View source for 9855

{{Short description|Natural number}}
{{infobox number
|number=9855
|divisor=1, 3, 5, 9, 15, 27, 45, 73, 135, 219, 365, 657, 1095, 1971, 3285, 9855
}}
'''9855 (nine thousand eight hundred fifty-five)''' is an odd, composite, four-digit number.  The number 9855 is the [[magic constant]] of an ''n'' × ''n'' normal [[magic square]] as well as [[Eight queens puzzle|n-Queens Problem]] for ''n'' = 27.<ref>{{Cite journal |last=ANDREWS |first=W.S. |date=1908 |title=MAGIC SQUARES AND CUBES Magic Squares and Pythagorean Numbers |url= https://zenodo.org/record/2156773|journal=Monist |volume=16 |issue=3 |pages=422–433 |doi=10.5840/monist190616319 |issn=0026-9662}}</ref> It can be expressed as the product of its prime factors:

:<math>9855 = 3^3 \times 5 \times 73</math><ref name=":0">{{Cite journal |last=Browne |first=C. A. |date=1906 |title=Magic Squares and Pythagorean Numbers |url=https://www.jstor.org/stable/27899667 |journal=The Monist |volume=16 |issue=3 |pages=422–433 |doi=10.5840/monist190616319 |jstor=27899667 |issn=0026-9662}}</ref>

'''9855''' is also the [[Magic constant]] of a [[Magic square]] of order 27.<ref>{{Cite web |last=Square |first=The Magic |title=The Magic Square Info |url=https://themagicsquare.info/ |access-date=2023-07-08 |website=The Magic Square Info |language=en-US}}</ref> In a magic square, the magic constant is the sum of numbers in each row, column, and diagonal, which is the same. For magic squares of order ''n'', the magic constant is given by the formula <math>\frac{n(n^2 + 1)}{2}</math>.<ref>{{Cite web |title=magic square constants : more terms |url=https://www.numbersaplenty.com/set/magic_constant/more.php |access-date=2023-07-08 |website=www.numbersaplenty.com}}</ref>

The magic constant '''9855'''<ref>{{Cite web |title=first 100 Magic Square series number |url=https://www.mymathtables.com/numbers/first-hundred-magic-square-series-table.html |access-date=2023-07-08 |website=www.mymathtables.com}}</ref> for the magic square of order 27 can be calculated<ref name=":0" /> as follows:

:<math>9855 = \frac{1}{27} \sum_{k=1}^{27^2} k = \frac{27 \times (27^2 + 1)}{2}</math>
This square contains the numbers 1 to 729, with 365 in the center. The square consists of 9 nine power magic squares. It has been noted that the number of days in 27 years (365 days per year) is 9855, the constant of the larger square.<ref>{{Cite book |last=Kelsey |first=Kenneth |title=A Magic Square for Plato |page=32}}</ref><ref name=":0" /> This was first discovered and solved by ancient Greeks: Aristotle understood this magic square, but it is noted from ''numeris Platonics nihil obscuris'' that Cicero was unable to solve it.<ref name=":0" /> The 27 years as alluded to by the square was mentioned in reference to Greek generation time.<ref name=":0" />

==References==
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[[Category:Integers]]