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1
Two and Three Dimensional Silver Cubes
Taeler Porter and Scott GildemeyerAdvisor Dr.
Abdollah Khodkar
Department of Mathematics,
Conclusions and Future Work We 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.
Introduction

1 2
3 1
Results
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
a solution 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.
2 dimensional order 2
1 5 4 7
3 1 2 6
6 7 3 5
2 4 1 3
1 3 2 6 11 10
4 1 10 3 9 5
5 2 6 8 10 11
8 4 5 7 2 3
9 7 3 11 1 8
7 9 4 2 6 1
2 dimensional order 4
2 dimensional order 6
7 5 2
2 4 6
3 2 1
3 1 4
5 3 7
2 6 3
6 4 3
1 2 4
4 7 5
3 dimensional order 3
Methods
10 8 6 5
8 10 3 1
9 7 4 2
3 1 2 4
1 3 9 7
4 2 7 9
5 6 1 3
6 5 8 10
2 4 7 9
3 1 9 7
6 5 10 8
5 6 3 1
8 10 1 3
10 8 5 6
1 3 2 4
7 9 4 2
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), 4854. 3 P. J. Wan,
Near-optimal conflict-free channel set
assignments for an optical clusterbased hypercube
network, J. Comb. Optim., 1 (1997), pp. 179186.
2n 1 , 3n 2
These are the two equations used to figure out
the highest consecutive number in the order n.
3 dimensional order 4

• 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.

1 2 3 4 5
6 1 10 11 7
7 10 1 13 2
8 7 13 12 9
9 11 6 8 13
10 3 2 7 4
12 9 5 10 8
8 6 4 3 12
1 2 11 4 13
2 5 13 6 11
11 5 13 1 6
9 11 12 2 3
13 3 7 9 8
2 12 4 3 10
7 4 1 12 9
3 dimensional order 5
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.
12 4 8 10 1
10 5 6 13 12
1 2 9 5 3
6 8 5 11 4
4 3 7 1 2
13 6 11 8 9
2 7 1 8 4
11 1 10 6 13
5 10 2 13 7
3 12 4 9 5
nd-1 _ ( of repetition) multiple of d This 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.
6 7 1 9 10 8
3 1 2 10 8 9
1 4 5 8 9 10
12 13 11 15 16 1
13 11 12 3 1 2
11 12 13 1 4 14
5 1 4 10 8 9
1 6 7 8 9 10
2 3 1 9 10 8
13 11 12 14 1 4
11 12 13 1 15 16
12 13 11 2 3 1
1 2 3 8 9 10
4 5 1 9 10 8
7 1 6 10 8 9
11 12 13 1 2 3
12 13 11 4 14 1
13 11 12 16 1 15

• 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.

Further information tporter1_at_my.westga.edu
Taeler Porter email sgildem1_at_my.westga.edu
Scott Gildemeyer email akhodkar_at_westga.edu
Abdollah Khodkar email
3 dimensional order 6
16 14 15 11 1 4
14 15 16 1 12 13
15 16 14 2 3 1
8 1 4 7 5 6
1 9 10 5 6 7
2 3 1 6 7 5
16 15 14 12 13 1
16 14 15 3 1 2
14 15 16 1 4 11
9 10 1 6 7 5
3 1 2 7 5 6
1 4 8 5 6 7
14 15 16 1 2 3
15 16 14 4 11 1
16 14 15 13 1 12
1 2 3 5 6 7
4 8 1 6 7 5
10 1 9 7 5 6