This is part two of some photos from my Grandparent’s house. I don’t know how much Yahtzee changed it’s box over the years, but this particular one is dated 1956 and is from the E. S. Lowe company. The price looks like it was $1.99, marked down to $1.17. That Wiki site says
Between 1956 and 1961, the game’s advertising slogan was changed from “The Game That Makes You Think While Having Fun” to “The Fun Game That Makes Thinking Fun!”
And on the action site, only the earliest ones appear to have this “alligator skin” cup.