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Use 10 Digits in 2 Fractions that Add to 1

Date: 05/26/2003 at 12:12:03
From: Walter Baeck
Subject: Use all 10 digits in 2 fractions that add to 1

I got this question on a quiz during the weekend; unfortunately, I 
couldn't find the answer for my team in the time that was available.

The idea is to obtain a sum of the form

   xx    xx
  --- + --- = 1
  xxx   xxx

(not necessarily with the number of digits at the 4 positions 
precisely as shown here). All digits 0-9 should be used exactly once.

How do you start with a problem that is so broad? Is this a well-known 
puzzle?

SPOILER

My intuition was to aim for the sum 1/2 + 1/2 = 1. Although I was 
unsuccessful, the answer given by the quiz master used this approach 
indeed:

     35   148
     -- + --- = 1
     70   296

Playing with it, I found trivial permutations that also work, like

     45   138
     -- + --- = 1
     90   276

I wonder if there is a vast array of possible solutions, or really 
just a handful. And I'm really interested by the question if all 
solutions need to be of "1/2 + 1/2" = 1 shape. Perhaps 1/3 + 2/3 = 1 
would also work, etc. ?

This question looks very similar to an answer in the Dr. Math 
archives:

   Lucky Seven Fractions Puzzle
   http://mathforum.org/library/drmath/view/55622.html 

The sum there is 7, however, and digit 0 is not used. I could not find 
my exact problem in a search, and I doubt that the methodical approach 
used in that answer can be imitated here.


Date: 05/27/2003 at 08:55:22
From: Doctor Mitteldorf
Subject: Re: Use all 10 digits in 2 fractions that add to 1

Dear Walter,

There are lots of solutions, most quite unexpected and difficult to
find, such as 91/273 + 056/84 = 1. I have little patience with
trial-and-error searches now that computers provide such an easy path
to victory, so I've written a short program to search exhaustively. 
Here is the program (DELPHI Pascal) and its output:

program Perms;
{$apptype CONSOLE}

var workstr  :string;
    fout     :text;

procedure Process(s :string);
          {Do what you like with the output here...}
          const tiny=1E-10;
          var ss :string;
              a,b,c,d,ta :integer;
          begin
          ss:=Copy(s,1,3);
          Val(ss,a,ta);
          ss:=Copy(s,4,3);
          Val(ss,b,ta);
          ss:=Copy(s,7,2);
          Val(ss,c,ta);
          ss:=Copy(s,9,2);
          Val(ss,d,ta);
          if (abs(a/b + c/d - 1)<tiny) then begin
             writeln(a,'/',b,' + ',c,'/',d,' = 1');
             writeln(fout,a,'/',b,' + ',c,'/',d,' = 1')
             end;
          if (abs(a/c + b/d - 1)<tiny) then begin
             writeln(a,'/',c,' + ',b,'/',d,' = 1');
             writeln(fout,a,'/',c,' + ',b,'/',d,' = 1')
             end;
          end;

procedure Permute_it(s :string);
          var ss    :string;
              i,len :integer;
          begin
          len:=length(s);
          if (len=0) then
            Process(workstr)
          else for i:=1 to len do
            begin
            workstr:=workstr+s[i];
            ss:=s;
            delete(ss,i,1);
            Permute_it(ss);
            Delete(workstr,length(workstr),1);
            end;
          end;

begin
assign(fout,'0-9Fract.txt'); rewrite(fout);
Permute_it('0123456789');
close(fout);
end.

24/138 + 57/69 = 1
27/189 + 54/63 = 1
28/146 + 59/73 = 1
32/184 + 57/69 = 1
34/158 + 62/79 = 1
37/629 + 48/51 = 1
42/637 + 85/91 = 1
42/651 + 87/93 = 1
48/156 + 27/39 = 1
52/364 + 78/91 = 1
52/376 + 81/94 = 1
58/174 + 26/39 = 1
58/174 + 62/93 = 1
61/793 + 48/52 = 1
64/192 + 38/57 = 1
67/134 + 29/58 = 1
69/138 + 27/54 = 1
73/146 + 29/58 = 1
73/219 + 56/84 = 1
79/158 + 23/46 = 1
79/158 + 32/64 = 1
87/192 + 35/64 = 1
89/267 + 34/51 = 1
91/273 + 56/84 = 1
91/832 + 57/64 = 1
93/186 + 27/54 = 1
93/651 + 72/84 = 1
96/312 + 54/78 = 1
96/324 + 57/81 = 1
105/623 + 74/89 = 1
109/327 + 56/84 = 1
135/270 + 48/96 = 1
138/276 + 45/90 = 1
140/368 + 57/92 = 1
143/528 + 70/96 = 1
145/290 + 38/76 = 1
148/296 + 35/70 = 1
169/507 + 32/48 = 1
185/370 + 46/92 = 1
186/372 + 45/90 = 1
204/867 + 39/51 = 1
207/549 + 38/61 = 1
208/793 + 45/61 = 1
231/609 + 54/87 = 1
267/534 + 9/18 = 1
269/538 + 7/14 = 1
269/807 + 34/51 = 1
273/406 + 19/58 = 1
273/546 + 9/18 = 1
276/345 + 18/90 = 1
284/710 + 39/65 = 1
287/369 + 10/45 = 1
293/586 + 7/14 = 1
306/459 + 27/81 = 1
307/614 + 29/58 = 1
307/921 + 56/84 = 1
308/462 + 19/57 = 1
309/618 + 27/54 = 1
310/465 + 29/87 = 1
315/609 + 42/87 = 1
327/654 + 9/18 = 1
329/658 + 7/14 = 1
351/702 + 48/96 = 1
357/408 + 12/96 = 1
369/574 + 10/28 = 1
372/465 + 18/90 = 1
375/480 + 21/96 = 1
381/762 + 45/90 = 1
386/579 + 4/12 = 1
391/476 + 5/28 = 1
405/729 + 36/81 = 1
416/598 + 7/23 = 1
417/695 + 32/80 = 1
426/710 + 38/95 = 1
451/902 + 38/76 = 1
473/528 + 10/96 = 1
481/962 + 35/70 = 1
485/970 + 13/26 = 1
485/970 + 16/32 = 1
485/970 + 31/62 = 1
486/972 + 15/30 = 1
504/623 + 17/89 = 1
507/819 + 24/63 = 1
531/708 + 24/96 = 1
540/972 + 36/81 = 1
609/783 + 12/54 = 1
630/945 + 27/81 = 1
638/957 + 4/12 = 1
678/904 + 13/52 = 1
715/832 + 9/64 = 1
728/936 + 10/45 = 1
735/840 + 12/96 = 1
748/935 + 12/60 = 1
754/928 + 3/16 = 1
845/923 + 6/71 = 1
864/912 + 3/57 = 1

- Doctor Mitteldorf, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
High School Calculators, Computers
High School Puzzles
Middle School Puzzles

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