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Introduction
Methods
Acknowledgments
We would like to acknowledge the National Science Foundation STEP grant #DUE-0336571 for the opportunity to research. Also, the GEMS Summer Fellows Program for the experience of research and skills.
Conclusions and Future WorkWe have learned to correctly use the formulas to help find possible numbers to place in the silver cells. Using these formulas we found different possibilities of two and three dimensional silver cubes. From our research we have found the three dimensional order of 7, found by Kevin Ventullo, which was the goal for this research. Currently we are working on finding the algorithm for the 7 x 7. The next step is to find the three dimensional order eleven and its algorithm.
Taeler Porter and Scott GildemeyerAdvisor: Dr. Abdollah Khodkar
Literature cited[1] F. J. MacWilliams and N. J. A. Sloane,
The theory of error-correcting codes II, North-Holland Publishing Co., Amsterdam, 1977.
[2] M. Mahdian and E. S. Mahmoodian, The roots of an IMO97 problem, Bull. Inst. Combin. Appl. 28 (2000), 48–54.
[3] P. J. Wan, Near-optimal conflict-free channel set assignments for an optical clusterbased hypercube network, J. Comb. Optim., 1 (1997), pp. 179–186.
Further information
[email protected] – Taeler Porter [email protected] – Scott Gildemeyer
[email protected] – Abdollah Khodkar
Two and Three Dimensional Silver Cubes
Department of Mathematics,
1 23 1
1 5 4 73 1 2 66 7 3 52 4 1 3
1 3 2 6 11 104 1 10 3 9 55 2 6 8 10 118 4 5 7 2 39 7 3 11 1 87 9 4 2 6 1
2 dimensional order 6
2 dimensional order 2
2 dimensional order 4
7 5 2
2 4 6
3 2 1
3 1 4
5 3 7
2 6 3
6 4 31 2 4
4 7 53 dimensional order 3
1 3 9 74 2 7 95 6 1 36 5 8 10
2 4 7 93 1 9 76 5 10 85 6 3 1
8 10 1 310 8 5 61 3 2 47 9 4 2
10 8 6 58 10 3 19 7 4 23 1 2 4
3 dimensional order 4
nd-1 _ (# of repetition)= multiple of dThis is the equation to find possibilities of silvers for the dimension d. If the equation is true, then there is a possibility that the number of repetitions will work in order to create the silver cube of order n.
Example•2 dimensional order of 4 with numbers repeating 3 times. 42 – (3) = multiple of 2
16 – 3 = 13
The repetition of 3 is not a possibility.
•3 dimensional order of 4 with numbers repeating 4 times. 42 – 4 = multiple of 3
16 – 4 = 12
The repetition of 4 is a possibility.
2n – 1 , 3n – 2
These are the two equations used to figure out the highest consecutive number in the order n.
Results
1 2 3 4 56 1 10 11 77 10 1 13 28 7 13 12 99 11 6 8 13
10 3 2 7 412 9 5 10 88 6 4 3 121 2 11 4 132 5 13 6 11
11 5 13 1 69 11 12 2 3
13 3 7 9 82 12 4 3 107 4 1 12 9
12 4 8 10 110 5 6 13 121 2 9 5 36 8 5 11 44 3 7 1 2
13 6 11 8 92 7 1 8 4
11 1 10 6 135 10 2 13 73 12 4 9 5
3 dimensional order 5
1 2 3 8 9 104 5 1 9 10 87 1 6 10 8 9
11 12 13 1 2 312 13 11 4 14 113 11 12 16 1 15
6 7 1 9 10 83 1 2 10 8 91 4 5 8 9 10
12 13 11 15 16 113 11 12 3 1 211 12 13 1 4 14
5 1 4 10 8 91 6 7 8 9 102 3 1 9 10 8
13 11 12 14 1 411 12 13 1 15 1612 13 11 2 3 1
14 15 16 1 2 315 16 14 4 11 116 14 15 13 1 121 2 3 5 6 74 8 1 6 7 5
10 1 9 7 5 6
16 15 14 12 13 116 14 15 3 1 214 15 16 1 4 119 10 1 6 7 53 1 2 7 5 61 4 8 5 6 7
16 14 15 11 1 414 15 16 1 12 1315 16 14 2 3 18 1 4 7 5 61 9 10 5 6 72 3 1 6 7 5
3 dimensional
order 6
An n × n matrix A is said to be silver if, for every i = 1, 2, . . . , n, each symbol in {1, 2, . . . , 2n − 1} appears either in the ith row or the ith column of A. A problem of the 38th International Mathematical Olympiad in 1997 introduced this definition and asked to prove that no silver matrix of order 1997 exists. In [2] the motivation behind this problem as well as asolution is presented: a silver matrix of order n exists if and only if n = 1 or n is even for two dimensional silver matrix. The next step was to find three dimensional silver matrix. All were found up to the 7 x 7. It was an open problem and we were chosen to do the research to find the 3 dimensional matrix of order 7.
Example
•2 dimensional order of 4. 2(4) – 1 = 7 so the cube would go 1 – 7.
•3 dimensional order of 4. 3(4) – 2 = 10 so the cube would go 1 – 10.
