TITLE: Sum of three consecutive prime numbers is $173$
QUESTION [6 upvotes]: If I tell you that the sum of three consecutive prime numbers is $173$, how quickly could we find the biggest of these numbers?

REPLY [5 votes]: It depends how much you know about the distribution of small primes. If $a,b$ are consecutive primes with $a<b\leq 89$ then $b-a\leq 6.$
If $p,q,r$ are consecutive primes with $p<q<r$  and $p+q+r=173$ then $q\leq 83.$ Otherwise $q\geq 89,$ but then $p+q+r>q+r>2 q\geq 178.$  So $r\leq 89.$
Therefore $q\leq p+6$ and $r\leq q+6\leq p+12.$ $$\text {Hence }\quad 173=p+q+r\leq p+(p+6)+(p+12)=3 p+18,$$ $$\text {giving }\quad p\geq 155/3.$$ $$\text {Also } \quad 173=r+q+p\geq r+(r-6)+(r-12)=3 r-18,$$ $$\text {giving }\quad r\leq 191/3.$$ There are exactly $3$ primes in the interval $[155/3,191/3],$ and they are $53,59,61.$ 
This is not how I solved it in my head. I just "looked around the neighborhood" of $173/3$.