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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.
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 ...
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]
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.
Labels like red and blue are only used when the number of colors is small, and normally it is understood that the labels are drawn from the integers {1, 2, 3, ...}. A coloring using at most k colors is called a (proper) k-coloring. The smallest number of colors needed to color a graph G is called its chromatic number, and is often denoted χ(G).
The average of these numbers is obviously n(n + 1) / 2 , and there are n(n + 1) / 2 of them, so their sum is ( n(n + 1) / 2 ) 2. Many early mathematicians have studied and provided proofs of Nicomachus's theorem. Stroeker (1995) claims that "every student of number theory surely must have marveled at this miraculous fact".
Secondary constipation, which is due to another cause, such as chronic conditions, like Type 2 diabetes, poor dietary choices or neurological conditions that may impact intestinal motility
The sum of the elements of a single row is twice the sum of the row preceding it. For example, row 0 (the topmost row) has a value of 1, row 1 has a value of 2, row 2 has a value of 4, and so forth. This is because every item in a row produces two items in the next row: one left and one right.