Latin squares have long been a staple object of study of combinatorics and recreational mathematics. A Latin square of size n is an 
It’s illustrative to move from the graphical definition of a Latin square to something more symbolic. A Latin square on an index set I is a function f assigning each element of 
Definition. A Latin square on a (finite) index set I is a subset S of 
Notice that “function-ness” and the conditions on the rows and columns are absorbed into one pleasantly symmetric condition. This explains the mysterious fourth paragraph of this Not About Apples post. (This definition also generalizes gracefully to orthogonal Latin squares; a list of k orthogonal squares corresponds naturally to a subset of 
Definition. A supersymmetric Latin square (ssLs) is a Latin square which (viewed a subset of
That is, a supersymmetric Latin square is in an orbit by itself under the
Example.
There are three supersymmetric Latin squares on {0,1,2}.
021 102 210 210 021 102 102 210 021
For ordinary Latin squares, there is another kind of symmetry. Rows can be permuted, columns can be permuted, and the symbols themselves can be permuted. That is, the symmetric group on the index set acts naturally on the set of Latin squares in three ways, by acting on the first, second, or third coordinate of the triples. This justifies the common convention of assuming without loss of generality that a Latin square has the symbols listed in order across the first row and down the first column; since each of these actions is free (though the composite action of 
This doesn’t work for supersymmetric Latin squares, though; these actions of 
Example.
The first ssLs listed above is stabilized by all of
I call two supersymmetric Latin squares on I equivalent if they are in the same orbit of this 
For (Edit: as noted in the comments, that’s just plain not true.) For 
There are infinitely many supersymmetric Latin squares. Let me briefly explain the construction of all the ones I know.
There is a natural product on Latin squares. If S (resp. S’) is a Latin square on I (resp. I’), then we define 
Since we have two ssLs’s of size 3, we can construct ssLs’s of size 9. There are four possibilities to consider: 
021687354 102768435 435102768 210876543 021687354 354021687 102768435 210876543 543210876 687354021 768435102 102768435 876543210 687354021 021687354 768435102 876543210 210876543 354021687 435102768 768435102 543210876 354021687 687354021 435102768 543210876 876543210
It’s not obvious, but the second and third squares are actually equivalent under e.g. the permutation 
The same approach enables us to construct two supersymmetric Latin squares 
Questions.
- Do supersymmetric Latin squares exist of any size which is not a power of 3?
- For size any positive power of 3, we have found two inequivalent ssLs’s. Is this a complete list?
- We could also consider less strict symmetry conditions, requiring only that the Latin square be invariant under some subgroup of
-symmetry, that the square be invariant under cyclic shifts. (I haven’t really thought about this yet, but it seems potentially interesting.)
came up frequently in the examples and homework. I overheard two of my students remarking on this phenomenon before class.
or something, I can just write down the integral and do it on my calculator.”
and 
. Must the set be all of $\mathbb{Z}^+$?
be the minimal m such that 
for some integer 
. Can we get any nontrivial estimates on how 
be positive integers. 
.
as counting the number of ways to choose a different (not necessary disjoint) b-element subsets of a set of c elements. For definiteness I can take the set of c elements to be 
, list the elements of a subset in increasing order, and list the subsets in lexicographical order. This suggests the following combinatorial formulation of the inequality.
matrices with entries in 
consist of those matrices in S for which the elements in each row are strictly increasing, and the rows are lexicographically strictly increasing. Let 
consist of those matrices in S for which the elements in each columns are strictly increasing, and the columns are lexicographically strictly increasing. Find an explicit bijection of 
are nonzero integers, at most one of which is negative, then the number
of inequivalent occurrences of three mutually tangent spheres of curvature 
with norm 
, up to multiplication by a unit. The bijection between algebraic integers with that norm and triples of spheres is natural and explicit.
on the positive integers as follows. If 
or 
, then 
if 
if 
, then 
; 
for all 
.
are nonzero integers, at most one of which is negative, then the number
of inequivalent occurrences of tangent spheres of curvature 
. Here 
is the number of solutions 
to the congruence 
, 
is the number of solutions 
to the congruence 
, and 
is 1 if 
and 
, otherwise 0.
-tuple of 
-tuples in ![R=[a,b]\times [c,d]](https://s0.wp.com/latex.php?latex=R%3D%5Ba%2Cb%5D%5Ctimes+%5Bc%2Cd%5D&bg=ffffff&fg=333333&s=0&c=20201002)
. For any real numbers 
, consider the function 
. The integral of this function over a rectangle, 
is relatively easy to compute: 
By the periodicity of the sine, this integral will vanish for every $\alpha,\beta$ if and only if the rectangle is semi-integral. If 
integrates to 0 on every subrectangle in the decomposition, then 
.
, of course). It’s not hard to check that this really is a topology, and that it has the following interesting properties.
is open; by 1, it is also closed if there are only finitely many primes.
is finite, so 
is not open, and 
are the curvatures of four mutually tangent circles iff they satisfy the Descartes equation 
.
, the possible curvatures of a fourth circle are the roots of the above equation, viewed as a quadratic in 
; in general there are two possibilities, 
. In particular, their sum is 
. If we know one curvature 
.
Cap Khoury's Whiteboard 