But I later realized it doesn’t matter what point we choose. What we’re really doing is adding one extra degree of freedom and using it to minimize the L2 norm of the derivative on [0, 4]. If we instead chose 1.5 instead, the middle steps might look different, but the desired polynomial that has the specified behavior at the other points (including values of the derivative) and which has minimal L2 norm would still be found. (I’m assuming that this polynomial is unique, but in fact I don’t know that this is true. If we’re in some pathological case where it’s not unique, then what I claim is *almost* true).

]]>Yes, that’s right. In the typical language, a “move” consists of one mouse-click-and-drag action. When there are open freecells, it is possible to move multiple cards using one mouse operation (as a shortcut to doing all the card moves individually). As an example, if all 4 freecells are open, you can move a 3-4-5-6 onto a 7 (with the correct suit combinations) with one “move”. Thus the “move” count differs from the “card move” count. Historically this came about from people tracking and sharing moves on usenet groups, and a condensed move notation emerged.

Further, I note that Freecell will automatically place cards on the ace-stack when there is no obstruction. These are also not counted as “moves”, since these are not mouse operations.

For more, I recommend checking out the links I gave in my MSE answer, and perhaps looking up some of the references or ideas you find from there. There’s quite a bit out there on Freecell — it may be that solvers have improved.

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Astoundingly enough, this has already been studied. And I’m almost embarrassed to say that I’m familiar with the result. I used to freecell a lot. And FYI, 11982 is the impossible Frecell game. But I recommend entering in games -1, -2, -3, etc too.

So here are some stats from some studies of freecell. Firstly, the depth of the aces, i.e. how many cards cover the aces, is not a good measure of difficulty. On average, 11.077 cards cover the aces (counting aces). Analyzing the dozens of thousands of deals, it takes an average of somewhere between 42.12 (from a solver that ran 1.5 million deals) and 46.33 (from a solver on 32000 deals, the original 32000) moves to solve. This is a hard measure, as this is based on the quality of the solver – and it is unknown whether these solvers were optimal.

An interesting player-based study showed that about 79% of deals are solved by a person on their first try. It also turns out that some people examine how many freecells (the four in the top left) are actually necessary to solve a game. The impossible 11982 can be solved with 5 freecells. Almost every game can be solved with 3. Over half can be solved with 2. And almost 100 can be solved without any freecells at all. Take that, freecell!

One of the big problems is that freecell games are not at all randomly assorted, and so pencil and paper solutions aren’t around. But lots of people have (surprisingly) cared about these questions, and so these results are all upper bounds. In short, about 45 moves is the average minimum.

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