[Combinatorics]
Problem: An integer $N$ is worth 1 point for each pair of digits it contains that forms a prime in its original order. For example, 6733 is worth 3 points (for 67,73 , and 73 again), and 20304 is worth 2 points (for 23 and 03). Compute the smallest positive integer that is worth exactly 11 points. [Note: Leading zeros are not allowed in the original integer.]

Solution: If a number $N$ has $k$ base 10 digits, then its maximum point value is $(k-1)+(k-2)+\cdots+1=$ $\frac{1}{2}(k-1)(k)$. So if $k \leq 5$, the number $N$ is worth at most 10 points. Therefore the desired number has at least six digits. If $100,000<N<101,000$, then $N$ is of the form $100 \underline{A} \underline{B} \underline{C}$, which could yield 12 possible primes, namely $1 \underline{A}, 1 \underline{B}, 1 \underline{C}, \underline{0} \underline{A}$ (twice), $0 \underline{B}$ (twice), $0 \underline{C}$ (twice), $\underline{A} \underline{B}, \underline{A} \underline{C}, \underline{B} \underline{C}$. So search for $N$ of the form $100 \underline{A} \underline{B} \underline{C}$, starting with lowest values first. Notice that if any of $A, B$, or $C$ is not a prime, at least two points are lost, and so all three numbers must be prime. Proceed by cases:

First consider the case $A=2$. Then $1 \underline{A}$ is composite, so all of $1 \underline{B}, 1 \underline{C}, \underline{A} \underline{B}, \underline{A} \underline{C}, \underline{B} \underline{C}$ must be prime. Considering now the values of $1 \underline{B}$ and $1 \underline{C}$, both $B$ and $C$ must be in the set $\{3,7\}$. Because 27 is composite, $B=C=3$, but then $\underline{B} \underline{C}=33$ is composite. So $A$ cannot equal 2 .

If $A=3$, then $B \neq 2$ because both 12 and 32 are composite. If $B=3,1 \underline{B}$ is prime but $\underline{A} \underline{B}=33$ is composite, so all of $C, 1 \underline{C}$, and $3 \underline{C}$ must be prime. These conditions are satisfied by $C=7$ and no other value. So $A=B=3$ and $C=7$, yielding $N=\mathbf{1 0 0 3 3 7}$.

Final answer: 100337