From the town of Leavenworth, Washington drive the Icicle Creek road to the Colchuck Lake trailhead, hike about four miles to the signed intersection and take the Stuart Lake option. Proceed to the Mineral Creek cutoff and make your way to the obvious Argonaut/Sherpa basin--a sprawling series of valleys leading to the respective basins for both peaks. Along the trail, you'll come to a meadow where both peaks and their connecting ridge system become very visible, and basically from this point you more or less bushwack your way to the desired basin--it's pretty tough to get lost here, although some 'shwack paths will be more efficient than others.
Once in the Sherpa Peak basin, the Couloir route becomes plainly visible, but just in case of confusion, see the attached photo--there are several gullies on the mountain, but this one is quick, moderately steep, and seems like it might stay in shape longer than some of the others.
Make your way up the low-angled slopes leading to the couloir, then head up through through the progressively steepening gully. Be prepared for hard snow and/or ice conditions. On the day we climbed it, the snow was hard enough to keep us on front points for several hundred feet--a second tool would have been welcome.
Finishing the gully, just take a right up to the obvious summit--Balanced Rock is already visible at this point. You'll enjoy moving easily along 3rd-class ledges and ramps up to a pentultimate, single low 5th-class pitch followed by an enjoyable, short scramble to the Balanced Rock itself.
Descent options are numerous, but if you don't want to back down the same route, you can do what we did: drop back to the notch at the top of the gully, then descend into the valley opposite the gully and make your way around several rocky ribs and back up to the ridge at Sherpa Pass, drop back into the basin and hike out [after a miserable forced bivy].
Be prepared for sustained 50-degree snow and/or ice in the couloir, followed by easy low 5th-class rock to the top. Those climbers with experience in such conditions will probably require only stout boots, traditional crampons and a good axe, possibly also a second tool. For the rock, a minimal rack to 3" will be more than adequate. Choice of rope will depend on your descent route preference.
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