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square number is 1 (solve the Diophantine equation x 2 = y 3 + 4y, where y is even); generalized pentagonal number is 171535 (solve the Diophantine equation x 2 = y 3 + 144y + 144, where y is divisible by 12); tetrahedral number is 2925. Note that 0 and 1 are the only normal magic constants of rational order which are also rational squares.
Again, the last number of a row represents the number of new vertices to be added to generate the next higher n-cube. In this triangle, the sum of the elements of row m is equal to 3 m. Again, to use the elements of row 4 as an example: 1 + 8 + 24 + 32 + 16 = 81, which is equal to =.
For odd square, since there are (n - 1)/2 same sided rows or columns, there are (n - 1)(n - 3)/8 pairs of such rows or columns that can be interchanged. Thus, there are 2 (n - 1)(n - 3)/8 × 2 (n - 1)(n - 3)/8 = 2 (n - 1)(n - 3)/4 equivalent magic squares obtained by combining such interchanges. Interchanging all the same sided rows flips each ...
Add the clues together, plus 1 for each "space" in between. For example, if the clue is 6 2 3, this step produces the sum 6 + 1 + 2 + 1 + 3 = 13. Subtract this number from the total available in the row (usually the width or height of the puzzle). For example, if the clue in step 1 is in a row 15 cells wide, the difference is 15 - 13 = 2.
As the sum of two odd numbers, the total number of dominoes—vertical and horizontal—must be even. But to cover the mutilated chessboard, 31 dominoes are needed, an odd number. [ 19 ] [ 20 ] Another method counts the edges of each color around the boundary of the mutilated chessboard.
The sum of the labels is 11, smaller than could be achieved using only two labels. In graph theory, a sum coloring of a graph is a labeling of its vertices by positive integers, with no two adjacent vertices having equal labels, that minimizes the sum of the labels. The minimum sum that can be achieved is called the chromatic sum of the graph. [1]
The objective is to fill the grid with numbers from 1 to 9 in a way that the following conditions are met: Each row, column, and nonet contains each number exactly once. The sum of all numbers in a cage must match the small number printed in its corner. No number appears more than once in a cage.
The number of structurally distinct Latin squares (i.e. the squares cannot be made identical by means of rotation, reflection, and/or permutation of the symbols) for n = 1 up to 7 is 1, 1, 1, 12, 192, 145164, 1524901344 respectively (sequence A264603 in the OEIS).