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
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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