How is it that when you turn the arrow to pull the string, how does the toy know which picture the arrow points to?
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It actually doesn’t, not at all.
At the heart of a see-and-say toy is a plastic disk with grooves carved in it and a metal needle, exactly like a record. It has one groove per picture on the front of the toy, and turning the arrow changes which groove the needle sits on.
Pulling the string winds up a spring inside the toy, and as it unwinds, the needle travels along the groove, playing the sound encoded in the wiggling of the groove.
I happen to know this because I opened one up years ago. It’s essentially a really cheap plastic record player. When you move the pointer to a particular position, the needle on the plastic record moves a set distance due to the tension of a spring. Little notches keep the needle in place until you pull the handle again.