When I was ten my grandmother taught me a form of solitaire that is very mathematical, requiring considerable strategy to win. In fact the game is called Calculation, though I did not know that at the time. I quickly became good at it, whence the game was too easy, so I developed a more advanced version, which is described below and available online.

Each play pile will have 13 cards when complete, culminating in a king. The first pile starts with ace and goes in sequence: ace 2 3 4 5 … king. The second pile starts with 2 and advances in steps of 2: 2 4 6 8 10 queen ace 3 5 7 9 jack king. The third pile holds multiples of 3: 3 6 9 queen 2 5 8 … king, and the fourth pile steps by 4: 4 8 queen 3 7 jack 2 … king. Since king corresponds to 13, a prime number, every pile must contain 13 cards to reach a king, and every pile covers all 13 cards, though in a different order. If you started with some other seed, such as 5, the pile would still progress through 13 different cards to reach a king: 5 10 2 7 queen 4 9 … 8 king. If the seed card is a queen, the pile is the reverse of the ace pile, running: queen jack 10 9 8 7 … 2 ace king. Similarly, jack is the reverse of 2, and so on. Well the seed cards are ace 2 3 4, at least in the basic version of the game.

Below each play pile is a stack, used for temporary storage. Cards can be placed on a stack, or taken off of a stack and moved to a play pile. Only the top card can be taken off the stack, which is why it is called a stack (push and pop). However, you can review the contents of a stack at any time, in case you have forgotten the cards beneath. You cannot, under any circumstances, preview the cards in the deck. That is the future; that is the unknown.

The game consists of a series of moves, where each move is one of the following.

Take the top card off the deck and place it on one of the four play piles or on one of the four stacks.

Take the top card off of one of the stacks and place it on any one of the play piles. Stacks and play piles do not correspond; there are simply four play piles and four stacks. There could be five stacks (making the game easier to win), or three stacks (making the game harder).

Once a move is made it cannot be reversed. There is no undo command.

The object of the game is to complete the four play piles using a series of moves as above. Upon success, the deck and the stacks will be empty. Conversely, if the deck is empty and the stacks are not, and no move can be made, then the game is lost.

If you happen to run across all four kings early, say halfway through the deck,
then the fourth stack has four kings, and you can place other cards on top of these kings with impunity.
I call this happy condition *kings early*,
and it definitely portends success.

If you have three kings down, you may feel comfortable placing some late cards on this stack, knowing there is only one more king to go. If you encounter a queen, a jack, and a 10, they could go on the three kings, and if the fourth king comes along and covers them up, that's not a problem. One of the play piles will finish, and lift this king, and then pile 1, that starts with an ace, will call for the 10 jack queen, and then you can play the last three kings to win. In general, any permutation that has two, three, or even four kings earlier in the deck than chance would dictate is fortuitous.

If the kings are the last four cards in the deck, that's lucky too,
since they are the last to be played,
except it isn't lucky at all, because you don't know where they are.
You're always afraid a king will come along and cap one of your stacks,
and you must plan for this contingency.
Perhaps you leave the fourth stack empty throughout the entire game, waiting for kings to come along.
Thus *kings late* is never in your favor.

At this point you should stop and play a few games to become familiar with the rules and some of the strategies. The rest of the article will make more sense if you know the game well.

A win using only three stacks is worth 4 points, and a win with three stacks and kings played at the end is worth 16. Following the power law, a win using two stacks is worth 64, or 256 if the last four cards played are kings. I achieved the latter three times in my life. A win using one stack is worth 1024, or 4096 with kings at the end, though this has never happened to me personally.

Two piles could start with the same seed, and that makes the game harder to win. Those two piles are always looking for the same cards, and if those cards happen to be late in the deck, you're in trouble. If three or four seeds are the same I simply write it off - deduct one point, shuffle the deck, and start over. It's just too hard to win when all the piles are calling for the same cards.

For the rest of this article, I will assume three stacks, but I will continue to refer to the piles as 1 2 3 and 4 to avoid further confusion in a topic that is already confusing enough. Pretend like you pulled ace 2 3 4 out of the deck, and these are the seed cards for your piles.

Life is better now, and solitaire is once again a recreational pastime, rather than a desperate attempt to keep my mind from wandering into dark places. There is however a little bit of OCD associated with the game. If I start playing I have to win before I can cook dinner, or go to bed, and I especially have to win before anyone leaves the house. Sometimes Wendy will sit patiently on the couch and ask, "Have you won yet?" I usually don't start to play unless I know I have time on my hands, with no near-term commitments. But then my mind conjures up another image.

A victim, perhaps a family member, is tied to a chair that slides up and down on a scaffold. She starts 5 meters off the ground, and each point, as per my point system, raises the chair one meter. When she reaches 50 meters she is set free, but if she ever descends below ground, i.e. the point total goes negative, she enters a pit of lions that eat her alive, or something equally gruesome. You need a lot of time on your hands to play this variant. Some 16-point wins are needed to reach the top, without too many losses along the way. One must be especially careful at the start, as she is quite close to the ground. A sportscaster provides running commentary on my card play, while a Color man fills in the gaps with interesting background information and statistics. Yes, this is a spectator sport. The crowd cheers every time the chair rises into the air, and falls silent when the chair descends. Of course there are some people in the stands, and at home, who are secretly hoping to see a grisly death, but they generally keep their antisocial passions to themselves.

Well that's enough of that; let's talk about some of the strategies that lead to success.

Continuing this example, it is perhaps better to place 5 on the first stack. 5 9 10 jack queen are all end cards as before, and you are still clean. Suppose the next card is 7. The only cards not yet seen are 7 8 and king, so 7 is certainly likely. Card counting is definitely part of a winning strategy. Place the 7 on the second stack and you are clean in three different ways:

{1} jack queen {3} 7 10 {4} 5 9

{1} queen {2} jack {3} 7 10 {4} 5 9

{1} queen {2} 5 7 9 jack {3} 10

Clean is always good, but clean in several ways is better as it provides more options. With the 7 so placed you can accommodate 2 4 9 or 8 (as a play card), ace or 3 (on the first stack), and 7 9 10 or jack (on the second stack).

ace →

{1} jack queen {3} 7 10 {4} ace 5 9

{1} queen {2} jack {3} 7 10 {4} ace 5 9

3 →

{1} queen {2} 3 5 7 9 jack {3} 10

7 →

{1} queen {2} 5 7 9 jack {3} 7 10

9 →

{1} queen {2} 5 7 9 jack {3} 10 {4} 9

{1} queen {2} 7 9 jack {3} 10 {4} 5 9

{1} queen {2} 9 jack {3} 7 10 {4} 5 9

10 →

{1} 10 jack queen {3} 7 10 {4} 5 9

jack →

{1} jack queen {2} jack {3} 7 10 {4} 5 9

{1} jack queen {2} 5 7 9 jack {3} 10

If, instead of the possibilities shown above, a 6 comes along, it is neither a play card nor an end card. Place it on the first stack. If a 4 comes along, you can play 2 4 6, but failing that, the quickest way to turn 6 into an end card is to place ace and 10 on the second stack. In other words, I'm hoping for a 4 right away, but if that doesn't happen let's shoot for an ace and a 10. I am considering the following unwind patterns for my stacks:

{1} jack queen {3} 7 10 {4} 6 ? ? 5 9

{1} queen {2} jack {3} 7 10 {4} 6 ? ? 5 9

Obviously the board is not clean any more. The trick is to steer it in the direction of clean as you draw more cards off the deck.

Try to play your cards in a manner that avoids becoming x hungry, although this is easier said than done. If the seed cards are 8 7 4 4, and you have played an ace on the 7, then the first pile already has an 8, and the next three piles are 8 hungry. Well that's just your Karma. Build your stacks carefully and hope the 8's come soon. If you choose your seed cards randomly, some configurations are harder than others.

Assume a 10 comes along, and it is placed on the third stack atop the king as you expected. The third stack is now 10 king, and has to end the 3 pile. You are now hoping for another king to place on the third stack, so you can perhaps 9 off, anticipating the completion of the 4 pile. Once in a while I must use the third stack to complete the 2 pile, and at that point I might say to myself, "Ok, it's time to jack off."

Keep a close count of the number of kings on the third stack. Each king admits end cards for exactly one pile. Each king allows you to x off for precisely one x. In the example above, the first king, on the floor of the third stack, allows you to 10 off, where 10 is the end card for the 3 pile. You can't use this strategy again until another king comes along and falls onto the third stack. At that point you might queen off, anticipating a queen to complete the ace pile. If all four kings are on the third stack, then you can use this stack for any purpose. This is an advantage of kings early.

If there are no kings down, you can still 10 off, but that is a bit risky. You need a king before the 10 comes along. The odds tilt in your favor if some 10's have already been played, or if the next 10 can act as a play card. Sometimes you are sure to lose the game if you don't x off, even onto an empty stack, so go ahead and take the chance. Hopefully a king will come and rescue you before x materializes.

This is merely an overview of the strategies I use to improve my chances of success. Grab a deck of cards and give it a whirl; I think this will become your favorite form of solitaire. Your smart phone probably includes Spider Solitaire - if only it could play Advanced Calculation as described above, including the victim in the chair who rises and falls commensurate with your play. If you would like to help me develop such an app I'll share the proceeds with you. An online game, acting as the prototype, can be found here.

Shuffling machines