[Guest]

a graceful ruler is one that: every number is unique and the subset sum of any consecutive numbers of the same length are also unique.

for example

1 2 4 3 - each number is unique.

3 6 7 4 - again, the sum of any 2 consecutive numbers is unique

7 9 8 6 - sum of any 3 consecutive numbers.

can you find a graceful ruler for the numbers 1-12?

[Guest]

1 2 3 4 5 6 7 8 9 10 11 12

[plasmid]

I think the OP intended to say that the ruler should be considered as a circular loop when you are taking sums, so for a ruler of length N one of the two-element sums would be element #N + element #1, and the three-element sums would include #(N-1) + #N + #1 and #N + #1 + #2. (At least, this assumption is based on the example given in the OP of the sums for ruler 1 2 4 3).

So

1 2 3 4 5 6 7 8 9 10 11 12

would include 12+1 and 6+7 to make it not a graceful ruler.

[Guest]

correct plasmid

[Guest]

I think you are confused with what a graceful ruler is. For a graceful ruler for the numbers 1-12 it would be as if you placed a limited number of united marks in a straight line and could get all the unit measurements only once with that ruler.

Spoiler wouldn't work for me but I found a graceful ruler for 1-13 [1,2,6,4]

Edited October 18, 2011 by tonyd

[Guest]

I don't get it, and I probably won't, but how is the set 1 2 4 3 unique? 1+2=3, so the three isn't really unique, is it?

I

[superprismatic]

1 4 2 7 3 5 6 8 9 10 11 12

pair sums: 5 6 9 10 8 11 14 17 19 21 23 13

triplet sums: 7 13 12 15 14 19 23 27 30 33 24 17

[Guest]

close supes, but not quite. with 12 numbers there are 11 lengths.

4+1+12+11+10 = 38

and

10+9+8+6+5 = 38

[superprismatic]

Oh, I misunderstood. So here's a corrected one:

1 2 5 3 6 8 4 7 9 10 11 12

3 7 8 9 14 12 11 16 19 21 23 13

8 10 14 17 18 19 20 26 30 33 24 15

11 16 22 21 25 28 30 37 42 34 26 20

17 24 26 28 34 38 41 49 43 36 31 23

25 28 33 37 44 49 53 50 45 41 34 29

29 35 42 47 55 61 54 52 50 44 40 37

36 44 52 58 67 62 56 57 53 50 48 41

45 54 63 70 68 64 61 60 59 58 52 48

55 65 75 71 70 69 64 66 67 62 59 57

66 77 76 73 75 72 70 74 71 69 68 67

[Guest]

congrats :-)