currently stuck on the problem 245, mind shedding some light? :p

thanks,

@Nathan: Here’s one hint from my solution to Project Euler Problem 208 (Robot Walks): In order to return back to the starting point, the geometry of the situation demands an equal number of each of the 5 “type” of arcs be used. This is essentially because of the Sin/Cos of 72 degree and other angles may not have forced this (more formal proofs of this is presented in the forum). You can also see from the figure given in the problem description. Starting northwards, number the 5 arcs as (a,b,c,d,e) for counter-clockwise movement and (A,B,C,D,E) for clockwise movement (a-A arcs are mirror images etc.) Then, the arcs encountered on the path in problem figure are:

aEabcCDEAeABdBdeabDbcCcde.

Notice that each of a-e appear thrice and each of A-E twice.

Now for the dynamic programming solution, just count all paths that begin with a/A (northward) given that there are occurrences of arcs a-e each and = occurrences of arcs A-E each. In fact, with this information, you can do away with dynamic programming altogether, and find a simple summation in terms of binomial coefficients etc.

I hope I didn’t give too much away to spoil the fun for you!

While I was glad on having solved it with dynamic programming more of less efficiently (~8 sec), I was really impressed with the purely combinatorial solutions posted by some members in the forum (do make sure to see them once you gain access to the forum).