From the 2016 International Mathematical and Logic Games Contest
Along the coast of Maths-land, the straight beach-front road contains a line of houses, all on the same side of the road. The houses are painted either blue or yellow and there is at least one house of each colour. Curiously enough, every pair of houses separated by ten other houses is painted the same colour, as is every pair separated by fifteen houses. What is the maximum number of houses on this road?
Bruteforcing this with a computer I found that $25$ is the maximum. I've been looking for a formal proof, to no avail. My combinatoric skills are very limited, perhaps someone can come up with a short proof...
One way to show that $25$ is the maximum would be to observe that the chain
$$11\to22\to6\to17\to1\to12\to23\to7\to18\to2\to13\to24\to8\to19\to3\to14\to25\to9\to20\to4\to15\to26\to10\to21\to5\to16$$
where each step in the chain either goes up $11$ or down $16$, accounts for all the numbers from $1$ to $26$. This shows that in any stretch of $26$ houses, all houses have the same color as the $11$th house.