Ken's POTW


Sums on a Triangle 
      a
     / \
    b - c
   / \ / \
  d - e - f
 / \ / \ / \
g - h - i - j
  1. Can you place the numbers 1-6 at the locations a-f on the triangle, such that every triangle (there are 5: abc, bde, bed, cef, adf) has a different sum, by adding the corners of each triangle?
    [This is not too hard, and it's interesting to note that once a solution is found, the additional three sums of the sides of the a-d-f triangle are also different. This is similar to the "Sums on a Cube" question of putting a different number on each side of a cube to create unique sums at the corners.]
  2. Can you place the numbers 1-10 at the locations a-j on the triangle, such that every triangle (there are 13) has a different sum? Consecutive sums?
Source: Original, but I wouldn't be surprised to find them somewhere else. Please feel free to let me know where, or let me know of a related puzzle.
Solutions were received from Kirk Bresniker: 
| 1. Can you place the numbers 1-6 at the locations a-f on the triangle,
|    such that every triangle (there are 5:  abc, bde, bed, cef, adf) has a
|    different sum, by adding the corners of each triangle?

Yes. Ignoring symmetries, there are 240 ways (none are consecutive). 
Here is one:

       1
     2   3
   4   6   5
sum of 0-1-2:   1+2+3 = 6
sum of 0-3-5:   1+4+5 = 10
sum of 1-3-4:   2+4+6 = 12
sum of 1-2-4:   2+3+6 = 11
sum of 2-4-5:   3+6+5 = 14

| 2. Can you place the numbers 1-10 at the locations a-j on the triangle,
|    such that every triangle (there are 13) has a different sum?

Ignoring symmetry, there are 7656 triangles with all different sums. Now
we can divide this by 6 (for the three orientations and their two
mirrors) to get 1276 triangles.

Of these 1276 triangles, there are 2 which also have sums which are also
consecutive:

         2
       1   10
     7   6   3
   5   8   9   4
sum of 0-1-2:   2+1+10 = 13
sum of 0-3-5:   2+7+3 = 12
sum of 0-6-9:   2+5+4 = 11
sum of 1-3-4:   1+7+6 = 14
sum of 1-2-4:   1+10+6 = 17
sum of 1-6-8:   1+5+9 = 15
sum of 2-4-5:   10+6+3 = 19
sum of 2-7-9:   10+8+4 = 22
sum of 3-6-7:   7+5+8 = 20
sum of 3-4-7:   7+6+8 = 21
sum of 4-7-8:   6+8+9 = 23
sum of 4-5-8:   6+3+9 = 18
sum of 5-8-9:   3+9+4 = 16

           *************
Sums are consecutive!
Each sum is different!

         6
       3   4
     2   5   10
   7   8   1   9
sum of 0-1-2:   6+3+4 = 13
sum of 0-3-5:   6+2+10 = 18
sum of 0-6-9:   6+7+9 = 22
sum of 1-3-4:   3+2+5 = 10
sum of 1-2-4:   3+4+5 = 12
sum of 1-6-8:   3+7+1 = 11
sum of 2-4-5:   4+5+10 = 19
sum of 2-7-9:   4+8+9 = 21
sum of 3-6-7:   2+7+8 = 17
sum of 3-4-7:   2+5+8 = 15
sum of 4-7-8:   5+8+1 = 14
sum of 4-5-8:   5+10+1 = 16
sum of 5-8-9:   10+1+9 = 20

          *************
Sums are consecutive!
Each sum is different!
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