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Here is a fact that I find interesting. Strictly speaking it is to say that a certain number counts three completely unrelated things, but it seems easy to make these into a combinatorial problem or to define an explicit bijection between two (or three) sets.

So, the concrete example is $168$, which

(1) the number of hours in a week;

(2) the number of primes under $1000$; and

(3) the size of the smallest simple group of Lie type.

$168$ is also $4 \times 42$, where $42$ is that famous number Douglas Adams wrote about. (How did he know?) Which is of course the reciprocal of the smallest positive number that can be written as $$1-\frac 1a -\frac 1b -\frac 1c$$ with $a,b,c\in \mathbb N$ and coincidentally (and relatedly) the largest size of the automorphism group of $C$ a smooth projective curve of genus at least $2$ is $42\cdot {\rm deg} K_C$ and of $S$ a smooth projective surface of general type is $42\cdot K_S^2$(42 K_S)^2$. So I guess one could add that$168$is also (4) the maximum value of$\frac 4{1 -\frac 1a -\frac 1b -\frac 1c}$with$a,b,c\in \mathbb N$; and (5) the largest possible size of the automorphism group of a smooth complex projective curve of genus$3$, but I admit the last two are a little artificial... 2 added 34 characters in body Here is a fact that I find interesting. Strictly speaking it is to say that a certain number counts three completely unrelated things, but it seems easy to make these into a combinatorial problem or to define an explicit bijection between two (or three) sets. So, the concrete example is$168$, which (1) the number of hours in a week; (2) the number of primes under$1000$; and (3) the size of the smallest simple group of Lie type.$168$is also$4 \times 42$, where$42$is that famous number Douglas Adams wrote about. (How did he know?) Which is of course the reciprocal of the smallest positive number that can be written as $$1-\frac 1a -\frac 1b -\frac 1c$$ with$a,b,c\in \mathbb N$and coincidentally (and relatedly) the largest size of the automorphism group of$C$a smooth projective curve of genus at least$2$is$42\cdot {\rm deg} K_C$and of$S$a smooth projective surface of general type is$42\cdot K_S^2$. So I guess one could add that$168$is also (4) the maximum value of$\frac 4{1 -\frac 1a -\frac 1b -\frac 1c}$with$a,b,c\in \mathbb N$; and (5) the largest possible size of the automorphism group of a smooth complex projective curve of genus$3$, but I admit the last two are a little artificial... 1 [made Community Wiki] Here is a fact that I find interesting. Strictly speaking it is to say that a certain number counts three completely unrelated things, but it seems easy to make these into a combinatorial problem or to define an explicit bijection between two (or three) sets. So, the concrete example is$168$, which (1) the number of hours in a week; (2) the number of primes under$1000$; and (3) the size of the smallest simple group of Lie type.$168$is also$4 \times 42$, where$42$is that famous number Douglas Adams wrote about. (How did he know?) Which is of course the reciprocal of the smallest positive number that can be written as $$1-\frac 1a -\frac 1b -\frac 1c$$ with$a,b,c\in \mathbb N$and coincidentally (and relatedly) the largest size of the automorphism group of$C$a smooth projective curve of genus at least$2$is$42\cdot {\rm deg} K_C$and of$S$a smooth projective surface of general type is$42\cdot K_S^2$. So I guess one could add that$168$is also (4) the maximum value of$\frac 4{1 -\frac 1a -\frac 1b -\frac 1c}$with$a,b,c\in \mathbb N$; and (5) the largest possible size of a smooth projective curve of genus$3\$,

but I admit the last two are a little artificial...