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Register a new userThe Classification of Magic SET Squares
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A magic SET square is a 3 by 3 table of SET cards such that each row, column, diagonal, and anti-diagonal is a set. We allow the following transformations of the square: shuffling features, shuffling values within the features, rotations and reflections of the square. Under these transformations, there are 21 types of magic SET squares. We calculate the number of squares of each type. In addition, we discuss a game of SET tic-tac-toe.
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Do you want to know what an anti-chiece Latin square is? Or what a non-consecutive toroidal modular Latin square is? We invented a ton of new types of Latin squares, some inspired by existing Sudoku variations. We cant wait to introduce them to you and answer important questions, such as: do they even exist? If so, under what conditions? What are some of their interesting properties? And how do we generate them?
A $Gamma$-magic rectangle set $MRS_{Gamma}(a, b; c)$ of order $abc$ is a collection of $c$ arrays $(atimes b)$ whose entries are elements of group $Gamma$, each appearing once, with all row sums in every rectangle equal to a constant $omegain Gamma$ and all column sums in every rectangle equal to a constant $delta in Gamma$. In this paper we prove that for ${a,b} eq{2^{alpha},2k+1}$ where $alpha$ and $k$ are some natural numbers, a $Gamma$-magic rectangle set MRS$_{Gamma}(a, b;c)$ exists if and only if $a$ and $b$ are both even or and $|Gamma|$ is odd or $Gamma$ has more than one involution. Moreover we obtain sufficient and necessary conditions for existence a $Gamma$-magic rectangle MRS$_{Gamma}(a, b)$=MRS$_{Gamma}(a, b;1)$.
Translated from the Latin original Novae demonstrationes circa resolutionem numerorum in quadrata (1774). E445 in the Enestrom index. See Chapter III, section XI of Weils Number theory: an approach through history. Also, a very clear proof of the four squares theorem based on Eulers is Theorem 370 in Hardy and Wright, An introduction to the theory of numbers, fifth ed. It uses Theorem 87 in Hardy and Wright, but otherwise does not assume anything else from their book. I translated most of the paper and checked those details a few months ago, but only finished last few parts now. If anything isnt clear please email me.
Using the standard Coxeter presentation for the symmetric group $S_n$, two reduced expressions for the same group element are said to be commutation equivalent if we can obtain one expression from the other by applying a finite sequence of commutations. The resulting equivalence classes of reduced expressions are called commutation classes. How many commutation classes are there for the longest element in $S_n$?
Graph labeling is a well-known and intensively investigated problem in graph theory. Sparse anti-magic squares are useful in constructing vertex-magic labeling for graphs. For positive integers $n,d$ and $d<n$, an $ntimes n$ array $A$ based on ${0,1,cdots,nd}$ is called emph{a sparse anti-magic square of order $n$ with density $d$}, denoted by SAMS$(n,d)$, if each element of ${1,2,cdots,nd}$ occurs exactly one entry of $A$, and its row-sums, column-sums and two main diagonal sums constitute a set of $2n+2$ consecutive integers. An SAMS$(n,d)$ is called emph{regular} if there are exactly $d$ positive entries in each row, each column and each main diagonal. In this paper, we investigate the existence of regular sparse anti-magic squares of order $nequiv1,5pmod 6$, and it is proved that for any $nequiv1,5pmod 6$, there exists a regular SAMS$(n,d)$ if and only if $2leq dleq n-1$.
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