We consider here the positive integers with respect to their unique decimal expansions, where each n ∈ ℕ is given by for some non-negative integer k and digit sequence αkαk-1 … α0. With slight abuse of notation, we also use n to denote αkαk-1 … α0. For such sequences of digits (as well as for the numbers represented by the corresponding expansions) we write x ⊲ y if x is a subsequence of y, which means that either x = y or x can be obtained from y by deleting some digits of y. For example, 514 ⊲ 352148. The main problem is as follows: Given a set S ⊂ ℕ, find the smallest possible set M ⊂ S such that for all s ∈ S there exists m ∈ M with m ⊲ s.

In 2011, Beeler and Hoilman generalized the game of peg solitaire to arbitrary connected graphs. In the same article, the authors proved some results on the solvability of Cartesian products, given solvable or distance 2-solvable graphs. We extend these results to Cartesian products of certain unsolvable graphs. In particular, we prove that ladders and grid graphs are solvable and, further, even the Cartesian product of two stars, which in a sense are the “most” unsolvable graphs.

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