[superprismatic]

Wolfgang's recent post, has me intrigued.

Suppose, you can pick any ten different values from the set {0,1,2,...,N} for the points A through J.

What is the smallest possible value of N for which it is possible for all five lines in the star to

add to the same value?

[Guest]

okay, we have 10 variables, 5 equations.

A+B+D+E = E +F +H +I = I +J +B +C = C +D +F +G = G+ H +J +A

since every equation must total the same value, each variable is paired with 3 other variables, and each variable is in two equations, we need a total of all 10 variables, *2, evenly divisable by 60.

then we either need all odd numbers, or all even numbers. (not entirely sure about this, but it makes the calculations easier.)

so we have...

(a +20a -2) *(10/2) = 60*n

21a -2= 12*n

so 12*n +2 needs to be evenly divisable by 21.

this never occurs. therefore we can rule out any +2 arithmetic sequence.

i'm sorely tempted to say that there is no set of numbers that work, but i dont see an easy proof of this.

my guess is, if there is such a sequence, the numbers are semi random values.

[superprismatic]

okay, we have 10 variables, 5 equations.

A+B+D+E = E +F +H +I = I +J +B +C = C +D +F +G = G+ H +J +A

since every equation must total the same value, each variable is paired with 3 other variables, and each variable is in two equations, we need a total of all 10 variables, *2, evenly divisable by 60.

then we either need all odd numbers, or all even numbers. (not entirely sure about this, but it makes the calculations easier.)

so we have...

(a +20a -2) *(10/2) = 60*n

21a -2= 12*n

so 12*n +2 needs to be evenly divisable by 21.

this never occurs. therefore we can rule out any +2 arithmetic sequence.

i'm sorely tempted to say that there is no set of numbers that work, but i dont see an easy proof of this.

my guess is, if there is such a sequence, the numbers are semi random values.

Anyway, here's a solution out of the set {0,1,2,...,21}:

A=10, B=8, C=12, D=6, E=20, F=17, G=9, H=4, I=3, J=21.

All lines sum to 44. But is 21 the smallest such number?

[plainglazed]

N=11 with A=9,B=2,C=11,D=5,E=4,F=1,G=3,H=8,I=7,J=0 all lines total 20.

am pretty sure N=9 is not possible. dont think there is an N=10 solution but just brute forced the above and am not certain i covered all the possibilities. had been solving for the lowest total instead of N before rereading the OP just now. Indeed an intriguing puzzle.

[Guest]

Agree with plainglazed

when n=11, there are solutions and the lowest sum = 20

120 ways to arrange the numbers to get that sum+solution:

A B C D E F G H I J

0 2 4 7 11 1 8 3 5 9

0 2 5 11 7 1 3 8 4 9

0 3 1 8 9 4 7 2 5 11

0 3 5 9 8 4 2 7 1 11

0 7 1 11 2 5 3 9 4 8

0 7 4 2 11 5 9 3 1 8

0 8 1 3 9 5 11 2 4 7

0 8 4 9 3 5 2 11 1 7

0 9 4 8 3 1 7 11 5 2

0 9 5 3 8 1 11 7 4 2

0 11 1 7 2 4 8 9 5 3

0 11 5 2 7 4 9 8 1 3

1 3 0 11 5 2 7 4 9 8

1 3 9 5 11 2 4 7 0 8

1 4 2 7 8 0 11 3 9 5

1 4 9 8 7 0 3 11 2 5

1 5 2 11 3 0 7 8 9 4

1 5 9 3 11 0 8 7 2 4

1 7 0 8 4 9 3 5 2 11

1 7 2 4 8 9 5 3 0 11

1 8 0 7 4 2 11 5 9 3

1 8 9 4 7 2 5 11 0 3

1 11 0 3 5 9 8 4 2 7

1 11 2 5 3 9 4 8 0 7

2 0 3 11 7 1 5 4 8 9

2 0 8 7 11 1 4 5 3 9

2 4 1 5 9 3 11 0 8 7

2 4 8 9 5 3 0 11 1 7

2 5 1 4 9 8 7 0 3 11

2 5 3 9 4 8 0 7 1 11

2 7 1 11 0 3 5 9 8 4

2 7 8 0 11 3 9 5 1 4

2 9 3 5 4 1 11 7 8 0

2 9 8 4 5 1 7 11 3 0

2 11 1 7 0 8 4 9 3 5

2 11 3 0 7 8 9 4 1 5

3 0 2 9 8 4 5 1 7 11

3 0 7 8 9 4 1 5 2 11

3 1 4 5 11 2 9 0 7 8

3 1 7 11 5 2 0 9 4 8

3 5 2 11 1 7 0 8 4 9

3 5 4 1 11 7 8 0 2 9

3 8 4 9 0 2 5 11 7 1

3 8 7 0 9 2 11 5 4 1

3 9 2 0 8 7 11 1 4 5

3 9 4 8 0 7 1 11 2 5

3 11 2 5 1 4 9 8 7 0

3 11 7 1 5 4 8 9 2 0

4 1 3 8 7 0 9 2 11 5

4 1 11 7 8 0 2 9 3 5

4 2 0 9 5 3 8 1 11 7

4 2 11 5 9 3 1 8 0 7

4 5 3 9 2 0 8 7 11 1

4 5 11 2 9 0 7 8 3 1

4 7 0 8 1 3 9 5 11 2

4 7 11 1 8 3 5 9 0 2

4 8 0 7 1 11 2 5 3 9

4 8 3 1 7 11 5 2 0 9

4 9 0 2 5 11 7 1 3 8

4 9 3 5 2 11 1 7 0 8

5 1 7 11 3 0 2 9 8 4

5 1 8 3 11 0 9 2 7 4

5 2 0 9 4 8 3 1 7 11

5 2 7 4 9 8 1 3 0 11

5 3 0 11 1 7 2 4 8 9

5 3 8 1 11 7 4 2 0 9

5 4 7 2 9 0 11 3 8 1

5 4 8 9 2 0 3 11 7 1

5 9 0 2 4 7 11 1 8 3

5 9 8 4 2 7 1 11 0 3

5 11 0 3 1 8 9 4 7 2

5 11 7 1 3 8 4 9 0 2

7 0 3 11 2 5 1 4 9 8

7 0 9 2 11 5 4 1 3 8

7 1 3 8 4 9 0 2 5 11

7 1 5 4 8 9 2 0 3 11

7 2 5 11 0 3 1 8 9 4

7 2 9 0 11 3 8 1 5 4

7 4 5 1 8 3 11 0 9 2

7 4 9 8 1 3 0 11 5 2

7 8 3 1 4 5 11 2 9 0

7 8 9 4 1 5 2 11 3 0

7 11 3 0 2 9 8 4 5 1

7 11 5 2 0 9 4 8 3 1

8 0 2 9 3 5 4 1 11 7

8 0 11 3 9 5 1 4 2 7

8 1 5 4 7 2 9 0 11 3

8 1 11 7 4 2 0 9 5 3

8 3 5 9 0 2 4 7 11 1

8 3 11 0 9 2 7 4 5 1

8 4 2 7 1 11 0 3 5 9

8 4 5 1 7 11 3 0 2 9

8 7 2 4 1 5 9 3 11 0

8 7 11 1 4 5 3 9 2 0

8 9 2 0 3 11 7 1 5 4

8 9 5 3 0 11 1 7 2 4

9 0 7 8 3 1 4 5 11 2

9 0 11 3 8 1 5 4 7 2

9 2 7 4 5 1 8 3 11 0

9 2 11 5 4 1 3 8 7 0

9 3 1 8 0 7 4 2 11 5

9 3 11 0 8 7 2 4 1 5

9 4 1 5 2 11 3 0 7 8

9 4 7 2 5 11 0 3 1 8

9 5 1 4 2 7 8 0 11 3

9 5 11 2 4 7 0 8 1 3

9 8 1 3 0 11 5 2 7 4

9 8 7 0 3 11 2 5 1 4

11 0 8 7 2 4 1 5 9 3

11 0 9 2 7 4 5 1 8 3

11 1 4 5 3 9 2 0 8 7

11 1 8 3 5 9 0 2 4 7

11 2 4 7 0 8 1 3 9 5

11 2 9 0 7 8 3 1 4 5

11 3 8 1 5 4 7 2 9 0

11 3 9 5 1 4 2 7 8 0

11 5 4 1 3 8 7 0 9 2

11 5 9 3 1 8 0 7 4 2

11 7 4 2 0 9 5 3 8 1

11 7 8 0 2 9 3 5 4 1

Edited August 12, 2011 by gadaju