Amusement devices: games – Puzzles – Jumping movement
Reexamination Certificate
2001-05-29
2003-09-30
Wong, Steven (Department: 3711)
Amusement devices: games
Puzzles
Jumping movement
Reexamination Certificate
active
06626431
ABSTRACT:
FIELD OF THE INVENTION
The invention relates to the field of puzzles and in particular to a mathematical puzzle in form of a cube with rotatable sections, whose goal is to rotate the pieces of the cubic puzzle so as to complete a “Magic Square” on each of the six faces of the puzzle.
The physical components of the puzzle include a cubic rotatable block type of puzzle that in its simplest form (a 3×3 array on each of the six faces) would be comprised of 27 individual sections. 26 of the sections are visible and the 27th is the core section in the center of the puzzle upon which the other sections rotate. Other means to achieve the rotation of the core section may be used without varying from the spirit of the invention.
This underlying structure may be described by reference to the RUBIK'S CUBE, (trademarked name for cubic puzzle) which would have the same number of rotating sections (26) that this puzzle would have in its simplest form. Moreover, the underlying physical construction of a center section in connection with the peripheral sections suggests that this sort of construction can be used in the present invention. There may be more sections (for higher order arrays, e.g. 4×4 and 5×5) but the underlying principle of a central section in connection with the peripheral sections that allow sections of the puzzle to move together would be established.
Each face of the puzzle is divided into 9 sections; in the case of Rubik's Cube, the 9 sections are supposed to all be of the same color when the puzzle is solved correctly. In the case of the invention herein, the 9 sections will form a “magic cube” when the sections are correctly lined up which means that all of the cells in the array on each of the six faces will add up to the same number. The cells will add up orthogonally (up/down or left/right) as well as diagonally “in space”, see below.
BACKGROUND OF THE INVENTION
It is first necessary to define what is meant by a magic cube for purposes of this invention and to determine what will constitute completing the puzzle.
W. S. Andrews first defined a magic square as a series of numbers so arranged in a square that the sum of each row and column and of both the corner diagonals shall by the same amount which may be termed the summation. (W. S. Andrews, Magic Squares and Cubes, 2nd edition, Dover Publication Inc. New York, 1917, p 1)
Martin Gardner defined a standard magic square as;
“ . . . a square array of positive integers from 1 through N
2
arranged so that the sum of every row, every column, and each of the two main diagonals is the same. N is the “order” of the square. It is easy to see that the magic constant is the sum of all the numbers divided by N. The formula is;
(1+2+3
. . . +N
2
)/
N
=½(
N
3
+N
)
The trivial square of order 1 is simply the number 1 and of course it is unique. It is equally trivial to prove that no order-2 square is possible” (Martin Gardner, Time Travel and Other Mathematical Bewilderments, W H Freeman and Co. New York, 1988, p 214.
By way of illustration;
FIG. 1
shows a 3×3 magic square. The sum of the numbers of any row column or diagonal (a diagonal drawn through the center cell) adds up to 15. Note also that the sum of any two opposite numbers (e.g. 3 and 7 in this example) in the magic square is 10 which is twice that of the center number (5 in this case) or N
2
+1.
FIG. 2
shows an example of a 4×4 magic square where the sum of any column, row or corner diagonal is 34. The sum of two opposite numbers is 17 which is the sum of the first number (1) and last number (16) of the series in this case.
FIG. 3
shows a 5×5 magic square. Again with the same properties when the columns, rows and diagonals are added up. The sum of two opposite numbers is twice that of the center number or n
2
+1.
As it turns out however, such magic squares do not exist that meet, precisely, these requirements when we move to three dimensional arrays. I.e. arrays that are arranged in space so that one can sum them up along different dimensions.
Again Gardner
“It is natural to extend the concept of magic squares to three dimension and even higher ones. A perfect magic cube is a cubical array of positive integers from 1 to N
3
such that every straight line of N cells adds up to a constant. These lines include the orthogonal and two main diagonals of every orthogonal cross section and the four space diagonals. The constant is;
(1+2+3
. . . +N
3
)/
N
2
=½(
N
4
+N
)
“There is of course, a unique perfect cube of order 1 and it is trivially true that there is none of order 2. Is there one of order 3? Unfortunately, 3 does not quite make it . . . Annoyed by the refusal of such a cube to exists, magic cube buffs have relaxed the requirements to define a species of semi-perfect cube that apparently does exist in all orders higher than 2. These are cubes where only the orthogonals and four space diagonals are magic. Let us call them Andrews cubes since W. S. Andrews devotes two chapters to them in his pioneering Magic Squares and Cubes.” (1917). The order 3 Andrews cube must be associative, with 14 in its center. There are four such cubes, not counting rotations and reflections. All are given by Andrews, although he seems not to have realized that they exhaust all basic types. (Gardner, Time Travel and Other Mathematical Bewilderments, p. 219).
It is with respect to this type of “Andrews Cube” that Gardner refers to that we will refer to as a “Magic Cube” for purposes of this invention. Note that this means that only the orthogonals (rows and columns illustrated by arrows
7
/
8
in
FIG. 8
) and the four “space diagonals” (arrow
3
in
FIG. 8
) meet the definition of the sums being the same. Ordinary diagonals (as one goes in a diagonal direction across the face of the cube) will not necessarily sum to the same number.
Note in contrast to “ordinary diagonals” that a “space diagonal” means a line drawn from a corner cell through the imaginary center (note again the center section is not visible to the player) and continuing in a straight line till it reaches the corner section that is opposite from the corner we started at. See arrow
3
running through the center of the 3×3×3 cube in FIG.
8
and having end points in a corner cube for a total of three numerical values.
There are four such center based, “space diagonals” in a cube and in a 3×3 array, this space diagonal must have 3 numbers that are summed together (just like the orthogonals in a 3×3×3 cube). Two of the numerical values corresponding to the corner sections of the puzzle and the other value corresponding to an imaginary center section for a total of three numberes to produce the “magic sum.” (A “corner section” means like cubic section
1
in FIGS.
6
/
8
)
The numerical value of the “center core” section may be imagined by the user because it cannot be seen when the puzzle is in normal use, and hence, there is no need to physically put a numerical value on that piece of the puzzle.
The same sort of relation holds with respect to 4×4×4 and higher order arrays. In the case of a 4×4×4 cube, the central core that cannot be seen will form a 2×2×2 cube (i.e. inside the larger 4×4×4 cube). See
FIG. 9
;
5
denotes the “central core” (normally unseen by the user).
The space diagonal in the 4×4×4 will thus cut through the center of this 2×2×2 core and thereby hit two members of the smaller 2×2×2 cube. So the same relationship holds, as above, only this time we will use four numbers to be summed up. This is true for each space diagonal as well as the orthogonals.
The same is true for 5×5×5 arrays with the central core in this case being 3×3×3 cube that is unseen by the user. In this case, a space diagonal will start/end at the two corners of the 5×5×5 and three cubes from the central 3×3×3 core will together form anot
Halvonik John P.
Wong Steven
LandOfFree
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