The strong law of small numbers is an informal warning about apparent numerical patterns. Its first formulation (Gardner 1980, Guy 1988, 1990) states "There aren't enough small numbers to meet the many demands made of them."
The second strong law of small numbers (Guy 1990) states that "When two numbers look equal, it ain't necessarily so." Guy (1988) gives 35 examples illustrating
these warnings, with further examples in Guy (1990). For example, example 36 (Guy
1990, p. 3) notes that the first few values of the interpolating polynomial 
(erroneously given in Guy (1990) with a coefficient 24 instead of 23) for 
, 2, ... are 1, 2, 4, 8, 16, .... Thus, the polynomial appears
to give the powers of 2, but then continues 31, 57, 99,
... (OEIS A000127). In fact, this sequence
gives the maximal number of regions obtained by joining 
points around a circle by chords
(circle division by chords).
Similarly, example 41 notes the curious fact that the function ![[e^((n-1)/2)]](/images/equations/StrongLawofSmallNumbers/Inline4.svg)
where ![[x]](/images/equations/StrongLawofSmallNumbers/Inline5.svg)
is the ceiling function gives the sequence
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ... (i.e., the first few Fibonacci
numbers) for 
, 1, ..., although it subsequently continues 91, 149, ...
(OEIS A005181), which are not Fibonacci
numbers.
Another example is provided by a near-identity of trinomial coefficients noticed by Euler (Guy 1990, p. 18).
