ILL 1992 Problem 25

(a) Show that the set $\mathbb N$ of all positive integers can be partitioned into three disjoint subsets $A, B$ , and $C$ satisfying the following conditions: $$A^2 = A, B^2 = C, C^2 = B,$$ $$AB = B, AC = C, BC = A,$$ where $HK$ stands for $\{hk | h \in H, k \in K\}$ for any two subsets $H, K$ of $\mathbb N$ , and $H^2$ denotes $HH.$ (b) Show that for every such partition of $\mathbb N$ , $\min\{n \in N | n \in A \text{ and } n + 1 \in A\}$ is less than or equal to $77.$