In studying mathematics, I sometimes come across examples of general facts that hold for all $n$ greater than some small number. One that comes to mind is the Abel–Ruffini theorem, which states that there is no general algebraic solution for polynomials of degree $n$ except when $n \leq 4$.
It seems that there are many interesting examples of these special accidents of structure that occur when the objects in question are "small enough", and I'd be interested in seeing more of them.
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Deciding whether a graph is $2$-colorable has an obvious polynomial-time algorithm.
Deciding whether a graph is $k$-colorable for $k \geq 3$ is already NP-Complete!
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The 26 sporadic groups is also an "accidental" occurrence. Since there is a finite number of them, after some large N, all the simple groups of order N or larger fit into a given category.
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The sequence of the maximal number of regions formed by drawing chords between all pairs of n points in arbitrary position on the border of a circle starts 1, 2, 4, 8, 16; and then the next term, of course, is... 31. (The canonical formula is $1+{n\choose2}+{n\choose4}$.) For more details, see http://oeis.org/A000127
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Another accident of small dimension: in all dimensions $\gt 4$, there are only three regular polytopes: the simplex, hypercube, and cross-polytope (the dual of the hypercube). In two dimensions there are infinitely many (all of the $n$-gons); in three dimensions you have two additional polyhedra, the dodecahedron and icosahedron; and in four dimensions there are three additional ones, the 120-cell and 600-cell (which are dual to each other) and the 24-cell (which is self-dual).
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I would include in this list a discussion of $G(n)$ and $g(n)$ in the Waring Problem.
The point is that when it comes to representing integers as sums of powers of non-negative integers, it seems to happen that some (smallish) integers require more powers to represent them due to some some (presumably unknown) peculiarity of small integers.
For example, in the case of representing integers as sums of cubes, it has been proved that 9 cubes are sufficient, and some numbers require 9, so that $g(3)=9$.
On the other hand, calculations suggest that almost all numbers from some point onwards are sums of at most 4 cubes (so that $G(3)$ might be 4, but this is not proved), and it appears that it is an "accident of small $n$" that there are some (smallish) numbers which require more than 4 cubes.
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The $n^\mathrm{th}$ cyclotomic polynomial is the minimal polynomial whose roots are the primitive $n^\mathrm{th}$ roots of unity, that is
$$\Phi_n(X) = \prod_{ {0\leq j < n} \atop {\gcd(j,n)=1}}(X - e^{2\pi i j / n})$$
The first few examples are: $$\Phi_1(X) = X - 1 $$ $$\Phi_2(X) = X + 1 $$ $$\Phi_3(X) = X^2 + X + 1$$
For all $n$ small enough, all the coefficients are $\pm 1$ or $0$. But if at least three odd primes divide $n$ (which requires at least $n\geq3\cdot5\cdot7 = 105$), other coefficients are possible.
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Orthogonal Latin squares exist for every order except 2 and 6. Euler conjectured from the small examples that they existed for any order not of the form $4k+2$, but he was mistaken.
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Similar to the answer about the $26$ sporadic groups and nontrivial $\mathrm{Out}(S_n)$ for $n=2,6$, there are a bunch of what are called exceptional isomorphisms that occur with low order/dimension. Basically, groups come in all sorts of special families (where there is a rule designating what groups are in the family and why) that are infinite, but these infinite families share a few finite cases between them.
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The ring of integers of $\mathbf{Q}[\sqrt d]$ is a principal ideal domain for $1 \leq d < 10$, but not for $d=10$.
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There are cyclic numbers for all bases $>4$, except for perfect squares, and $6$ (if you exclude leading zeros).
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The alternating group $A_n$ is simple for $n\neq 4$.
This is related to the example given by the thread creator since it allows $S_4$ to be solvable, thus guaranteeing that all polynomials of degree $4$ or less are resolvable with only the field operations plus the extraction of roots.
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I wouldn't really call them accidents, but here are two simple related results from algebra:
There exists algebraic extensions of $\mathbb{R}$ of dimension $n$ only for $n=1$ and $n=2$.
Also it is not possible to construct an Division Ring over $\mathbb{R}$ of dimension $n$, excepting when $n=4$.
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$\mathbb{R}^n$ has a unique smooth structure except when $n=4$. Furthermore, the cardinality of [diffeomorphism classes of] smooth manifolds that are homeomorphic but not diffeomorphic to $\mathbb{R}^4$ is $\mathfrak{c}=2^{\aleph_0}$.
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Although there is an infinite family of generalized Petersen graphs with arbitrarily many vertices, there are exactly seven of these that are edge-transitive, the largest having 48 vertices.
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For all $n>2$, the symmetric group $S_n$ can be generated by two elements, $c$ and $b$, where $c$ has order 3 and $b$ has order 2…
…except for $n=5, 6,$ or $8$.
Miller, G.A., “On the groups generated by two operators”, Bull. Amer. Math. Soc. 7 (1901), p. 14–32.
You might enjoy the article The Strong Law of Small Numbers by Richard K. Guy.
He has other publications in which he partly recycles the title.