We give a new, simplified and detailed account of the correspondence between levels of the Sherali-Adams relaxation of graph isomorphism and levels of pebble-game equivalence with counting (higher-dimensional Weisfeiler-Lehman colour refinement). The correspondence between basic colour refinement and fractional isomorphism, due to Tinhofer [22,23] and Ramana, Scheinerman and Ullman [17], is re-interpreted as the base level of Sherali-Adams and generalised to higher levels in this sense by Atserias and Maneva [1] and Malkin [14], who prove that the two resulting hierarchies interleave. In carrying this analysis further, we here give (a) a precise characterisation of the level k Sherali-Adams relaxation in terms of a modified counting pebble game; (b) a variant of the Sherali-Adams levels that precisely match the k-pebble counting game; (c) a proof that the interleaving between these two hierarchies is strict. We also investigate the variation based on boolean arithmetic instead of real/rational arithmetic and obtain analogous correspondences and separations for plain k-pebble equivalence (without counting). Our results are driven by considerably simplified accounts of the underlying combinatorics and linear algebra. §1. Introduction. We study a surprising connection between equivalence in finite variable logics and a linear programming approach to the graph isomorphism problem. This connection has recently been uncovered by Atserias and Maneva [1] and, independently, Malkin [14], building on earlier work of Tinhofer [22, 23] and Ramana, Scheinerman and Ullman [17] that just concerns the 2-variable case.Finite variable logics play a central role in finite model theory. Most important for this paper are finite variable logics with counting, which have been specifically studied in connection with the question for a logical characterisation of polynomial time and in connection with the graph isomorphism problem (e.g., [6,8,9,12,13,16]). Equivalence in finite variable logics can be characterised in terms of simple combinatorial games known as pebble games. Specifically, C k -equivalence can be characterised by the bijective k-pebble game introduced by Hella [10]. Cai, Fürer and Immerman [6] observed that C k -equivalence exactly corresponds to indistinguishability by the k-dimensional Weisfeiler-Lehman (WL) algorithm, 1 a combinatorial graph isomorphism algorithm that goes back to work of Weisfeiler and Lehman in the 1970s (for example, [24]; see [6] for an account of the history of the algorithm). The 2-dimensional version of the WL algorithm precisely corresponds to an even simpler isomorphism algorithm known as colour refinement.