Use 10 Digits in 2 Fractions that Add to 1Date: 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/ |
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