Sit adjacent to your opponent, facing at a right angle, rather than opposite your opponent. Take one stash and two sets of cards numbered 1-4. Deal out one set of cards 1-4 from left to right in front of you, so that the eight cards from you and your opponent label the rows and columns of a virtual 4x4 square grid. The other two sets of cards will be used later. Choose or determine a starting player.
Intersect is played in two stages: the placement stage and the removal stage.
Alternate turns. On your turn, place a pyramid of any size from your stash onto one of the squares in the virtual grid formed by the cards on the table. There are two restrictions on placement:
Take an unused set of cards numbered 1-4 into your hand. Secretly select a card from your hand and place it face down in front of you. Each card corresponds to the same numbered card on the table in front of you, indicating one of four columns of the virtual grid. When both players have selected a card, simultaneously reveal them. The columns indicated by the two cards intersect at one square in the virtual grid; if the topmost pyramid in this square belongs to you, remove it and place it in front of you, and then discard another card from your hand face up. (If the square is empty, then neither player discards a card.) Both players then select another card from their remaining hands face down, and the game continues. If at any time your hand is empty, immediately put all four cards back into your hand; in particular, if you win a pyramid with the last card in your hand, you must pick up all four cards, discard one face up, and then continue with the remaining three.
When you have at least three pyramids of each size in front of you (three large, three medium, and three small), the game is over and you win!
I invented this game in August 2005, inspired by the game-choosing game described in Piers Anthony's Apprentice Adept series of novels. I read these long ago, and I can't even remember if that game had simultaneous selection or not, but I remember thinking that the mechanism of each player choosing a row or column might make an interesting game. I'm not sure I was successful, though.
I submitted Intersect to the Fifth Ice Game Design Competition. The results have still not been posted, but since there were only 5 entries I'm guaranteed to do better than my previous entry!