Problem

Source: Iranian National Math Olympiad (Final exam) 2006

Tags: analytic geometry, algorithm, pigeonhole principle, geometry, combinatorics proposed, combinatorics



The image shown below is a cross with length 2. If length of a cross of length $k$ it is called a $k$-cross. (Each $k$-cross ahs $6k+1$ squares.) Invalid image file a) Prove that space can be tiled with $1$-crosses. b) Prove that space can be tiled with $2$-crosses. c) Prove that for $k\geq5$ space can not be tiled with $k$-crosses.