(Nz[n]*Pn[n]);
var int: i; constraint i=sum(n in 1..S)(Gz[n]*Pn[n]); constraint n<i; constraint n*g=i*e;
var int: g; constraint g=sum(n in 1..S)(if n=a then 0 elseif n>a then Gz[n]*Pn[n1] else Gz[n]*Pn[n] endif);
var int: e; constraint e=sum(n in 1..S)(if n=l then 0 elseif n>l then Nz[n]*Pn[n1] else Nz[n]*Pn[n] endif);
var 1..S: l; constraint Nz[l]=w;
var 1..S: a; constraint Gz[a]=w;
var 1..9: w; constraint member(Nz,w) /\ member(Gz,w);
output [show(n)++"/"++show(i)++" becomes "++show(e)++"/"++show(g)++" when "++show(w)++" is omitted"]
The Tasks[edit]
 Displaying 12 solutions
 minizinc numsolutions 12 DS=2
 Output:
16/64 becomes 1/4 when 6 is omitted  26/65 becomes 2/5 when 6 is omitted  19/95 becomes 1/5 when 9 is omitted  49/98 becomes 4/8 when 9 is omitted  ==========
 minizinc numsolutions 12 DS=3
 Output:
132/231 becomes 12/21 when 3 is omitted  134/536 becomes 14/56 when 3 is omitted  134/938 becomes 14/98 when 3 is omitted  136/238 becomes 16/28 when 3 is omitted  138/345 becomes 18/45 when 3 is omitted  139/695 becomes 13/65 when 9 is omitted  143/341 becomes 13/31 when 4 is omitted  146/365 becomes 14/35 when 6 is omitted  149/298 becomes 14/28 when 9 is omitted  149/596 becomes 14/56 when 9 is omitted  149/894 becomes 14/84 when 9 is omitted  154/253 becomes 14/23 when 5 is omitted 
 minizinc numsolutions 12 DS=4
 Output:
2147/3164 becomes 247/364 when 1 is omitted  2314/3916 becomes 234/396 when 1 is omitted  2147/5198 becomes 247/598 when 1 is omitted  3164/5198 becomes 364/598 when 1 is omitted  2314/6319 becomes 234/639 when 1 is omitted  3916/6319 becomes 396/639 when 1 is omitted  5129/7136 becomes 529/736 when 1 is omitted  3129/7152 becomes 329/752 when 1 is omitted  4913/7514 becomes 493/754 when 1 is omitted  7168/8176 becomes 768/876 when 1 is omitted  5129/9143 becomes 529/943 when 1 is omitted  7136/9143 becomes 736/943 when 1 is omitted 
 minizinc numsolutions 12 DS=5
 Output:
21356/31472 becomes 2356/3472 when 1 is omitted  21394/31528 becomes 2394/3528 when 1 is omitted  21546/31752 becomes 2546/3752 when 1 is omitted  21679/31948 becomes 2679/3948 when 1 is omitted  21698/31976 becomes 2698/3976 when 1 is omitted  25714/34615 becomes 2574/3465 when 1 is omitted  27615/34716 becomes 2765/3476 when 1 is omitted  25917/34719 becomes 2597/3479 when 1 is omitted  25916/36518 becomes 2596/3658 when 1 is omitted  31276/41329 becomes 3276/4329 when 1 is omitted  21375/41625 becomes 2375/4625 when 1 is omitted  31584/41736 becomes 3584/4736 when 1 is omitted 
 minizinc numsolutions 12 DS=6
 Output:
123495/172893 becomes 12345/17283 when 9 is omitted  123594/164792 becomes 12354/16472 when 9 is omitted  123654/163758 becomes 12654/16758 when 3 is omitted  124678/135679 becomes 12478/13579 when 6 is omitted  124768/164872 becomes 12768/16872 when 4 is omitted  125349/149352 becomes 12549/14952 when 3 is omitted  125394/146293 becomes 12534/14623 when 9 is omitted  125937/127936 becomes 12537/12736 when 9 is omitted  125694/167592 becomes 12564/16752 when 9 is omitted  125769/135786 becomes 12769/13786 when 5 is omitted  125769/165837 becomes 12769/16837 when 5 is omitted  125934/146923 becomes 12534/14623 when 9 is omitted 
 Count number of solutions
 minizinc allsolutions s DS=3
 Output:
%%%mznstat: nSolutions=122
 minizinc allsolutions s DS=4
 Output:
%%%mznstat: nSolutions=660
 minizinc allsolutions s DS=5
 Output:
%%%mznstat: nSolutions=5087
Pascal[edit]
Using a permutation k out of n with k <= n
Inserting a record with this number and all numbers with one digit removed of that number.So only once calculated.Trade off is big size and no cache friendly local access.
program FracRedu;
{$IFDEF FPC}
{$MODE DELPHI}
{$OPTIMIZATION ON,ALL}
{$ELSE}
{$APPTYPE CONSOLE}
{$ENDIF}
uses
SysUtils;
type
tdigit = 0..9;
const
cMaskDgt: array [tdigit] of Uint32 = (1, 2, 4, 8, 16, 32, 64, 128, 256, 512
{,1024,2048,4096,8193,16384,32768});
cMaxDigits = High(tdigit);
type
tPermfield = array[tdigit] of uint32;
tpPermfield = ^tPermfield;
tDigitCnt = array[tdigit] of Uint32;
tErg = record
numUsedDigits : Uint32;
numUnusedDigit : array[tdigit] of Uint32;
numNormal : Uint64;// so sqr of number stays in Uint64
dummy : array[0..7] of byte;//> sizeof(tErg) = 64
end;
tpErg = ^tErg;
var
Erg: array of tErg;
pf_x, pf_y: tPermfield;
DigitCnt :tDigitCnt;
permcnt, UsedDigits,Anzahl: NativeUint;
function Fakultaet(i: integer): integer;
begin
Result := 1;
while i > 1 do
begin
Result := Result * i;
Dec(i);
end;
end;
procedure OutErg(dgt: Uint32;pi,pJ:tpErg);
begin
writeln(dgt:3,' ', pi^.numUnusedDigit[dgt],'/',pj^.numUnusedDigit[dgt]
,' = ',pi^.numNormal,'/',pj^.numNormal);
end;
function Check(pI,pJ : tpErg;Nud :Word):integer;
var
dgt: NativeInt;
Begin
result := 0;
dgt := 1;
NUD := NUD SHR 1;
repeat
IF NUD AND 1 <> 0 then
Begin
If pI^.numNormal*pJ^.numUnusedDigit[dgt] = pJ^.numNormal*pI^.numUnusedDigit[dgt] then
Begin
inc(result);
inc(DigitCnt[dgt]);
IF Anzahl < 110 then
OutErg(dgt,pI,pJ);
end;
end;
inc(dgt);
NUD := NUD SHR 1;
until NUD = 0;
end;
procedure CheckWithOne(pI : tpErg;j,Nud:Uint32);
var
pJ : tpErg;
l : NativeUInt;
Begin
pJ := pI;
if UsedDigits <5 then
Begin
for j := j+1 to permcnt do
begin
inc(pJ);
//digits used by both numbers
l := NUD AND pJ^.numUsedDigits;
IF l <> 0 then
inc(Anzahl,Check(pI,pJ,l));
end;
end
else
Begin
for j := j+1 to permcnt do
begin
inc(pJ);
l := NUD AND pJ^.numUsedDigits;
inc(Anzahl,Check(pI,pJ,l));
end;
end;
end;
procedure SearchMultiple;
var
pI : tpErg;
i : NativeUInt;
begin
pI := @Erg[0];
for i := 0 to permcnt do
Begin
CheckWithOne(pI,i,pI^.numUsedDigits);
inc(pI);
end;
end;
function BinomCoeff(n, k: byte): longint;
var
i: longint;
begin
{n ueber k = n ueber (nk) , also kuerzere Version waehlen}
if k > n div 2 then
k := n  k;
Result := 1;
if k <= n then
for i := 1 to k do
Result := Result * (n  i + 1) div i;{geht immer ohne Rest }
end;
procedure InsertToErg(var E: tErg; const x: tPermfield);
var
n : Uint64;
k,i,j,dgt,nud: NativeInt;
begin
// k of PermKoutofN is reduced by one for 9 digits
k := UsedDigits;
n := 0;
nud := 0;
for i := 1 to k do
begin
dgt := x[i];
nud := nud or cMaskDgt[dgt];
n := n * 10 + dgt;
end;
with E do
begin
numUsedDigits := nud;
numNormal := n;
end;
//calc all numbers with one removed digit
For J := k downto 1 do
Begin
n := 0;
for i := 1 to j1 do
n := n * 10 + x[i];
for i := j+1 to k do
n := n * 10 + x[i];
E.numUnusedDigit[x[j]] := n;
end;
end;
procedure PermKoutofN(k, n: nativeInt);
var
x, y: tpPermfield;
i, yi, tmp: NativeInt;
begin
//initialise
x := @pf_x;
y := @pf_y;
permcnt := 0;
if k > n then
k := n;
if k = n then
k := k  1;
for i := 1 to n do
x^[i] := i;
for i := 1 to k do
y^[i] := i;
InserttoErg(Erg[permcnt], x^);
i := k;
repeat
yi := y^[i];
if yi < n then
begin
Inc(permcnt);
Inc(yi);
y^[i] := yi;
tmp := x^[i];
x^[i] := x^[yi];
x^[yi] := tmp;
i := k;
InserttoErg(Erg[permcnt], x^);
end
else
begin
repeat
tmp := x^[i];
x^[i] := x^[yi];
x^[yi] := tmp;
Dec(yi);
until yi <= i;
y^[i] := yi;
Dec(i);
end;
until (i = 0);
end;
procedure OutDigitCount;
var
i : tDigit;
Begin
writeln('omitted digits 1 to 9');
For i := 1 to 9do
write(DigitCnt[i]:UsedDigits);
writeln;
end;
procedure ClearDigitCount;
var
i : tDigit;
Begin
For i := low(DigitCnt) to high(DigitCnt) do
DigitCnt[i] := 0;
end;
var
t1, t0: TDateTime;
begin
For UsedDigits := 8 to 9 do
Begin
writeln('Used digits ',UsedDigits);
T0 := now;
ClearDigitCount;
setlength(Erg, Fakultaet(UsedDigits) * BinomCoeff(cMaxDigits, UsedDigits));
Anzahl := 0;
permcnt := 0;
PermKoutOfN(UsedDigits, cMaxDigits);
SearchMultiple;
T1 := now;
writeln('Found solutions ',Anzahl);
OutDigitCount;
writeln('time taken ',FormatDateTime('HH:NN:SS.zzz', T1  T0));
setlength(Erg, 0);
writeln;
end;
end.
 Output:
{ /* inserted by hand / solutions Used digits 2 count of different numbers 72 / 4 Used digits 3 count of different numbers 504 / 122 Used digits 4 count of different numbers 3024 / 660 Used digits 5 count of different numbers 15120 / 5087 Used digits 6 count of different numbers 60480 / 9778 Used digits 7 count of different numbers 181440 / 40163 Used digits 8 count of different numbers 362880 / 17722 Used digits 9 count of different numbers 362880 / 92413 */ } Used digits 2 6 1/4 = 16/64 9 1/5 = 19/95 6 2/5 = 26/65 9 4/8 = 49/98 Found solutions 4 omitted digits 1 to 9 0 0 0 0 0 2 0 0 2 time taken 00:00:00.000 Used digits 3 3 12/21 = 132/231 3 14/56 = 134/536 3 14/98 = 134/938 3 16/28 = 136/238 3 18/45 = 138/345 9 13/65 = 139/695 4 13/31 = 143/341 6 14/35 = 146/365 9 14/28 = 149/298 9 14/56 = 149/596 9 14/84 = 149/894 5 14/23 = 154/253 Found solutions 122 omitted digits 1 to 9 0 0 9 1 6 15 16 15 60 time taken 00:00:00.004 Used digits 4 3 124/496 = 1234/4936 9 123/615 = 1239/6195 4 126/369 = 1246/3649 9 124/248 = 1249/2498 9 125/625 = 1259/6295 9 127/635 = 1279/6395 3 128/512 = 1283/5132 9 127/254 = 1297/2594 9 127/381 = 1297/3891 9 128/256 = 1298/2596 9 128/384 = 1298/3894 9 128/512 = 1298/5192 Found solutions 660 omitted digits 1 to 9 14 25 92 14 29 63 16 17 390 time taken 00:00:00.060 Used digits 5 9 1234/2468 = 12349/24698 5 1236/6798 = 12356/67958 3 1258/1462 = 12358/14362 3 1258/1564 = 12358/15364 3 1258/1768 = 12358/17368 3 1258/1972 = 12358/19372 3 1258/2176 = 12358/21376 3 1258/2584 = 12358/25384 9 1235/6175 = 12359/61795 2 1364/3596 = 12364/32596 9 1237/6185 = 12379/61895 2 1386/3654 = 12386/32654 Found solutions 5087 omitted digits 1 to 9 75 40 376 78 209 379 591 351 2988 time taken 00:00:01.787 Used digits 6 9 12345/61725 = 123459/617295 3 12468/49872 = 123468/493872 3 12469/17524 = 123469/173524 3 12469/19546 = 123469/193546 3 12469/21568 = 123469/213568 3 12469/28645 = 123469/283645 3 12469/49876 = 123469/493876 3 12469/57964 = 123469/573964 9 12347/61735 = 123479/617395 9 12345/17283 = 123495/172893 5 12348/67914 = 123548/679514 2 13574/35786 = 123574/325786 Found solutions 9778 omitted digits 1 to 9 230 256 921 186 317 751 262 205 6650 time taken 00:00:31.858 Used digits 7 3 124569/498276 = 1234569/4938276 3 124579/195286 = 1234579/1935286 3 124579/245791 = 1234579/2435791 3 124579/286195 = 1234579/2836195 3 124579/457912 = 1234579/4537912 3 124579/528619 = 1234579/5238619 3 124579/579124 = 1234579/5739124 3 124579/619528 = 1234579/6139528 9 123457/617285 = 1234579/6172895 9 123457/617285 = 1234597/6172985 9 123465/617325 = 1234659/6173295 3 124678/498712 = 1234678/4938712 Found solutions 40163 omitted digits 1 to 9 333 191 1368 278 498 1094 3657 1434 31310 time taken 00:04:54.703 Used digits 8 3 1245679/2457691 = 12345679/24357691 6 1234579/2435791 = 12345679/24357691 3 1245679/4982716 = 12345679/49382716 3 1245679/6194728 = 12345679/61394728 9 1234567/6172835 = 12345679/61728395 3 1245689/4982756 = 12345689/49382756 9 1234567/6172835 = 12345967/61729835 9 1234657/6173285 = 12346579/61732895 9 1234657/6173285 = 12346597/61732985 3 1246789/4987156 = 12346789/49387156 9 1234685/6173425 = 12346859/61734295 3 1246879/4987516 = 12346879/49387516 Found solutions 17233 omitted digits 1 to 9 247 233 888 288 355 710 425 193 13894 time taken 00:18:58.784 Used digits 9 3 12456789/49827156 = 123456789/493827156 3 12456879/49827516 = 123456879/493827516 9 12345687/61728435 = 123456879/617284395 9 12345687/61728435 = 123456987/617284935 9 12345687/61728435 = 123459687/617298435 9 12346857/61734285 = 123468579/617342895 9 12346857/61734285 = 123468597/617342985 9 12346857/61734285 = 123469857/617349285 9 12347685/61738425 = 123476859/617384295 9 12347685/61738425 = 123476985/617384925 5 12347896/67913428 = 123478956/679134258 9 12347685/61738425 = 123479685/617398425 Found solutions 92413 omitted digits 1 to 9 266 110 1008 131 324 737 300 159 89378 time taken 00:13:04.511 /* go version go1.10.3 gccgo (Debian 8.3.06) 8.3.0 linux/amd64 6digit fractions found:9778, omitted [230 256 921 186 317 751 262 205 6650] Took 1m38.85577279s */
Perl[edit]
use strict;
use warnings;
use feature 'say';
use List::Util qw<sum uniq uniqnum head tail>;
for my $exp (map { $_  1 } <2 3 4>) {
my %reduced;
my $start = sum map { 10 ** $_ * ($exp  $_ + 1) } 0..$exp;
my $end = 10**($exp+1)  1 + sum map { 10 ** $_ * ($exp  $_) } 0..$exp1;
for my $den ($start .. $end1) {
next if $den =~ /0/ or (uniqnum split '', $den) <= $exp;
for my $num ($start .. $den1) {
next if $num =~ /0/ or (uniqnum split '', $num) <= $exp;
my %i;
map { $i{$_}++ } (uniq head 1, split '',$den), uniq tail 1, split '',$num;
my @set = grep { $_ if $i{$_} > 1 } keys %i;
next if @set < 1;
for (@set) {
(my $ne = $num) =~ s/$_//;
(my $de = $den) =~ s/$_//;
if ($ne/$de == $num/$den) {
$reduced{"$num/$den:$_"} = "$ne/$de";
}
}
}
}
my $digit = $exp + 1;
say "\n" . +%reduced . " $digitdigit reducible fractions:";
for my $n (1..9) {
my $cnt = scalar grep { /:$n/ } keys %reduced;
say "$cnt with removed $n" if $cnt;
}
say "\n 12 (or all, if less) $digitdigit reducible fractions:";
for my $f (head 12, sort keys %reduced) {
printf " %s => %s removed %s\n", substr($f,0,$digit*2+1), $reduced{$f}, substr($f,1)
}
}
 Output:
4 2digit reducible fractions: 2 with removed 6 2 with removed 9 12 (or all, if less) 2digit reducible fractions: 16/64 => 1/4 removed 6 19/95 => 1/5 removed 9 26/65 => 2/5 removed 6 49/98 => 4/8 removed 9 122 3digit reducible fractions: 9 with removed 3 1 with removed 4 6 with removed 5 15 with removed 6 16 with removed 7 15 with removed 8 60 with removed 9 12 (or all, if less) 3digit reducible fractions: 132/231 => 12/21 removed 3 134/536 => 14/56 removed 3 134/938 => 14/98 removed 3 136/238 => 16/28 removed 3 138/345 => 18/45 removed 3 139/695 => 13/65 removed 9 143/341 => 13/31 removed 4 146/365 => 14/35 removed 6 149/298 => 14/28 removed 9 149/596 => 14/56 removed 9 149/894 => 14/84 removed 9 154/253 => 14/23 removed 5 660 4digit reducible fractions: 14 with removed 1 25 with removed 2 92 with removed 3 14 with removed 4 29 with removed 5 63 with removed 6 16 with removed 7 17 with removed 8 390 with removed 9 12 (or all, if less) 4digit reducible fractions: 1234/4936 => 124/496 removed 3 1239/6195 => 123/615 removed 9 1246/3649 => 126/369 removed 4 1249/2498 => 124/248 removed 9 1259/6295 => 125/625 removed 9 1279/6395 => 127/635 removed 9 1283/5132 => 128/512 removed 3 1297/2594 => 127/254 removed 9 1297/3891 => 127/381 removed 9 1298/2596 => 128/256 removed 9 1298/3894 => 128/384 removed 9 1298/5192 => 128/512 removed 9
Phix[edit]
function to_n(sequence digits, integer remove_digit=0)
if remove_digit!=0 then
integer d = find(remove_digit,digits)
digits[d..d] = {}
end if
integer res = digits[1]
for i=2 to length(digits) do
res = res*10+digits[i]
end for
return res
end function
function ndigits(integer n)
 generate numbers with unique digits efficiently
 and store them in an array for multiple reuse,
 along with an array of the removeddigit values.
sequence res = {},
digits = tagset(n),
used = repeat(1,n)&repeat(0,9n)
while true do
sequence nine = repeat(0,9)
for i=1 to length(used) do
if used[i] then
nine[i] = to_n(digits,i)
end if
end for
res = append(res,{to_n(digits),nine})
bool found = false
for i=n to 1 by 1 do
integer d = digits[i]
if not used[d] then ?9/0 end if
used[d] = 0
for j=d+1 to 9 do
if not used[j] then
used[j] = 1
digits[i] = j
for k=i+1 to n do
digits[k] = find(0,used)
used[digits[k]] = 1
end for
found = true
exit
end if
end for
if found then exit end if
end for
if not found then exit end if
end while
return res
end function
atom t0 = time(),
t1 = time()+1
for n=2 to 6 do
sequence d = ndigits(n)
integer count = 0
sequence omitted = repeat(0,9)
for i=1 to length(d)1 do
{integer xn, sequence rn} = d[i]
for j=i+1 to length(d) do
{integer xd, sequence rd} = d[j]
for k=1 to 9 do
integer yn = rn[k], yd = rd[k]
if yn!=0 and yd!=0 and xn/xd = yn/yd then
count += 1
omitted[k] += 1
if count<=12 then
printf(1,"%d/%d => %d/%d (removed %d)\n",{xn,xd,yn,yd,k})
elsif time()>t1 then
printf(1,"working (%d/%d)...\r",{i,length(d)})
t1 = time()+1
end if
end if
end for
end for
end for
printf(1,"%ddigit fractions found:%d, omitted %v\n\n",{n,count,omitted})
end for
?elapsed(time()t0)
 Output:
16/64 => 1/4 (removed 6) 19/95 => 1/5 (removed 9) 26/65 => 2/5 (removed 6) 49/98 => 4/8 (removed 9) 2digit fractions found:4, omitted {0,0,0,0,0,2,0,0,2} 132/231 => 12/21 (removed 3) 134/536 => 14/56 (removed 3) 134/938 => 14/98 (removed 3) 136/238 => 16/28 (removed 3) 138/345 => 18/45 (removed 3) 139/695 => 13/65 (removed 9) 143/341 => 13/31 (removed 4) 146/365 => 14/35 (removed 6) 149/298 => 14/28 (removed 9) 149/596 => 14/56 (removed 9) 149/894 => 14/84 (removed 9) 154/253 => 14/23 (removed 5) 3digit fractions found:122, omitted {0,0,9,1,6,15,16,15,60} 1234/4936 => 124/496 (removed 3) 1239/6195 => 123/615 (removed 9) 1246/3649 => 126/369 (removed 4) 1249/2498 => 124/248 (removed 9) 1259/6295 => 125/625 (removed 9) 1279/6395 => 127/635 (removed 9) 1283/5132 => 128/512 (removed 3) 1297/2594 => 127/254 (removed 9) 1297/3891 => 127/381 (removed 9) 1298/2596 => 128/256 (removed 9) 1298/3894 => 128/384 (removed 9) 1298/5192 => 128/512 (removed 9) 4digit fractions found:660, omitted {14,25,92,14,29,63,16,17,390} 12349/24698 => 1234/2468 (removed 9) 12356/67958 => 1236/6798 (removed 5) 12358/14362 => 1258/1462 (removed 3) 12358/15364 => 1258/1564 (removed 3) 12358/17368 => 1258/1768 (removed 3) 12358/19372 => 1258/1972 (removed 3) 12358/21376 => 1258/2176 (removed 3) 12358/25384 => 1258/2584 (removed 3) 12359/61795 => 1235/6175 (removed 9) 12364/32596 => 1364/3596 (removed 2) 12379/61895 => 1237/6185 (removed 9) 12386/32654 => 1386/3654 (removed 2) 5digit fractions found:5087, omitted {75,40,376,78,209,379,591,351,2988} 123459/617295 => 12345/61725 (removed 9) 123468/493872 => 12468/49872 (removed 3) 123469/173524 => 12469/17524 (removed 3) 123469/193546 => 12469/19546 (removed 3) 123469/213568 => 12469/21568 (removed 3) 123469/283645 => 12469/28645 (removed 3) 123469/493876 => 12469/49876 (removed 3) 123469/573964 => 12469/57964 (removed 3) 123479/617395 => 12347/61735 (removed 9) 123495/172893 => 12345/17283 (removed 9) 123548/679514 => 12348/67914 (removed 5) 123574/325786 => 13574/35786 (removed 2) 6digit fractions found:9778, omitted {230,256,921,186,317,751,262,205,6650} "10 minutes and 13s"
Python[edit]
def indexOf(haystack, needle):
idx = 0
for straw in haystack:
if straw == needle:
return idx
else:
idx += 1
return 1
def getDigits(n, le, digits):
while n > 0:
r = n % 10
if r == 0 or indexOf(digits, r) >= 0:
return False
le = 1
digits[le] = r
n = int(n / 10)
return True
def removeDigit(digits, le, idx):
pows = [1, 10, 100, 1000, 10000]
sum = 0
pow = pows[le  2]
i = 0
while i < le:
if i == idx:
i += 1
continue
sum = sum + digits[i] * pow
pow = int(pow / 10)
i += 1
return sum
def main():
lims = [ [ 12, 97 ], [ 123, 986 ], [ 1234, 9875 ], [ 12345, 98764 ] ]
count = [0 for i in range(5)]
omitted = [[0 for i in range(10)] for j in range(5)]
i = 0
while i < len(lims):
n = lims[i][0]
while n < lims[i][1]:
nDigits = [0 for k in range(i + 2)]
nOk = getDigits(n, i + 2, nDigits)
if not nOk:
n += 1
continue
d = n + 1
while d <= lims[i][1] + 1:
dDigits = [0 for k in range(i + 2)]
dOk = getDigits(d, i + 2, dDigits)
if not dOk:
d += 1
continue
nix = 0
while nix < len(nDigits):
digit = nDigits[nix]
dix = indexOf(dDigits, digit)
if dix >= 0:
rn = removeDigit(nDigits, i + 2, nix)
rd = removeDigit(dDigits, i + 2, dix)
if (1.0 * n / d) == (1.0 * rn / rd):
count[i] += 1
omitted[i][digit] += 1
if count[i] <= 12:
print "%d/%d = %d/%d by omitting %d's" % (n, d, rn, rd, digit)
nix += 1
d += 1
n += 1
i += 1
i = 2
while i <= 5:
print "There are %d %ddigit fractions of which:" % (count[i  2], i)
j = 1
while j <= 9:
if omitted[i  2][j] == 0:
j += 1
continue
print "%6s have %d's omitted" % (omitted[i  2][j], j)
j += 1
i += 1
return None
main()
 Output:
16/64 = 1/4 by omitting 6's 19/95 = 1/5 by omitting 9's 26/65 = 2/5 by omitting 6's 49/98 = 4/8 by omitting 9's 132/231 = 12/21 by omitting 3's 134/536 = 14/56 by omitting 3's 134/938 = 14/98 by omitting 3's 136/238 = 16/28 by omitting 3's 138/345 = 18/45 by omitting 3's 139/695 = 13/65 by omitting 9's 143/341 = 13/31 by omitting 4's 146/365 = 14/35 by omitting 6's 149/298 = 14/28 by omitting 9's 149/596 = 14/56 by omitting 9's 149/894 = 14/84 by omitting 9's 154/253 = 14/23 by omitting 5's 1234/4936 = 124/496 by omitting 3's 1239/6195 = 123/615 by omitting 9's 1246/3649 = 126/369 by omitting 4's 1249/2498 = 124/248 by omitting 9's 1259/6295 = 125/625 by omitting 9's 1279/6395 = 127/635 by omitting 9's 1283/5132 = 128/512 by omitting 3's 1297/2594 = 127/254 by omitting 9's 1297/3891 = 127/381 by omitting 9's 1298/2596 = 128/256 by omitting 9's 1298/3894 = 128/384 by omitting 9's 1298/5192 = 128/512 by omitting 9's 12349/24698 = 1234/2468 by omitting 9's 12356/67958 = 1236/6798 by omitting 5's 12358/14362 = 1258/1462 by omitting 3's 12358/15364 = 1258/1564 by omitting 3's 12358/17368 = 1258/1768 by omitting 3's 12358/19372 = 1258/1972 by omitting 3's 12358/21376 = 1258/2176 by omitting 3's 12358/25384 = 1258/2584 by omitting 3's 12359/61795 = 1235/6175 by omitting 9's 12364/32596 = 1364/3596 by omitting 2's 12379/61895 = 1237/6185 by omitting 9's 12386/32654 = 1386/3654 by omitting 2's There are 4 2digit fractions of which: 2 have 6's omitted 2 have 9's omitted There are 122 3digit fractions of which: 9 have 3's omitted 1 have 4's omitted 6 have 5's omitted 15 have 6's omitted 16 have 7's omitted 15 have 8's omitted 60 have 9's omitted There are 660 4digit fractions of which: 14 have 1's omitted 25 have 2's omitted 92 have 3's omitted 14 have 4's omitted 29 have 5's omitted 63 have 6's omitted 16 have 7's omitted 17 have 8's omitted 390 have 9's omitted There are 5087 5digit fractions of which: 75 have 1's omitted 40 have 2's omitted 376 have 3's omitted 78 have 4's omitted 209 have 5's omitted 379 have 6's omitted 591 have 7's omitted 351 have 8's omitted 2988 have 9's omitted
Racket[edit]
Racket's generator is horribly slow, so I roll my own more efficient generator. Pretty much using continuationpassing style, but then using macro to make it appear that we are writing in the direct style.
#lang racket
(require racket/generator
syntax/parse/define)
(definesyntaxparser for**
[(_ [x:id {~datum <} (e ...)] rst ...) #'(e ... (λ (x) (for** rst ...)))]
[(_ e ...) #'(begin e ...)])
(define (permutations xs n yield #:lower [lower #f])
(let loop ([xs xs] [n n] [acc '()] [lower lower])
(cond
[(= n 0) (yield (reverse acc))]
[else (for ([x (inlist xs)] #:when (or (not lower) (>= x (first lower))))
(loop (remove x xs)
(sub1 n)
(cons x acc)
(and lower (= x (first lower)) (rest lower))))])))
(define (list>number xs) (foldl (λ (e acc) (+ (* 10 acc) e)) 0 xs))
(define (calc n)
(define rng (range 1 10))
(ingenerator
(for** [numer < (permutations rng n)]
[denom < (permutations rng n #:lower numer)]
(for* (#:when (not (equal? numer denom))
[crossed (inlist numer)]
#:when (member crossed denom)
[numer* (invalue (list>number (remove crossed numer)))]
[denom* (invalue (list>number (remove crossed denom)))]
[numer** (invalue (list>number numer))]
[denom** (invalue (list>number denom))]
#:when (= (* numer** denom*) (* numer* denom**)))
(yield (list numer** denom** numer* denom* crossed))))))
(define (enumerate n)
(for ([x (calc n)] [i (inrange 12)])
(apply printf "~a/~a = ~a/~a (~a crossed out)\n" x))
(newline))
(define (stats n)
(define digits (makehash))
(for ([x (calc n)]) (hashupdate! digits (last x) add1 0))
(printf "There are ~a ~adigit fractions of which:\n" (for/sum ([(k v) (inhash digits)]) v) n)
(for ([digit (inlist (sort (hash>list digits) < #:key car))])
(printf " The digit ~a was crossed out ~a times\n" (car digit) (cdr digit)))
(newline))
(define (main)
(enumerate 2)
(enumerate 3)
(enumerate 4)
(enumerate 5)
(stats 2)
(stats 3)
(stats 4)
(stats 5))
(main)
 Output:
16/64 = 1/4 (6 crossed out) 19/95 = 1/5 (9 crossed out) 26/65 = 2/5 (6 crossed out) 49/98 = 4/8 (9 crossed out) 132/231 = 12/21 (3 crossed out) 134/536 = 14/56 (3 crossed out) 134/938 = 14/98 (3 crossed out) 136/238 = 16/28 (3 crossed out) 138/345 = 18/45 (3 crossed out) 139/695 = 13/65 (9 crossed out) 143/341 = 13/31 (4 crossed out) 146/365 = 14/35 (6 crossed out) 149/298 = 14/28 (9 crossed out) 149/596 = 14/56 (9 crossed out) 149/894 = 14/84 (9 crossed out) 154/253 = 14/23 (5 crossed out) 1234/4936 = 124/496 (3 crossed out) 1239/6195 = 123/615 (9 crossed out) 1246/3649 = 126/369 (4 crossed out) 1249/2498 = 124/248 (9 crossed out) 1259/6295 = 125/625 (9 crossed out) 1279/6395 = 127/635 (9 crossed out) 1283/5132 = 128/512 (3 crossed out) 1297/2594 = 127/254 (9 crossed out) 1297/3891 = 127/381 (9 crossed out) 1298/2596 = 128/256 (9 crossed out) 1298/3894 = 128/384 (9 crossed out) 1298/5192 = 128/512 (9 crossed out) 12349/24698 = 1234/2468 (9 crossed out) 12356/67958 = 1236/6798 (5 crossed out) 12358/14362 = 1258/1462 (3 crossed out) 12358/15364 = 1258/1564 (3 crossed out) 12358/17368 = 1258/1768 (3 crossed out) 12358/19372 = 1258/1972 (3 crossed out) 12358/21376 = 1258/2176 (3 crossed out) 12358/25384 = 1258/2584 (3 crossed out) 12359/61795 = 1235/6175 (9 crossed out) 12364/32596 = 1364/3596 (2 crossed out) 12379/61895 = 1237/6185 (9 crossed out) 12386/32654 = 1386/3654 (2 crossed out) There are 4 2digit fractions of which: The digit 6 was crossed out 2 times The digit 9 was crossed out 2 times There are 122 3digit fractions of which: The digit 3 was crossed out 9 times The digit 4 was crossed out 1 times The digit 5 was crossed out 6 times The digit 6 was crossed out 15 times The digit 7 was crossed out 16 times The digit 8 was crossed out 15 times The digit 9 was crossed out 60 times There are 660 4digit fractions of which: The digit 1 was crossed out 14 times The digit 2 was crossed out 25 times The digit 3 was crossed out 92 times The digit 4 was crossed out 14 times The digit 5 was crossed out 29 times The digit 6 was crossed out 63 times The digit 7 was crossed out 16 times The digit 8 was crossed out 17 times The digit 9 was crossed out 390 times There are 5087 5digit fractions of which: The digit 1 was crossed out 75 times The digit 2 was crossed out 40 times The digit 3 was crossed out 376 times The digit 4 was crossed out 78 times The digit 5 was crossed out 209 times The digit 6 was crossed out 379 times The digit 7 was crossed out 591 times The digit 8 was crossed out 351 times The digit 9 was crossed out 2988 times
Raku[edit]
(formerly Perl 6)
my %reduced;
my $digits = 2..4;
for $digits.map: *  1 > $exp {
my $start = sum (0..$exp).map( { 10 ** $_ * ($exp  $_ + 1) });
my $end = 10**($exp+1)  sum (^$exp).map( { 10 ** $_ * ($exp  $_) } )  1;
($start ..^ $end).race(:8degree, :3batch).map: > $den {
next if $den.contains: '0';
next if $den.comb.unique <= $exp;
for $start ..^ $den > $num {
next if $num.contains: '0';
next if $num.comb.unique <= $exp;
my $set = ($den.comb.head(*  1).Set ∩ $num.comb.skip(1).Set);
next if $set.elems < 1;
for $set.keys {
my $ne = $num.trans: $_ => '', :delete;
my $de = $den.trans: $_ => '', :delete;
if $ne / $de == $num / $den {
print "\b" x 40, "$num/$den:$_ => $ne/$de";
%reduced{"$num/$den:$_"} = "$ne/$de";
}
}
}
}
print "\b" x 40, ' ' x 40, "\b" x 40;
my $digit = $exp +1;
my %d = %reduced.pairs.grep: { .key.chars == ($digit * 2 + 3) };
say "\n({+%d}) $digit digit reduceable fractions:";
for 1..9 {
my $cnt = +%d.pairs.grep( *.key.contains: ":$_" );
next unless $cnt;
say " $cnt with removed $_";
}
say "\n 12 Random (or all, if less) $digit digit reduceable fractions:";
say " {.key.substr(0, $digit * 2 + 1)} => {.value} removed {.key.substr(*  1)}"
for %d.pairs.pick(12).sort;
}
 Sample output:
(4) 2 digit reduceable fractions: 2 with removed 6 2 with removed 9 12 Random (or all, if less) 2 digit reduceable fractions: 16/64 => 1/4 removed 6 19/95 => 1/5 removed 9 26/65 => 2/5 removed 6 49/98 => 4/8 removed 9 (122) 3 digit reduceable fractions: 9 with removed 3 1 with removed 4 6 with removed 5 15 with removed 6 16 with removed 7 15 with removed 8 60 with removed 9 12 Random (or all, if less) 3 digit reduceable fractions: 149/298 => 14/28 removed 9 154/352 => 14/32 removed 5 165/264 => 15/24 removed 6 176/275 => 16/25 removed 7 187/286 => 17/26 removed 8 194/291 => 14/21 removed 9 286/385 => 26/35 removed 8 286/682 => 26/62 removed 8 374/572 => 34/52 removed 7 473/572 => 43/52 removed 7 492/984 => 42/84 removed 9 594/693 => 54/63 removed 9 (660) 4 digit reduceable fractions: 14 with removed 1 25 with removed 2 92 with removed 3 14 with removed 4 29 with removed 5 63 with removed 6 16 with removed 7 17 with removed 8 390 with removed 9 12 Random (or all, if less) 4 digit reduceable fractions: 1348/4381 => 148/481 removed 3 1598/3196 => 158/316 removed 9 1783/7132 => 178/712 removed 3 1978/5934 => 178/534 removed 9 2971/5942 => 271/542 removed 9 2974/5948 => 274/548 removed 9 3584/4592 => 384/492 removed 5 3791/5798 => 391/598 removed 7 3968/7936 => 368/736 removed 9 4329/9324 => 429/924 removed 3 4936/9872 => 436/872 removed 9 6327/8325 => 627/825 removed 3
REXX[edit]
/*REXX pgm reduces fractions by "crossing out" matching digits in nominator&denominator.*/
parse arg high show . /*obtain optional arguments from the CL*/
if high==''  high=="," then high= 4 /*Not specified? Then use the default.*/
if show==''  show=="," then show= 12 /* " " " " " " */
say center(' some samples of reduced fractions by crossing out digits ', 79, "═")
$.=0 /*placeholder array for counts; init. 0*/
do L=2 to high; say /*do 2dig fractions to HIGHdig fract.*/
lim= 10**L  1 /*calculate the upper limit just once. */
do n=10**(L1) to lim /*generate some N digit fractions. */
if pos(0, n) \==0 then iterate /*Does it have a zero? Then skip it.*/
if hasDup(n) then iterate /* " " " " dup? " " " */
do d=n+1 to lim /*only process likesized #'s */
if pos(0, d)\==0 then iterate /*Have a zero? Then skip it. */
if verify(d, n, 'M')==0 then iterate /*No digs in common? Skip it.*/
if hasDup(d) then iterate /*Any digs are dups? " " */
q= n/d /*compute quotient just once. */
do e=1 for L; xo= substr(n, e, 1) /*try crossing out each digit.*/
nn= space( translate(n, , xo), 0) /*elide from the numerator. */
dd= space( translate(d, , xo), 0) /* " " " denominator. */
if nn/dd \== q then iterate /*Not the same quotient? Skip.*/
$.L= $.L + 1 /*Eureka! We found one. */
$.L.xo= $.L.xo + 1 /*count the silly reduction. */
if $.L>show then iterate /*Too many found? Don't show.*/
say center(n'/'d " = " nn'/'dd " by crossing out the" xo"'s.", 79)
end /*e*/
end /*d*/
end /*n*/
end /*L*/
say; @with= ' with crossedout' /* [↓] show counts for any reductions.*/
do k=1 for 9 /*traipse through each cross─out digit.*/
if $.k==0 then iterate /*Is this a zero count? Then skip it. */
say; say center('There are ' $.k " "k'digit fractions.', 79, "═")
@for= ' For ' /*literal for SAY indentation (below). */
do #=1 for 9; if $.k.#==0 then iterate
say @for k"digit fractions, there are " right($.k.#, k1) @with #"'s."
end /*#*/
end /*k*/
exit /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
hasDup: parse arg x; /* if L<2 then return 0 */ /*L will never be 1.*/
do i=1 for L1; if pos(substr(x,i,1), substr(x,i+1)) \== 0 then return 1
end /*i*/; return 0
 output when using the input of: 5 12
══════════ some samples of reduced fractions by crossing out digits ═══════════ 16/64 = 1/4 by crossing out the 6's. 19/95 = 1/5 by crossing out the 9's. 26/65 = 2/5 by crossing out the 6's. 49/98 = 4/8 by crossing out the 9's. 132/231 = 12/21 by crossing out the 3's. 134/536 = 14/56 by crossing out the 3's. 134/938 = 14/98 by crossing out the 3's. 136/238 = 16/28 by crossing out the 3's. 138/345 = 18/45 by crossing out the 3's. 139/695 = 13/65 by crossing out the 9's. 143/341 = 13/31 by crossing out the 4's. 146/365 = 14/35 by crossing out the 6's. 149/298 = 14/28 by crossing out the 9's. 149/596 = 14/56 by crossing out the 9's. 149/894 = 14/84 by crossing out the 9's. 154/253 = 14/23 by crossing out the 5's. 1234/4936 = 124/496 by crossing out the 3's. 1239/6195 = 123/615 by crossing out the 9's. 1246/3649 = 126/369 by crossing out the 4's. 1249/2498 = 124/248 by crossing out the 9's. 1259/6295 = 125/625 by crossing out the 9's. 1279/6395 = 127/635 by crossing out the 9's. 1283/5132 = 128/512 by crossing out the 3's. 1297/2594 = 127/254 by crossing out the 9's. 1297/3891 = 127/381 by crossing out the 9's. 1298/2596 = 128/256 by crossing out the 9's. 1298/3894 = 128/384 by crossing out the 9's. 1298/5192 = 128/512 by crossing out the 9's. 12349/24698 = 1234/2468 by crossing out the 9's. 12356/67958 = 1236/6798 by crossing out the 5's. 12358/14362 = 1258/1462 by crossing out the 3's. 12358/15364 = 1258/1564 by crossing out the 3's. 12358/17368 = 1258/1768 by crossing out the 3's. 12358/19372 = 1258/1972 by crossing out the 3's. 12358/21376 = 1258/2176 by crossing out the 3's. 12358/25384 = 1258/2584 by crossing out the 3's. 12359/61795 = 1235/6175 by crossing out the 9's. 12364/32596 = 1364/3596 by crossing out the 2's. 12379/61895 = 1237/6185 by crossing out the 9's. 12386/32654 = 1386/3654 by crossing out the 2's. ═══════════════════════There are 4 2digit fractions.════════════════════════ For 2digit fractions, there are 2 with crossedout 6's. For 2digit fractions, there are 2 with crossedout 9's. ══════════════════════There are 122 3digit fractions.═══════════════════════ For 3digit fractions, there are 9 with crossedout 3's. For 3digit fractions, there are 1 with crossedout 4's. For 3digit fractions, there are 6 with crossedout 5's. For 3digit fractions, there are 15 with crossedout 6's. For 3digit fractions, there are 16 with crossedout 7's. For 3digit fractions, there are 15 with crossedout 8's. For 3digit fractions, there are 60 with crossedout 9's. ══════════════════════There are 660 4digit fractions.═══════════════════════ For 4digit fractions, there are 14 with crossedout 1's. For 4digit fractions, there are 25 with crossedout 2's. For 4digit fractions, there are 92 with crossedout 3's. For 4digit fractions, there are 14 with crossedout 4's. For 4digit fractions, there are 29 with crossedout 5's. For 4digit fractions, there are 63 with crossedout 6's. For 4digit fractions, there are 16 with crossedout 7's. For 4digit fractions, there are 17 with crossedout 8's. For 4digit fractions, there are 390 with crossedout 9's. ══════════════════════There are 5087 5digit fractions.══════════════════════ For 5digit fractions, there are 75 with crossedout 1's. For 5digit fractions, there are 40 with crossedout 2's. For 5digit fractions, there are 376 with crossedout 3's. For 5digit fractions, there are 78 with crossedout 4's. For 5digit fractions, there are 209 with crossedout 5's. For 5digit fractions, there are 379 with crossedout 6's. For 5digit fractions, there are 591 with crossedout 7's. For 5digit fractions, there are 351 with crossedout 8's. For 5digit fractions, there are 2988 with crossedout 9's.
Ruby[edit]
def indexOf(haystack, needle)
idx = 0
for straw in haystack
if straw == needle then
return idx
else
idx = idx + 1
end
end
return 1
end
def getDigits(n, le, digits)
while n > 0
r = n % 10
if r == 0 or indexOf(digits, r) >= 0 then
return false
end
le = le  1
digits[le] = r
n = (n / 10).floor
end
return true
end
POWS = [1, 10, 100, 1000, 10000]
def removeDigit(digits, le, idx)
sum = 0
pow = POWS[le  2]
i = 0
while i < le
if i == idx then
i = i + 1
next
end
sum = sum + digits[i] * pow
pow = (pow / 10).floor
i = i + 1
end
return sum
end
def main
lims = [ [ 12, 97 ], [ 123, 986 ], [ 1234, 9875 ], [ 12345, 98764 ] ]
count = Array.new(5, 0)
omitted = Array.new(5) { Array.new(10, 0) }
i = 0
for lim in lims
n = lim[0]
while n < lim[1]
nDigits = [0] * (i + 2)
nOk = getDigits(n, i + 2, nDigits)
if not nOk then
n = n + 1
next
end
d = n + 1
while d <= lim[1] + 1
dDigits = [0] * (i + 2)
dOk = getDigits(d, i + 2, dDigits)
if not dOk then
d = d + 1
next
end
nix = 0
while nix < nDigits.length
digit = nDigits[nix]
dix = indexOf(dDigits, digit)
if dix >= 0 then
rn = removeDigit(nDigits, i + 2, nix)
rd = removeDigit(dDigits, i + 2, dix)
if (1.0 * n / d) == (1.0 * rn / rd) then
count[i] = count[i] + 1
omitted[i][digit] = omitted[i][digit] + 1
if count[i] <= 12 then
print "%d/%d = %d/%d by omitting %d's\n" % [n, d, rn, rd, digit]
end
end
end
nix = nix + 1
end
d = d + 1
end
n = n + 1
end
print "\n"
i = i + 1
end
i = 2
while i <= 5
print "There are %d %ddigit fractions of which:\n" % [count[i  2], i]
j = 1
while j <= 9
if omitted[i  2][j] == 0 then
j = j + 1
next
end
print "%6s have %d's omitted\n" % [omitted[i  2][j], j]
j = j + 1
end
print "\n"
i = i + 1
end
end
main()
 Output:
16/64 = 1/4 by omitting 6's 19/95 = 1/5 by omitting 9's 26/65 = 2/5 by omitting 6's 49/98 = 4/8 by omitting 9's 132/231 = 12/21 by omitting 3's 134/536 = 14/56 by omitting 3's 134/938 = 14/98 by omitting 3's 136/238 = 16/28 by omitting 3's 138/345 = 18/45 by omitting 3's 139/695 = 13/65 by omitting 9's 143/341 = 13/31 by omitting 4's 146/365 = 14/35 by omitting 6's 149/298 = 14/28 by omitting 9's 149/596 = 14/56 by omitting 9's 149/894 = 14/84 by omitting 9's 154/253 = 14/23 by omitting 5's 1234/4936 = 124/496 by omitting 3's 1239/6195 = 123/615 by omitting 9's 1246/3649 = 126/369 by omitting 4's 1249/2498 = 124/248 by omitting 9's 1259/6295 = 125/625 by omitting 9's 1279/6395 = 127/635 by omitting 9's 1283/5132 = 128/512 by omitting 3's 1297/2594 = 127/254 by omitting 9's 1297/3891 = 127/381 by omitting 9's 1298/2596 = 128/256 by omitting 9's 1298/3894 = 128/384 by omitting 9's 1298/5192 = 128/512 by omitting 9's 12349/24698 = 1234/2468 by omitting 9's 12356/67958 = 1236/6798 by omitting 5's 12358/14362 = 1258/1462 by omitting 3's 12358/15364 = 1258/1564 by omitting 3's 12358/17368 = 1258/1768 by omitting 3's 12358/19372 = 1258/1972 by omitting 3's 12358/21376 = 1258/2176 by omitting 3's 12358/25384 = 1258/2584 by omitting 3's 12359/61795 = 1235/6175 by omitting 9's 12364/32596 = 1364/3596 by omitting 2's 12379/61895 = 1237/6185 by omitting 9's 12386/32654 = 1386/3654 by omitting 2's There are 4 2digit fractions of which: 2 have 6's omitted 2 have 9's omitted There are 122 3digit fractions of which: 9 have 3's omitted 1 have 4's omitted 6 have 5's omitted 15 have 6's omitted 16 have 7's omitted 15 have 8's omitted 60 have 9's omitted There are 660 4digit fractions of which: 14 have 1's omitted 25 have 2's omitted 92 have 3's omitted 14 have 4's omitted 29 have 5's omitted 63 have 6's omitted 16 have 7's omitted 17 have 8's omitted 390 have 9's omitted There are 5087 5digit fractions of which: 75 have 1's omitted 40 have 2's omitted 376 have 3's omitted 78 have 4's omitted 209 have 5's omitted 379 have 6's omitted 591 have 7's omitted 351 have 8's omitted 2988 have 9's omitted
Visual Basic .NET[edit]
Module Module1
Function IndexOf(n As Integer, s As Integer()) As Integer
For ii = 1 To s.Length
Dim i = ii  1
If s(i) = n Then
Return i
End If
Next
Return 1
End Function
Function GetDigits(n As Integer, le As Integer, digits As Integer()) As Boolean
While n > 0
Dim r = n Mod 10
If r = 0 OrElse IndexOf(r, digits) >= 0 Then
Return False
End If
le = 1
digits(le) = r
n \= 10
End While
Return True
End Function
Function RemoveDigit(digits As Integer(), le As Integer, idx As Integer) As Integer
Dim pows = {1, 10, 100, 1000, 10000}
Dim sum = 0
Dim pow = pows(le  2)
For ii = 1 To le
Dim i = ii  1
If i = idx Then
Continue For
End If
sum += digits(i) * pow
pow \= 10
Next
Return sum
End Function
Sub Main()
Dim lims = {{12, 97}, {123, 986}, {1234, 9875}, {12345, 98764}}
Dim count(5) As Integer
Dim omitted(5, 10) As Integer
Dim upperBound = lims.GetLength(0)
For ii = 1 To upperBound
Dim i = ii  1
Dim nDigits(i + 2  1) As Integer
Dim dDigits(i + 2  1) As Integer
Dim blank(i + 2  1) As Integer
For n = lims(i, 0) To lims(i, 1)
blank.CopyTo(nDigits, 0)
Dim nOk = GetDigits(n, i + 2, nDigits)
If Not nOk Then
Continue For
End If
For d = n + 1 To lims(i, 1) + 1
blank.CopyTo(dDigits, 0)
Dim dOk = GetDigits(d, i + 2, dDigits)
If Not dOk Then
Continue For
End If
For nixt = 1 To nDigits.Length
Dim nix = nixt  1
Dim digit = nDigits(nix)
Dim dix = IndexOf(digit, dDigits)
If dix >= 0 Then
Dim rn = RemoveDigit(nDigits, i + 2, nix)
Dim rd = RemoveDigit(dDigits, i + 2, dix)
If (n / d) = (rn / rd) Then
count(i) += 1
omitted(i, digit) += 1
If count(i) <= 12 Then
Console.WriteLine("{0}/{1} = {2}/{3} by omitting {4}'s", n, d, rn, rd, digit)
End If
End If
End If
Next
Next
Next
Console.WriteLine()
Next
For i = 2 To 5
Console.WriteLine("There are {0} {1}digit fractions of which:", count(i  2), i)
For j = 1 To 9
If omitted(i  2, j) = 0 Then
Continue For
End If
Console.WriteLine("{0,6} have {1}'s omitted", omitted(i  2, j), j)
Next
Console.WriteLine()
Next
End Sub
End Module
 Output:
16/64 = 1/4 by omitting 6's 19/95 = 1/5 by omitting 9's 26/65 = 2/5 by omitting 6's 49/98 = 4/8 by omitting 9's 132/231 = 12/21 by omitting 3's 134/536 = 14/56 by omitting 3's 134/938 = 14/98 by omitting 3's 136/238 = 16/28 by omitting 3's 138/345 = 18/45 by omitting 3's 139/695 = 13/65 by omitting 9's 143/341 = 13/31 by omitting 4's 146/365 = 14/35 by omitting 6's 149/298 = 14/28 by omitting 9's 149/596 = 14/56 by omitting 9's 149/894 = 14/84 by omitting 9's 154/253 = 14/23 by omitting 5's 1234/4936 = 124/496 by omitting 3's 1239/6195 = 123/615 by omitting 9's 1246/3649 = 126/369 by omitting 4's 1249/2498 = 124/248 by omitting 9's 1259/6295 = 125/625 by omitting 9's 1279/6395 = 127/635 by omitting 9's 1283/5132 = 128/512 by omitting 3's 1297/2594 = 127/254 by omitting 9's 1297/3891 = 127/381 by omitting 9's 1298/2596 = 128/256 by omitting 9's 1298/3894 = 128/384 by omitting 9's 1298/5192 = 128/512 by omitting 9's 12349/24698 = 1234/2468 by omitting 9's 12356/67958 = 1236/6798 by omitting 5's 12358/14362 = 1258/1462 by omitting 3's 12358/15364 = 1258/1564 by omitting 3's 12358/17368 = 1258/1768 by omitting 3's 12358/19372 = 1258/1972 by omitting 3's 12358/21376 = 1258/2176 by omitting 3's 12358/25384 = 1258/2584 by omitting 3's 12359/61795 = 1235/6175 by omitting 9's 12364/32596 = 1364/3596 by omitting 2's 12379/61895 = 1237/6185 by omitting 9's 12386/32654 = 1386/3654 by omitting 2's There are 4 2digit fractions of which: 2 have 6's omitted 2 have 9's omitted There are 122 3digit fractions of which: 9 have 3's omitted 1 have 4's omitted 6 have 5's omitted 15 have 6's omitted 16 have 7's omitted 15 have 8's omitted 60 have 9's omitted There are 660 4digit fractions of which: 14 have 1's omitted 25 have 2's omitted 92 have 3's omitted 14 have 4's omitted 29 have 5's omitted 63 have 6's omitted 16 have 7's omitted 17 have 8's omitted 390 have 9's omitted There are 5087 5digit fractions of which: 75 have 1's omitted 40 have 2's omitted 376 have 3's omitted 78 have 4's omitted 209 have 5's omitted 379 have 6's omitted 591 have 7's omitted 351 have 8's omitted 2988 have 9's omitted
zkl[edit]
fcn toInt(digits,remove_digit=0){
if(remove_digit!=0) digits=digits.copy().del(digits.index(remove_digit));
digits.reduce(fcn(s,d){ s*10 + d });
}
fcn nDigits(n){
// generate numbers with unique digits efficiently
// and store them in an array for multiple reuse,
// along with an array of the removeddigit values.
res,digits := List(), n.pump(List(),'+(1)); // 1,2,3,4..n
used := List.createLong(n,1).extend(List.createLong(9n,0));
while(True){
nine:=List.createLong(9,0);
foreach i in (used.len()){ if(used[i]) nine[i]=toInt(digits,i+1) }
res.append(T(toInt(digits),nine));
found:=False;
foreach i in ([n1..0, 1]){
d:=digits[i];
if(not used[d1]) println("ack!");
used[d1]=0;
foreach j in ([d..8]){
if(not used[j]){
used[j]=1;
digits[i]=j+1;
foreach k in ([i+1..n1]){
digits[k] = used.find(0) + 1;
used[digits[k]  1]=1;
}
found=True;
break;
}
}
if(found) break;
}//foreach i
if(not found) break;
}//while
res
}
foreach n in ([2..5]){
rs,rsz,count,omitted := nDigits(n),rs.len()1, 0, List.createLong(9,0);
foreach i in (rsz){
xn,rn := rs[i];
foreach j in ([i+1..rsz]){
xd,rd := rs[j];
foreach k in ([0..8]){
yn,yd := rn[k],rd[k];
if(yn!=0 and yd!=0 and
xn.toFloat()/xd.toFloat() == yn.toFloat()/yd.toFloat()){
count+=1;
omitted[k]+=1;
if(count<=12)
println("%d/%d > %d/%d (removed %d)".fmt(xn,xd,yn,yd,k+1));
}
}
}
}
println("%ddigit fractions found: %d, omitted %s\n"
.fmt(n,count,omitted.concat(",")));
}
 Output:
16/64 > 1/4 (removed 6) 19/95 > 1/5 (removed 9) 26/65 > 2/5 (removed 6) 49/98 > 4/8 (removed 9) 2digit fractions found: 4, omitted 0,0,0,0,0,2,0,0,2 132/231 > 12/21 (removed 3) 134/536 > 14/56 (removed 3) 134/938 > 14/98 (removed 3) 136/238 > 16/28 (removed 3) 138/345 > 18/45 (removed 3) 139/695 > 13/65 (removed 9) 143/341 > 13/31 (removed 4) 146/365 > 14/35 (removed 6) 149/298 > 14/28 (removed 9) 149/596 > 14/56 (removed 9) 149/894 > 14/84 (removed 9) 154/253 > 14/23 (removed 5) 3digit fractions found: 122, omitted 0,0,9,1,6,15,16,15,60 1234/4936 > 124/496 (removed 3) 1239/6195 > 123/615 (removed 9) 1246/3649 > 126/369 (removed 4) 1249/2498 > 124/248 (removed 9) 1259/6295 > 125/625 (removed 9) 1279/6395 > 127/635 (removed 9) 1283/5132 > 128/512 (removed 3) 1297/2594 > 127/254 (removed 9) 1297/3891 > 127/381 (removed 9) 1298/2596 > 128/256 (removed 9) 1298/3894 > 128/384 (removed 9) 1298/5192 > 128/512 (removed 9) 4digit fractions found: 660, omitted 14,25,92,14,29,63,16,17,390 12349/24698 > 1234/2468 (removed 9) 12356/67958 > 1236/6798 (removed 5) 12358/14362 > 1258/1462 (removed 3) 12358/15364 > 1258/1564 (removed 3) 12358/17368 > 1258/1768 (removed 3) 12358/19372 > 1258/1972 (removed 3) 12358/21376 > 1258/2176 (removed 3) 12358/25384 > 1258/2584 (removed 3) 12359/61795 > 1235/6175 (removed 9) 12364/32596 > 1364/3596 (removed 2) 12379/61895 > 1237/6185 (removed 9) 12386/32654 > 1386/3654 (removed 2) 5digit fractions found: 5087, omitted 75,40,376,78,209,379,591,351,2988