Flamme rouge for the spatially constrained, and other interesting race tracks

Welcome back to the world of cycle racing!

In a previous article, I took on myself to determine how many unique race track could be build using the pieces available in the game Flamme Rouge, a game of cyclist racing.

While doing so, I encountered all kind of interesting ideas, and had a bit of time to reflect on what makes a race track interesting…

So today, I decided to make a quick post about those!

(As a side note, just for fun, I generated some images for this blog post using Dalle 2, the title image being one of those. All the image not directly related to the analysis are from Dalle2. While not flawless, I found they fit right in for a blog using computers to analyze board games).

Flamme Rouge takes up a lot of table real estate.

One thing someone quickly realize when setting the Flamme Rouge game, is how much space it can take. Since I was exploring possible shapes of race tracks, it made sense to keep some simple statistics about the final size and area covered by each track.

At first, I naively decided to just check for the minimum and maximum rectangular boxes that would contain the track. I though this was straightforward. I was wrong.

Dalle 2 image

When drawing a racetrack, I calculate screen coordinates for a bunch of points for each track piece.

The simplest approach would then be just to find the maximum and minimum values of the track coordinates in X and Y, obtaining a 2d-box able to contain the track.

Here what it would look like:

I was just forgetting one thing: When you are at your table considering your maybe too big race track, there is one thing that you will instinctively do to make it fit: rotate it!

So you would actually end up looking at the same race track fitting a smaller rectangle:

If your screen is your table, you can also rotate your track. But by how much?

Computer don’t have eyes and a brain evolved over millennia to process spatial information instinctively. To find the real answer, so you need to try… All the possible ways.

Since all the possible way is rather exhaustive, the most reasonable approach I found was to rotate each track by increment of 5 degrees.

The only issue being: Rotation are costly in term of computer processing. So to revisit the whole set of possible tracks from the previous article would have taken more than 8 of my computer days… Well, that was unacceptable, so I spent several days finding optimizations lowering the whole thing to 3 days! (That way, I can re-run it, in the too often seen case were you realize you forgot something after the first run! So running the whole thing 3 time take 9 days instead of 24.)

It seems absurd, and it is. But at the same time, it kind of simply is the way!

Let me show you what is the result….

Are the tracks in this article playable?

!! Yes !!

All the shape shown here can be built. The only thing left to decide is where to put the uphill and downhill sections of the track. You should keep the uphill section to a max of 3-4 and the downhill section to 2-3 maximum, but those are just rules of thumb. You can experiment with different variants and see how it goes.

The smallest racetracks

Independently of your table shape, it is always interesting to find the smallest tracks possible, as they will probably fit on any table.

But what do we look for to find the smallest track? Even that is not that simple!

The simplest idea is to find the tracks that fit in a box with the smallest area. Which I have done here, the absolute smallest being the following:

Flamme Rouge track fitting in the smallest area
Flamme Rouge track fitting in a box with the smallest area.

Or if you prefer the real thing (As you can see real tracks are a bit more roomy than my strict representation):

Here are some more that I extracted from the list of 50 tracks fitting in the smallest possible boxed area:

Most of those tracks are fitting in elongated rectangles, most would fit in a box of about 90 x 30 cm (35.4 x 11.8 inches). Here I only show you six, but you can imagine there are multiple variations of those tracks, only changing the location of a few turns without changing the overall shape.

Optimizations will tend to do that.Very good at finding a good ratio, often minimizing one variable, but you end up with a certain type of results and sadly, not much variations.

But what if your table is not a rectangle?

Could you find a more practical track? I looked for that, trying this time to minimize the length of both axis. Which is the same than trying to find the track fitting in the smallest possible square box.

The resulting tracks are much more satisfying to look at (and more varied!)

Here is the absolute smallest track (fitting in a square box of 55 cm or 21.6535 inches). This represent quite a smaller footprint than the suggested tracks included in the game!

As I said, this method tends to produce interestingly convoluted race tracks, as you can see below. And while the more linear one can be fun, alternating those with more twisted tracks can would certainly give a warm feeling of variety to a racing night!

The interesting thing here, is that I kept track of the smallest square containing each track, and can present to you some additional amusing graphs!

How many racetrack for your table ?

So given that I was tracking the smallest box containing each race track, I kept a tally of how racetracks were fitting on different table size (Basically I kept an histogram of the maximum length on each box, so you are guarantee to fit on a squaretable)

Here is the histogram showing the additional number of racetrack that you can fit as the table gets larger.

A graph showing how many new racetrack can be build as we increase the side length of our table

If you prefer to know the total, here is the same data but this time, showing the total number of fitting racetrack as the table gets larger.

Cumulative number of tracks for a table side length in mm

Once again, for the fun of completeness, here is a similar histogram, but this time only considering the area of the enclosing box. So a mix of different box size, some thin and long, other more squared.

The largest racetracks

Finally, given that we had all this data, it made sense to keep track of the largest race track as well. Fair warning, in Flamme Rouge case, those answer are nor particularly interesting, but for the sake of completeness here they are:

As for the smallest racetrack, we can consider two definition of largest.

  • The racetrack fitting in the biggest boxed area.
  • The racetrack fitting in a box with the longest dimension

For the biggest area, the racetracks all look mostly like a large L, much bigger than the smaller box tracks I added as a size comparaison.

For the longest axis, they end up mostly as a long line, with variations on turn placement like the following:

So not very original, but nonetheless interesting in a quest for size extreme (and the sake of completeness…)

In Conclusion

While anyone could just play with the pieces to build a fitting racetrack, I think it was fun to find the smallest racetrack possible.

From a gaming perspective, to get an idea of how big your table really need to be to be able to play.

I think everyone can appreciate how modular boards are a curse and a blessing. Not every combinaison are potentially interesting in term of variety or utilisation of space.

From an optimization point of view: seeing the impact on the final shape of how we define the ‘smallest’ track is a perfect illustration of the difficulty of optimizing anything.

You have to be very careful on how you measure things, and often you have to try different approaches. The first results often being not at all what you expected.

Hopefully, you found some interesting insights in this. And maybe you can adopt on the smaller racetrack for your own use.

If you ever read until this point, (thanks!) if you would like to have an explicit list of small racetrack, let me know, I may produce one and add it here!

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