This game may not actually be an actual solitaire, as it might have been something my grandmother had me do when I was wee to keep me out of her hair. But if anyone DOES know it, I would dearly like to know if I am forgetting some of the rules or playing it wrong.
Play: using standard 52 card deck, deal 4 cards, one into each of 4 piles. If some of the cards have the same rank then move the matching cards to the rightmost, repeating for all matches. Then deal another 4 cards, one onto each stack, and repeat. If all 4 on top of a stack are the same card (four of a kind) you withdraw them from the deck entirely. You repeat until you have no more cards.
For example, if you dealt A 2 3 4, nothing would move. If you dealt 2 3 3 4 on top of that, the left 3 would move onto the right three. This would reveal the 2 that it had covered, and the 2 on top of the leftmost stack would shift to the right.
Once all cards are dealt, you pick them back up in sequence (This is the part I remember the least. I THINK that the leftmost pile ends up on the 'bottom' of the new stack, and the rightmost pile on the top) and then deal through again until you win.