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By "young father" do you mean that you're going to professional baseball games with your children and want to archive these, or are you scoring youth baseball games that your children play in?

I bought the Stuart Miller book scorebook two years ago and take it to professional games. Has spaces for 10 innings and two subs per batting order position. Not terribly much room for notes and no place for balls and strikes. (When scoring my son's youth games, I keep track of balls and strikes in order to have a pitch count, but at a pro game for fun it's just too much to keep track of anyways.) I like it well enough, especially the hard covers that make a good writing surface and spiral binding to keep it flat. The diamonds take up most of each score box, and if I were to buy a new book today, I think I'd want to get one with smaller diamonds; I've started using dashes instead of 1B/2B and trying to record where a ball is hit using Project Scoresheet location codes, and I'd like more room for that. Definitely more archivable than sporting goods store books.

A few others to consider (I haven't seen any IRL though): The Eephus League Halfliner, which looks clean and elegant. There's the I Love to Score book, which has more intricate boxes; not sure of the binding. Rich Burk is another broadcaster selling his own design scorebooks. Andres Wirkmaa, who literally wrote the book about scorekeeping, sells his scorebook design. And Baseball by the Numbers has probably the most elaborate score boxes I've seen.

Welcome to the sub!

I've kept my scorecards in a scorebook, so I can't speak specifically for this, but I have had a ticket scrapbook before. Photo corners could be a really clutch idea if you wanted to mount the scorecards on a bigger piece of paper (this is the first result on Amazon, but I always got mine from a craft store). Could even mount the tickets on the same page if it's big enough and you've got em around.