More Sums on a Triangle 

      .
     /a\
    .---.
   /b\c/d\
  .---.---.
 /e\f/g\h/i\
.---.---.---.
  1. Can you place the numbers 1-9 in the locations a-i on the triangle, such that every row of 5 smaller triangles (there are 3: abcef, acdhi, efghi) has the same sum? How many different sums are possible?
  2. Can you place the numbers 1-9 in the locations a-i on the triangle, such that the six rows (three of 5 triangles, three of 3 triangles) have different sums? Consecutive sums?

Source: Original.

Solutions were received from Louis Elrod, Nick Baxter and Bert Sevenhant: 
  1.        .
      /1\
     .---.
    /7\6/9\
   .---.---.
  /3\5/8\4/2\
 .---.---.---.
    Louis Elrod (Sum=22) 
           .
      /9\
     .---.
    /3\4/1\
   .---.---.
  /7\5/2\6/8\
 .---.---.---.
    Bert Sevenhant (Sum=28)     		Adding all three rows is the same as adding 1-9 twice, minus the three side positions. Letting those be 1,2,3 or 7,8,9, and dividing by 3 to get the common sum, we find the sums can only range from 22 through 28. Nick Baxter has theorized that 23 and 27 have no solutions. 
  2.        .
      /2\
     .---.
    /7\5/8\
   .---.---.
  /4\3/9\1/6\
 .---.---.---.
    Nick Baxter     		Nick says "[Getting] consecutive sums was more challenging." The sums are 3:(18,19,20), 5:(21,22,23).
On 4/13/98, Craig Gentry sent: For the first question, the following list should be exhaustive (up to rotations, flips, switches between positions "a" and "c", and the function that maps the number x to 10-x (and hence maps Elrod's triangle to Sevenhant's)). 
       .
      /8\
     .---.
    /1\6/2\
   .---.---.
  /9\4/3\5/7\     28
 .---.---.---.


       .
      /9\
     .---.
    /1\5/4\
   .---.---.
  /8\3/7\2/6\     26
 .---.---.---.


       .
      /8\
     .---.
    /1\6/4\
   .---.---.
  /9\2/7\3/5\     26
 .---.---.---.


       .
      /9\
     .---.
    /2\5/3\
   .---.---.
  /6\4/7\1/8\     26
 .---.---.---.


       .
      /8\
     .---.
    /2\6/3\
   .---.---.
  /9\1/7\4/5\     26
 .---.---.---.


       .
      /8\
     .---.
    /1\6/5\
   .---.---.
  /7\3/9\2/4\     25
 .---.---.---.


       .
      /9\
     .---.
    /2\4/5\
   .---.---.
  /7\3/8\1/6\     25
 .---.---.---.


       .
      /7\
     .---.
    /2\6/5\
   .---.---.
  /9\1/8\3/4\     25
 .---.---.---.


       .
      /8\
     .---.
    /3\4/5\
   .---.---.
  /9\1/7\2/6\     25
 .---.---.---.


       .
      /9\
     .---.
    /4\2/5\
   .---.---.
  /7\3/6\1/8\     25
 .---.---.---.


       .
      /8\
     .---.
    /4\3/5\
   .---.---.
  /9\1/6\2/7\     25
 .---.---.---.
There are at least 2 other solutions for problem 2: 
       .
      /3\
     .---.
    /8\5/9\
   .---.---.
  /1\7/6\4/2\
 .---.---.---.
       .
      /3\
     .---.
    /9\7/6\
   .---.---.
  /1\4/8\5/2\
 .---.---.---.
It's impossible for any of the sums to be smaller than 18, because it would force the b, d, and g positions to be impossibly large. Craig Gentry
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