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So far I've had 3 K31's, 2 with birch stock (from the early '50's) and 1 with a walnut stock made in 1938 and "P" (privat) marked. All 3 shot very accurately (MOA after tuning with spacers between barrel bands). Some friends talked me out of the 2 birch stocked ones but I kept onto the walnut stock.
https://www.milsurps.com/images/impo...03/K31II-1.jpg
https://www.milsurps.com/images/impo...kctzpkqs-1.jpg
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You really can't go wrong with any K31 so long as you make sure you check it over for the usual defects. Every one I've seen (UK) has been really bashed about on the wood whether walnut or birch but mechanically they always seem sweet.
Go for the scope mount and fit a fixed power scope, mine wears a Weaver K6, just operate the bolt gently to avoid singing the scope or wrap some insulation tape around the tube to protect it.