There is nothing listed as a Remington Boys rifle although the No. 4 was built on the lightweight frame and was available in .22, .25 Stevens, and .32 calibers. About 50,000 were made between 1890 and 1933. Values go from $75 in poor condition to $750 in VG condition. Also, the No. 6 was a lightweight, small rifle designed expressly for young boys and was chambered in .22 rimfire and .32 short or long as well as a smoothbore for use with shot shells. It was a take down model with a knurled knob on the underside of the receiver. Early models had a case colored frame and these models are worth about 10% more. Over 250,000 were made between 1902 and 1903. Values are $75 poor to $550 VG.
Please check the link for the Manufacture Date codes details on the Remington Society of America.
1920
1914
1913
1920
1919
late 1919
1920
1915
1912
1920
1912