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We present the results of an interferometric study of 38 millimeter-wave lines of ¹²CH₃OH in the vicinity of the massive star-forming region W3(OH/H₂O). These lines cover a wide range of excitation energies and line strengths, allowing for a detailed study of excitation mechanisms and opacities. In this paper we concentrate on the region around the water maser source W3(H₂O) and a region extending about 30" to the south and west of the hydroxyl maser source W3(OH). The methanol-emitting region around W3(H₂O) has an extent of 2".0x1".2 (4400x2600 AU). The density is of the order of 10⁷/cm3, sufficient to thermalize most of the methanol lines. The kinetic temperature is approximately 140 K and the methanol fractional abundance greater than 10^–6, indicative of a high degree of grain mantle evaporation. The W3(H₂O) source contains substructure, with peaks corresponding to the TW source and, separated from it by 2500 AU in projection, a combination of Wyrowski's sources B and C. The kinematics is consistent with these being distinct protostellar cores in a wide binary orbit and a dynamical mass for the region of a few tens of M_☉. The extended methanol emission to the southwest of W3(OH) is seen strongly only from the lowest excitation lines and from lines known elsewhere to be Class I methanol masers, namely, the 84.5 GHz 5_–1-4₀E and 95.2 GHz 8₀-7₁A⁺ lines. This suggests that this region, like Class I maser sources, is dominated by collisional excitation. Within this region there are two compact clumps, which we denote as swA and swB, each about 15" (0.16 pc projected distance) away from W3(OH). Excitation analysis of these clumps indicates the presence of lines with inverted populations but only weak amplification. The sources swA and swB appear to have kinetic temperatures of the order of 50-100 K and densities of the order of 10⁵-10⁶/cm3. The methanol fractional abundance for the warmer clump is of the order of 10^–7, suggestive of partial grain mantle evaporation. The clumping occurs on mass scales of the order of 1 M_☉.