Does anyone know what this peg game is?
It’s the French and Swedish version of (Peg) Solitaire.

https://www.mastersofgames.com/rules/solitaire-rules.htm


  The Rules of Solitaire
  
  Equipment
  
  The game of Solitaire is most commonly played on a 33 point board (as pictured above left) in a cross shape with 32 pegs, marbles or pieces. In France and Sweden a 37 point board is more common with 36 pegs, marbles or pieces.
  
  Solitaire can also be played on other shapes of board - two of the most interesting are a 41 point board (take the 33 point board and add 3 extra points at the 4 ends of the cross) and the 45 point board (take the 37 point board and add a single point in the middle of each of the 4 square ends - to makes a square).
  
  Preparation and Objective
  
  The game is set up so that pieces fill every hole except the middle hole.
  
  The objective is to remove every piece except one, with the final piece ending up in the centre hole. Solitaire is played by one person and is therefore technically not a game at all, but a puzzle.
  
  Basic Play
  
  The player makes successive capturing moves, removing a single piece each turn until is it impossible to make any more capturing moves.
  
  Each turn, the player captures a piece by jumping over that piece orthogonally (not diagonally) from one adjacent point to the vacant adjacent point on the other side.
  
  Therefore, the first turn can be made only by jumping a piece into the middle hole from one of 4 possible points.
  
  Advanced Play
  
  Once you have mastered the basic game, target a different hole as the hole that the final piece should finish in. You can also aim to get certain patterns of pieces left over.
  
  Interestingly, it has been deduced that the 37 point board is less complex than the 33 point board.