superprismatic Posted August 2, 2011 Report Share Posted August 2, 2011 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? Quote Link to comment Share on other sites More sharing options...
0 Guest Posted August 2, 2011 Report Share Posted August 2, 2011 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. Quote Link to comment Share on other sites More sharing options...
0 superprismatic Posted August 2, 2011 Author Report Share Posted August 2, 2011 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? Quote Link to comment Share on other sites More sharing options...
0 plainglazed Posted August 3, 2011 Report Share Posted August 3, 2011 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. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted August 12, 2011 Report Share Posted August 12, 2011 (edited) 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 Quote Link to comment Share on other sites More sharing options...
Question
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?
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