parmenides51 wrote: Find all functions $f : N\rightarrow N - \{1\}$ satisfying $f (n)+ f (n+1)= f (n+2) +f (n+3) -168$ for all $n \in N$ . Writing $g(n)=f(n)+f(n+1)$, equation is $g(n+2)=g(n)+168$ and so : $g(2n)=168n+a$ and $g(2n+1)=168n+b$ for some integers $a,b$ This implies $f(2n)+f(2n+1)=168n+a$ And $f(2n+1)+f(2n+2)=168n+b$ Subtracting, we get $f(2n+2)=f(2n)+(b-a)$ and so $f(2n)=(b-a)n+c$ And so $f(2n+1)=168n+a-f(2n)=(168+a-b)n+a-c$ It remains to check the constraints in order to have codomain $\mathbb N\setminus\{1\}$ and we get the solution : $\boxed{f(2n)=an+b\text{ and }f(2n+1)=(168-a)n+c}$ which indeed are solutions, whatever are integers $a\in\{0,1,2,...,168\}$, $b>1-a$ and $c>1$

I am creating a huge problem set for functional equations and this problem really got my eye. The modified version of this problem (in which there is a product between $f(n+2)$ and $f(n+3)$ instead of a sum) has appeared on the forum so many times. This problem has been posted at least 15 times (which is a new record) on AoPS fora since 2005, which means it has been posted at least twice in at least one year. The fun thing is that this is indeed a good example of pigeonhole principle because the problem has been posted on Dec. 6 and also on Dec. 10 in 2012! Links to the modified problem (sorted by date, olders first) 2005: https://artofproblemsolving.com/community/c6h35248p219690 2008: https://artofproblemsolving.com/community/c6h241564p1329198 2008: https://artofproblemsolving.com/community/c6h241720p1330072 2009: https://artofproblemsolving.com/community/c6h303012p1639996 2009: https://artofproblemsolving.com/community/c6h311579p1680539 2009: https://artofproblemsolving.com/community/c6h316532 (Dec. 6) 2009: https://artofproblemsolving.com/community/c6h317565p1707638 (Dec. 10!) 2011: https://artofproblemsolving.com/community/c6h436083p2460307 2011: https://artofproblemsolving.com/community/c6h436923p2464444 2011: https://artofproblemsolving.com/community/c6h446599p2513419 2012: https://artofproblemsolving.com/community/c6h467768p2619059 2012: https://artofproblemsolving.com/community/c6h491662p2757405 2014: https://artofproblemsolving.com/community/c6h570003p3346779 2015: https://artofproblemsolving.com/community/q2h1148868p5426963 2016: https://artofproblemsolving.com/community/c6h1316855p7077580 Links to similar problems: (sorted by date, olders first) 2005: https://artofproblemsolving.com/community/c6h56591 2007: https://artofproblemsolving.com/community/c6h169128 2012: https://artofproblemsolving.com/community/c6h456682 2017: http://artofproblemsolving.com/community/c6h1521856p9094742 (this problem)