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Get the right product (Posted on 2015-12-01) Difficulty: 3 of 5
For a positive integer n > 2, consider the n-1 fractions 2/1; 3/2; ...; n/(n-1).

The product of these fractions equals n, but if you reciprocate (i.e. turn upside down) some of the fractions, the product will change.

How can you make the product equal 1?
Find all values of n for which this is possible.

No Solution Yet Submitted by Ady TZIDON    
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Some Thoughts Possible for 9 | Comment 4 of 5 |
The sequence  1  1  0  0  0  0  0  0  1 below is interpreted as

2/1 and 3/2 are left alone, while the rest are inverted (reciprocated). the product is 1.

2/1 * 3/2 = 3

3/4 * 4/5 * ... * 8/9 = 1/3

The product is 1.

Note that other occurrences of 1 are also found below, with the sequence code being interpreted the same way as to which fractions are inverted.

All the possible products:

 1//9 
 9//64 
 64//441 
 9//49 
 49//324 
 49//256 
 16//81 
 1//4 
 4//25 
 81//400 
 256//1225 
 324//1225 
 49//225 
 441//1600 
 64//225 
 9//25 
 25//144 
 225//1024 
 100//441 
 225//784 
 1225//5184 
 1225//4096 
 25//81 
 25//64 
 1//4 
 81//256 
 16//49 
 81//196 
 49//144 
 441//1024 
 4//9 
 9//16 
 16//81 
 1//4 
 1024//3969 
 16//49 
 196//729 
 49//144 
 256//729 
 4//9 
 64//225 
 9//25 
 4096//11025 
 576//1225 
 784//2025 
 49//100 
 1024//2025 
 16//25 
 25//81 
 25//64 
 1600//3969 
 25//49 
 1225//2916 
 1225//2304 
 400//729 
 25//36 
 4//9 
 9//16 
 256//441 
 36//49 
 49//81 
 49//64 
 64//81 
 1 
 0  0  1  1  1  1  1  1  1 
 1//4 
 81//256 
 16//49 
 81//196 
 49//144 
 441//1024 
 4//9 
 9//16 
 9//25 
 729//1600 
 576//1225 
 729//1225 
 49//100 
 3969//6400 
 16//25 
 81//100 
 25//64 
 2025//4096 
 25//49 
 2025//3136 
 1225//2304 
 11025//16384 
 25//36 
 225//256 
 9//16 
 729//1024 
 36//49 
 729//784 
 49//64 
 3969//4096 
 1 
 0  1  0  1  1  1  1  0  1 
 81//64 
 4//9 
 9//16 
 256//441 
 36//49 
 49//81 
 49//64 
 64//81 
 1 
 0  1  1  0  0  1  1  1  1 
 16//25 
 81//100 
 1024//1225 
 1296//1225 
 196//225 
 441//400 
 256//225 
 36//25 
 25//36 
 225//256 
 400//441 
 225//196 
 1225//1296 
 1225//1024 
 100//81 
 25//16 
 1 
 0  1  1  1  1  0  0  0  1 
 81//64 
 64//49 
 81//49 
 49//36 
 441//256 
 16//9 
 9//4 
 4//9 
 9//16 
 256//441 
 36//49 
 49//81 
 49//64 
 64//81 
 1 
 1  0  0  0  0  1  1  1  1 
 16//25 
 81//100 
 1024//1225 
 1296//1225 
 196//225 
 441//400 
 256//225 
 36//25 
 25//36 
 225//256 
 400//441 
 225//196 
 1225//1296 
 1225//1024 
 100//81 
 25//16 
 1 
 1  0  0  1  1  0  0  0  1 
 81//64 
 64//49 
 81//49 
 49//36 
 441//256 
 16//9 
 9//4 
 64//81 
 1 
 1  0  1  0  0  0  0  1  1 
 4096//3969 
 64//49 
 784//729 
 49//36 
 1024//729 
 16//9 
 256//225 
 36//25 
 16384//11025 
 2304//1225 
 3136//2025 
 49//25 
 4096//2025 
 64//25 
 100//81 
 25//16 
 6400//3969 
 100//49 
 1225//729 
 1225//576 
 1600//729 
 25//9 
 16//9 
 9//4 
 1024//441 
 144//49 
 196//81 
 49//16 
 256//81 
 4 
 1  0  1  1  1  1  1  1  4 
 1 
 1  1  0  0  0  0  0  0  1 
 81//64 
 64//49 
 81//49 
 49//36 
 441//256 
 16//9 
 9//4 
 36//25 
 729//400 
 2304//1225 
 2916//1225 
 49//25 
 3969//1600 
 64//25 
 81//25 
 25//16 
 2025//1024 
 100//49 
 2025//784 
 1225//576 
 11025//4096 
 25//9 
 225//64 
 9//4 
 729//256 
 144//49 
 729//196 
 49//16 
 3969//1024 
 4 
 1  1  0  1  1  1  1  0  4 
 81//16 
 16//9 
 9//4 
 1024//441 
 144//49 
 196//81 
 49//16 
 256//81 
 4 
 1  1  1  0  0  1  1  1  4 
 64//25 
 81//25 
 4096//1225 
 5184//1225 
 784//225 
 441//100 
 1024//225 
 144//25 
 25//9 
 225//64 
 1600//441 
 225//49 
 1225//324 
 1225//256 
 400//81 
 25//4 
 4 
 1  1  1  1  1  0  0  0  4 
 81//16 
 256//49 
 324//49 
 49//9 
 441//64 
 64//9 
 9 
 1  1  1  1  1  1  1  1  9 
 


     5 kill "riteprod.txt"
    10   dim Fr(9),h(9)
    20   for I=1 to 8
    30       Fr(I)=(I+1)//I
    40   next
    45   open "riteprod.txt" for output as #2
    46   tot=1
    50   gosub *addOn(1)
    55   close #2
    60   end
    70   *addOn(wh)
    75     local newfr,newfrNo
    80       for newfrNo=0 to 1
    85           h(wh)=newfrNo
    90           if newfrNo=1 then newfr=fr(wh):else newfr=1//fr(wh):endif
   100           tot=tot*newfr
   110           if wh=8 then
   120             :print #2,tot
   122             :if tot=int(tot) then
   123               :for i=1 to 8
   124                 : print h(i);:print #2,h(i);
   125               :next
   126               :print tot:print #2,tot
   127             :endif
   130            :else
   140             :gosub *addOn(wh+1)
   150           :endif
   160           tot=tot//newfr
   170       next
  180   return

Edited on December 2, 2015, 11:18 am
  Posted by Charlie on 2015-12-02 11:11:35

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