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Is this the Cracker Barrel peg game?
One of the many variants. It’s an old camp game too.
I had a little travel size peg game like this I'd play on road trips when I was a kid. Totally forgot about this. Cool website
Isn't there a limited subset of solutions for this and beyond that it's mechanical and repetitive? So it's not that interesting?

I think an interesting variant would be varying the shape of the board, so all the sudden it's not a one trick poney.

Additionally, maybe add special pegs that could jump over two, combos that give a power up that let you slide an entire row or column by one etc.

disclaimer: I make games.

Yet it's still deeply engrossing. I remember buying the triangular version from Cracker Barrel and bringing on a beach vacation. Everyone played the heck out of it.

Growing up we had one in the same cross configuration we see here. My grandfather made it from wood with a router, stained and sealed it. It was on a circular piece of wood with a groove running around it that the jumped marbles could be put in.

There is a limited subset of solutions, but that doesn’t make it uninteresting to the casual person waiting for their server at cracker barrel or relaxing at some summer camp.

This is a classic game made by wood shop students. It’s nice to have a clean electronic implementation.

This. I've only ever played this at Cracker Barrel, but it's doing my mind loads of good to just veg out and try to figure this out at home. I definitely had the thought that, yeah, once you figure out what the solution(s) are, the magic's gone, but I definitely don't play this enough for it to be ruined, and it's gonna take me forever to get there; I keep getting stuck with three left!

Edit: Two! Finally! lol

They play this in Nauvoo Illinois at the summer vacation spot for members of the Christ of Jesus Christ of Latter-day Saints. Sister missionaries taught me a song to solve this and leave the last leg in the middle.
A two-player version of this is the ancient Hawaiian game of Konane.

It can be played on a checker board if you have enough pieces. The Polynesians made do with white and black rocks, and I played quite a few matches with decorative spray-painted pine cones.

https://en.wikipedia.org/wiki/K%C5%8Dnane

There are a number of 2-player games that use a solitare board. My favorite is Asalto[1], which is a variant of fox-and-geese.

Played it with 24 pennies and two dimes on a hand-drawn board once when I was stuck somewhere with no games.

1: https://en.wikipedia.org/wiki/Asalto

I used to play that game a lot when I was a kid and never finished it once.

I just tried this and won the first time, with the last peg in the centre hole!

This made my day.

The above comment does not need the context of this HN page/sumission in order to be a complete wholesome story in itself. Nice.
Hehe, next time I'll try and make it a haiku
Oh! I had a real world version of this game as a kid. I eventually figured out the algorithm for winning, then perfected it to win with the last marble in the center hole. I lost interest after, but this brings up some fond memories. I'll try and reproduce my childhood performance :)
Are there good heuristics for this game or is it essentially just a matter of brute forcing the game tree?
Not the OP, but it’s partly a parity problem—if a marble starts in a position with (say) an even row # and odd column #, then no matter how it jumps, it will always have an even row # and odd column #.
This is weird because this is the kind of game you know you have played at least a couple of times a long time ago before, you instantly know how to play it but I personally cannot remember where nor when.
Not sure where you are from but if you go to any Cracker Barrel in the US one of these puzzles can be found on your table.
A solution that I found is this:

1) Remove the edges until you have a little house;

2) Rotate on a side of the house until you have a cross;

3) Ending left to the reader.