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Many of those old rolling block Remingtons had the serial number on the left side of the tang; you have to remove the stock to read it. I have a Marlin Model 27S from around 1913 that is the same way, it has a serial number but the stock has to come off to read it. In both instances the sellers were unaware and marked the sales tickets as "NSN."
https://www.milsurps.com/images/impo...1568rbvi-1.jpg
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Mr Tickle, You know, as rolling block rifles go-- that one looks in great shape. I would not mess with it if it were mine. Even with scrubbed markings it is quite collectible and could be good shooter. For the cost of barrel work and gunsmithing you could probably come close to buying a replica for shooting. Salt Flat
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For the cost of rebarreling you could buy some brass and dies and just shoot it.
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Use it as it is... Enjoy it and leave it unmolested for future generations.