Patchwork is a 2-player game by Uwe Rosenberg which uses known mechanisms (tile-fitting, a time row, a circular card row, economics with multiple resources) and combines them in a refreshing way. Like many 2-player games, it can get confrontational as denying your opponent nice things is as good as getting nice things yourself. There has been some discussion about which tiles are "nice", which ones are adequately priced, and which ones are the real bargains.

The cost of tiles in buttons should somehow be dependent on their other features, which are their size and shape, their cost in time, and their button yield. So let's look at some of the features, and see if statistics can help us there. Images are clickable to inspect them at their original size.

# Return of investment, and how time is money

Button yield should be a major factor. A patch purchased on your first turn pays off eight times. If only I could afford that fancy piece of cloth adorned with three buttons. My start capital of a mere five buttons, however, only permits me to purchase half of the double button producers, and none of the triple-buttoned cash cows. Let's see how purchase price relates to expected gain.

There is quite an association, as could be expected.

One other factor determining the price of a patch should be the time that goes into sewing it into the design. You have 59 spaces to cover before your 9 by 9 quilt has to be finished, so all things being equal, the lower the time, the higher that cost should be.

Most illustrative is a pair of tiles where all things *are* equal, and that's the two corner triominoes, which have a time and cost of (3, 1) and (1, 3), respectively.

Let's see how the correspondence between time and cost looks in general.

Now this is a lot less clear-cut. If anything, time is positively associated with cost. How can that be? Apparently, other factors, namely the button yield, play a greater role and are masking the effect here. Let's separate the plots by their button yield.

This is a lot more insightful now. We can now apply linear regression to our data and see how exactly time and yield affect button prices. A linear regression of patch costs on button yield and time used scores an adjusted R² of 77 per cent, which means 77 per cent of the variance can be explained by those to cost factors. Here are the effect estimates with their confidence intervals.

parameter | estimate | 2.5 % | 97.5 % |

(Intercept) | 3.643 | 2.497 | 4.789 |

buttons | 3.444 | 2.725 | 4.162 |

time | -1.098 | -1.573 | -0.624 |

We can see that each button sewn on a patch hikes up the price by about 3.4 buttons … which means that those patches are worth acquiring as long as your time counter has at least four buttons ahead of it. Time and button cost seem to have similar weight, as was exemplified by those two corner triominoes.

I think this is the moment to remind ourselves that most tiles are situational. Sometimes you need badly to hike up your income, lest you'd be screwed for choice for the next few moves. Sometimes you want to take *more* time to sew that patch in to gain that leather patch before your opponent, conveniently fixing a gap. Whatever we end up with in this analysis, it doesn't spare us thinking about what is most helpful in the current situation. The idea of the exercise is rather to get an idea about how Uwe got to price the tiles the way they are.

# Size does matter, or does it?

While 77 per cent explained variance is pretty good (and would have me gloating and raving if I achieved this at my job working with medical data) we notice that there are cost gaps between similar tiles left. So what could be another factor? Size, of course, as each empty square eats two of your hard-earned buttons at the end of the game. So tile size should count for a lot in linear regression, right?

parameter | estimate | 2.5 % | 97.5 % |

(Intercept) | 3.6 | 1.609 | 5.59 |

buttons | 3.443 | 2.711 | 4.175 |

time | -1.1 | -1.587 | -0.613 |

size | 0.01 | -0.356 | 0.376 |

Wrong.

Size does not matter.

But how can that be? Possibly Uwe thought that big tiles are more situational and harder to fit in? Let's think of a measure that reflects the "awkwardness" of a tile. The least awkward one is the rectangle. So why not draw a rectangular bounding box around each tile and count the number of squares outside the tile, the number of "holes" in the shape? The "H" heptomino would have two holes by that measure, the only octomino in play four, and the cactus hexomino is discounted by its six holes which make it hard to fit in, placing its cost at the price of a measly domino with no holes. So let's see how the number of holes affects the price.

parameter | estimate | 2.5 % | 97.5 % |

(Intercept) | 2.502 | 0.725 | 4.28 |

buttons | 3.606 | 2.985 | 4.227 |

time | -1.22 | -1.634 | -0.806 |

size | 0.515 | 0.1 | 0.93 |

holes | -0.467 | -0.725 | -0.209 |

And lo and behold, not only does the number of holes affect the price, but when considering the number of holes in addition to size, size enters the equation all of a sudden!

Another thing could be considered. Some tile shapes are invariant to flipping or rotating. Some of them don't change their appearance at all, no matter how you twist and turn them (I'm looking at you, x pentomino). Some of the tiles are versa-tiles, others stubbornly cling to one way to be built in. Why not differentiate the tiles according to the number of different shapes they can assume, i.e. 1, 2, 4, or 8? A higher versatility would not necessarily mean a higher use in your current situation, as you can only fit the patch in one way, but it means it is of use more often. Let's see how versatility influences pricing.

parameter | estimate | 2.5 % | 97.5 % |

(Intercept) | 2.544 | 0.417 | 4.671 |

buttons | 3.608 | 2.973 | 4.242 |

time | -1.219 | -1.642 | -0.796 |

size | 0.514 | 0.09 | 0.937 |

holes | -0.467 | -0.73 | -0.203 |

orient | -0.015 | -0.405 | 0.376 |

Again, the answer is "not very much". Nor does it affect the influence of the shape parameters "size" and "holes", which are more or less invariant to the inclusion of the orientation parameter. But maybe I have forgotten another term associated with price whose inclusion makes the orientation parameter stand out.

# Bargains, and button sinks

So, which are the bargains and which patches would we rather avoid? This can be answered by a technique called "residual plot", which is a scatter plot of the deviation of the price from the price we expect if we believe in the model (which is, in this case, the model without the "orient" parameter). The x axis of the plot reflects the expected price.

What we can see is that the latin cross hexomino, given away for free, is worth every button spent on it, and more. There are two tiles which the model would suggest could have been given away for free. Those are the "Z" hexomino with its eight holes and the corner triomino eating three of your precious time. The most extortionate price seems to be 10 buttons for the long L hexomino. In this case, however, the "number of holes" metric seems to be a bit harsh as the "L" shape is rarely unwelcome. Even so, more than half of the patches (within the zone between -.5 and .5) seem to be adequately priced, as raising or lowering the price by one button would only increase their contribution to the total variance.

# Conclusion

There are a couple of interesting lessons I see here:

- Size looks underpriced, if you consider the value of two buttons per unfinished square. The large tiles come with some downsides, however. One of them is than once you have obtained the 7 by 7 prize, you can't fit more than one tile with a 3 by 3 or larger bounding box in. It seems to me that forgoing the 7 by 7 in favour of collecting large tiles up to the end game is a viable strategy.
- Even in simple models like linear regression it is possible that the effect of some parameters may mask or confound the effect of other parameters. Multiple predictors may influence each other in a non-obvious way. Medical literature [is http://www.ncbi.nlm.nih.gov/pubmed/22106764] rife with predictive models built by selecting the parameters by their influence in simple models where no other explanatory variables are considered. With that approach, we'd have been stuck here with button yield and time as meaningful explanatory variables, with time showing an effect in the marginal model which is *opposite* to the effect in all models with button yield included. A great example of ecological fallacy, by the way.