## Point distribution in reversed grid races

In motor racing, as in other similar competitions, it makes sense for the amount of points given to be a decreasing function of position at the finish line: for example, the current Formula 1 scoring system awards

 25 18 15 12 10 8 6 4 2 1

points to positions 1-10. Same system is used in FIA Formula 2 and Formula 3 races held on Saturdays. However, their Sunday race has partially reversed grid: those who finished 1-2-…-8 on Saturday start 8-7-…-1 on Sunday, while 9-10-… start where they finished. Can this reversal make it profitable to give up a position on Saturday?

The Sunday payouts are smaller: only top 8 earn points, in the amounts

 15 12 10 8 6 4 2 1

– that is, same as the Saturday sequence without the first two terms. If the Sunday race has no position changes (which is not out of question, considering F2 venues include Monaco and Budapest) the totals amounts earned by those in positions 1-10 on Saturday would be:

 26 20 19 18 18 18 18 19 2 1

By this logic, finishing 8th on Saturday would be slightly better than finishing 7th. And of course, there is a huge difference between being 8th and 9th on Saturday. Let us see what happens in reality, when overtakes do occur.

## 2019 Formula 2 season

For each position 1-10 on Saturday, the table states the points earned on Saturday, average points earned on Sunday, and average Saturday-Sunday total. Bonus points for pole position and fastest lap are not included, in order to focus on the effect of the finish position alone.

Sat Pos Sat Pts Sun Pts Total
1 25 5.6 30.6
2 18 5.6 23.6
3 15 5.5 20.5
4 12 5.7 17.7
5 10 8.1 18.1
6 8 5.7 13.7
7 6 6.9 12.9
8 4 8.2 12.2
9 2 1.5 3.5
10 1 1.4 2.4

Finishing 5th on Saturday is on average more profitable than finishing 4th. The gambit here is that losing 2 points on Saturday, one gets on the second row of the starting grid on Sunday (while the 4th place on Saturday becomes 5th, hence the 3rd row, on Sunday). The second row start gives an opportunity to quickly overtake the potentially slower drivers on the front row (after all, they finished 7-8 on Saturday) and take the lead. And indeed, three of the Sunday races of the 2019 F2 season were won by the driver who finished 5th on Saturday. It was a different driver each time (de Vries in Barcelona, Sette Câmara in Spielberg, and Aitken in Silverstone), so it does not look like anyone is intentionally executing this gambit.

Finishing 8th on Saturday maximizes the expected Sunday payout; in particular, 4 of the Sunday races were won by the driver who finished 8th on Saturday: Hubert did it twice in Monte Carlo and Le Castellet, then Schumacher in Budapest, and Aitken in Sochi. But when Saturday points are included, finishing 8th becomes less profitable than higher positions, although it is nearly the same as 6th or 7th.

## Graphing Formula 1 seasons 1960-1984

The earlier post Graphing Formula 1 seasons 1985-2018 had its scope limited to 1985-2018 because of how strange the earliest days of the sport were. Several times in the 1950s there were two drivers sharing the race win, or sharing the second place, or both. This does not really work for my approach of visualizing the results by a graph with edges connecting the drivers finishing in positions 1 and 2. But from 1960 onward, every Formula 1 race (with one exception in 1983) had exactly one driver finishing first, and exactly one driver finishing second. So these seasons can still be drawn as graphs, which is done below. Some features not seen in 1985-2018 range are highlighted below.

• Trees: 1961, 1963, 1971. From 1972 onward, every season has a triangle.
• Disconnected forest: 1966
• Bipartite non-tree graphs: 1960, 1969, 1970. All have girth 4.
• Maximal girth: 5 in 1980
• Most vertices: 14 in 1982
• Three connected components: 1960, 1967, 1968
• One edge away from 5-clique and non-planarity: 1973 [also occurred in 2019]
• 1991 and 1998 remain the only pair of isomorphic seasons in the range 1960-present.

Apart from graph-theoretical observations, this period is strewn with driver fatalities in a way that would be unimaginable in modern motorsport. I tried to keep some balance between highs and lows in these brief summaries. No videos of fatal crashes appear here.

## 1960 season

(The layout could be better.) Both two small components have something to do with banked oval circuits, something not normally associated with Formula 1 today. The year 1960 was the last year when Indianapolis 500 was a part of Formula 1 championship; it contributed the Rathmann-Ward component after 29 lead changes in the race. Monza race was inconsequential for the championship, which was already won by Brabham. As for Hill-Ginther, the inclusion of Monza’s old banked oval in the Formula 1 race track led to the race being boycotted by several teams, allowing the otherwise uncompetitive Ferrari team to finish 1-2-3.

Phil Hill gets to keep the initial, to avoid confusion with Graham Hill who will appear on this page soon and will stay around for much longer.

## 1961 season

Our first tree. One would guess that Gurney should be the winner, but he finished 4th in the championship won by Hill. The teammates Hill and von Trips scored three 1-2 finishes in the season, which only one of them would survive. The fatal crash of von Trips in Monza ended the use of the 10km Monza circuit in Formula 1.

## 1962 season

The first appearance of Graham Hill on this page is also the last appearance of Phil Hill. The former won his first championship. At 1962 French Grand Prix, the absence of Ferrari drivers and multiple retirements combined to create the small component.

## 1963 season

Another tree. Also the record gap between the largest and second-largest vertex degrees, Clark with 6 vs Ginther with 2. No surprise here: Clark won 7 out of 10 races.

## 1964 season

A very close one: Hill collected more points than Surtees, but only the six best results counted for the championship, which went to Surtees.

## 1965 season

A single triangle prevents this from being a tree, but it’s one of the most distinguished triangles one could imagine: Clark-Hill-Stewart. They finished 1-2-3 in the season, which was Stewart’s first season in F1.

As if Formula 1 was not dangerous enough in the 1960s, in 1965 French Grand Prix came to Circuit de Charade winding around an extinct volcano, with no run-off areas and with volcanic rocks falling on the track.

## 1966 season

The only disconnected acyclic graph in the catalog. At its center, Brabham won his third and final championship. The season opener at Monaco created the small component, with the teams scrambling to adapt to the new engine specifications (3L instead of 1.5L):

Although Stewart won the opener, he would only finish 4th and 5th for the rest of the season.

## 1967 season

Another three-component year: Rodríguez-Love comes from the season opener in South Africa, and Gurney-Stewart from Spa-Francorchamps. The triangle Hulme-Brabham-Clark finished 1-2-3 in the driver standings. Hulme somehow managed the feat without a single pole position.

## 1968 season

The last (ever?) three-component graph, although the layout does not make this clear. The Siffert-Amon component is not particularly notable, other than being the first victory by a Swiss driver. The Ickx-Surtees component was created at French Grand Prix, the place of Schlesser’s fatal accident.

Five Grand Prix drivers died in racing accidents in 1968, including Clark who won the season opener. Safety measures would begin to be introduced next season at the insistence of several drivers led by Stewart.

At the center of the large symmetric component, Hill won the championship.

## 1969 season

Stewart’s first championship. The seasons 1969-1970 produced the only two connected graphs of girth 4.

## 1970 season

Stewart and Rindt again appear in a 4-cycle in a girth-4 graph. This time Rindt won the championship, but it was awarded posthumously. On the brighter side, Fittipaldi made his F1 debut this year, and took his first win at the U.S. Grand Prix. The first Grand Prix for a Brazilian driver, and definitely not the last.

## 1971 season

The last acyclic graph in F1 history (so far). The natural guess is correct: Stewart was the champion. Fittipaldi is again on an edge of the graph – his appearance is due solely to his 2nd place in Austria, where Siffert took the last win of his career.

Both Siffert and Rodríguez, who appear at distance 2 from the center of the graph, died in separate racing accidents during the year.

## 1972 season

Just two years after Fittipaldi became the first Brazilian driver to win an F1 race, he became the youngest (to that point) F1 champion.

## 1973 season

This is the closest Formula 1 ever came to a non-planar graph: the only edge missing from a 5-clique is Cevert-Revson. One can imagine a few ways in which a 5-clique could be completed. One was the Dutch Grand Prix, where Cevert was second – if Revson won instead of being 4th. Instead the event was noted for the death of Roger Williamson which better fire safety measures would have prevented.

The final chance to complete the 5-clique was the U.S. Grand Prix, where Revson progressed from last place at start to 5th at finish. But by that point Cevert was already dead. As for Revson, he would be killed in a testing accident a few months later.

This was Stewart’s last championship and last season in F1.

## 1974 season

A rare graph of diameter 6, which shares this record with the 1962 and 2009 seasons. The champion, Fittipaldi, is at the center of a wheel subgraph. Three of his neighbors are future champions.

## 1975 season

Lauda’s “unbelievable year” in which he won the championship by a wide margin. His only retirement of the season, in Spain, is responsible for the small component Mass-Ickx (poorly placed on the layout). The concerns over the safety of the circuit led to Fittipaldi not taking part in the race. The race cost the lives of five spectators and was ended after 29 laps instead of the scheduled 75.

The Silverstone race was shortened as well, but for a different reason: a strong hail storm. It turned out to be Fittipaldi’s last race victory.

## 1976 season

Hunt won by 1 point over Lauda in a season that is difficult to summarize. Lauda had a near fatal crash at the old 22.8 km Nürburgring circuit, which luckily did not end his career.

The following race was in Austria where Ferrari withdrew in protest against Lauda’s disqualification in Spain (and Lauda was in no condition to race anyhow). This race created the Watson-Laffite component and still remains the last F1 race without Ferrari.

## 1977 season

Lauda won the championship despite sitting out the last two races of the season, and despite winning only 3 of the races (versus 4 won by Andretti). The season had more than its share of fatal accidents, but I prefer to highlight the Swedish Grand Prix, which was the first victory for Laffite, as well as first for a French team.

Laffite’s victory was unexpected enough that the race organizers did not arrange for La Marseillaise performance during the podium ceremony. Well, better late than never:

## 1978 season

Andretti won the championship, and remains the last American driver to do so. Peterson appears on the graph for the last time – he died following an accident at Monza. Fittipaldi finished 2nd at his home race, marking his final appearance on these graphs – although with two grandsons currently racing, we might see the name Fittipaldi in F1 again. In other family notes, the season marked important steps for two drivers who became both F1 champions and fathers of F1 champions: first win of Gilles Villeneuve and first race of Keke Rosberg. Villeneuve’s first victory came at his home race.

## 1979 season

Schechter won the championship for Ferrari, the last driver to do so until Schumacher in 2000.

## 1980 season

The only non-bipartite triangle-free graph here; it is a 5-cycle with four appendages. The mostly-French cycle of Jones-Piquet-Arnoux-Laffite-Reutermann finished 1-2-6-4-3 in the championship. This was also the debut season of Prost, who does not appear on this graph, but is present on a dozen of the graphs that follow (continuing into 1985-present).

## 1981 season

The graph offers little clue to who might win the championship (Piquet did). The French Grand Prix was interrupted by heavy rain when Piquet had the lead. But since less than 75% of the distance was covered, the race was restarted, and Prost won on the strength of the shorter second stint. His first victory could be considered a fluke at the time, but he had 50 more afterwards.

## 1982 season

In a messy season that tied the record for diameter 6, Rosberg won despite scoring just one race victory – a situation made possible by a career-ending injury to Pironi, who led the championship at the time of his crash. Villeneuve scored his final victory in San Marino, two weeks before he was killed during qualifying in Belgium. On the brighter side, Lauda un-retired and won twice, preventing the graph from splitting into two sizable components. Without him, Villeneuve-Pironi-Piquet-Patrese would have been the largest small component in F1 history.

## 1983 season

A very close one: Piquet by 2 points over Prost. The small component Watson-Lauda comes from the United States Grand Prix where they started 22nd and 23rd, respectively. Winning from 22nd grid position… has not happened in F1 since, and is unlikely to happen anytime soon, given there are fewer than 22 cars nowadays.

Should this small component even exist? Piquet, Rosberg, and Lauda finished 1-2-3 in Brazil but Rosberg was disqualified for a push start. Ordinarily, that would mean that Lauda becomes 2nd, creating a Piquet-Lauda edge, and thus connecting the graph. But no… instead of Lauda and others being promoted, the second place simply was not awarded to anyone. So, oddly enough, this race contributes no edge to the graph.

## 1984 season

This time, it’s Lauda over Prost by 0.5 points. How frustrating that had to be, especially considering that Prost won 7 races versus Lauda’s 5. The fractional points came from the rain-stopped race at Monaco.

The Monaco race also contributed the Prost-Senna edge to this graph, in Senna’s first season.

## Graphing Formula 1 seasons 1985-2018

This post summarizes Formula 1 championships (1985-2018) by way of graphs: the outcome of each race is represented by an edge between the drivers who finished #1 and #2. The graph is undirected (no distinction between the winner and 2nd place is made), and simple (no record of multiple edges is kept). This erases some of the information, but depending on how much you care about F1, the graphs may still be enough to bring back some memories.

All graph-theoretical “records” are based on 1985-2018 data only, 2019 season being the subject of a separate post: Year 2019 in Formula 1 and feeder series. Some highlights:

• Most vertices: 12 in 1997
• Fewest vertices: 5 in 2000 and 2011
• Most edges: 16 in 2012
• Fewest edges: 6 in 1988, 2002, 2011, and 2015
• Largest maximal degree: 6 in 1990, 1997, 2004, and 2012
• Smallest maximal degree: 3 in 1996
• Largest minimal degree: 2 in 1989, 2016, and 2018
• Largest diameter: 6 in 2009
• Smallest diameter: 2 in 1993, 2000, 2001, 2002, 2007, 2011, and 2016
• Disconnected: 1985, 1991, 1996, 1998, 1999, 2006, and 2008
• Isomorphic seasons: 1991 and 1998
• Hamiltonian cycle: 2016 and 2018
• Triangle-free: none (hence no trees and no bipartite graphs)

Appropriately, both Hamiltonian cycles include Hamilton.

## 1985 season

This was the year of Senna’s first race victory, but the championship went to Prost, who shared maximal vertex degree (4) with Rosberg (Keke Rosberg, of course, not his son Nico Rosberg). This is also one of the few seasons with a disconnected graph. A small connected component, such as Angelis-Boutsen here, likely indicates something weird… in this case, the 1985 San Marino Grand Prix at Imola where Senna ran out of fuel and Prost was disqualified.

## 1986 season

Prost won again, this time with vertex degree 5.

## 1987 season

The four-way battle between Mansell, Piquet, Prost, and Senna fell just short of creating a complete subgraph on four vertices. Their best chance of creating ${K_4}$ was at Detroit, where Senna won and Prost was 3rd. Piquet won the championship.

## 1988 season

The graph is smaller than the previous ones, but is actually larger than one would expect, considering that Senna and Prost combined for 15 wins in 16 races. Berger extended this graph by his win at Monza, in the season otherwise dominated by McLaren. The graph also suggests that Prost should win the championship, and he would have if the champion was determined by the total of all points earned as it is now. But only the best 11 results counted then, and Senna won by that metric.

## 1989 season

Again just an edge short of ${K_4}$ subgraph, but this time it was not a four-way battle at all. Berger only finished 3 races (but in top two every time). Senna and Mansell also had too many retirements to challenge Prost for the championship. This is the first time we see a graph with no vertices of degree 1. But there is no Hamiltonian cycle here.

## 1990 season

The first time we see a degree of vertex 6, and the second time Senna is the champion.

## 1991 season

Another disconnected graph, with Piquet scoring his last career victory in Canada under strange circumstances: Mansell’s car stopped on the last lap when he led by almost a minute and was already waving to the crowd.

If such a mishap also happened at Silverstone, where Mansell, Berger, and Prost finished 1-2-3, we would have ${K_4}$ as a subgraph. Senna won the championship for the last time.

## 1992 season

Sorry about Schumacher’s name being cut off… this was the year of his first race win, at Spa-Francorchamps. Meanwhile, Mansell utterly dominated the championship.

## 1993 season

The first time we get a graph of diameter 2. It suggests Hill was the winner, but in reality he finished third in the championship, with Prost winning for the last time in his career.

## 1994 season

The year of Senna’s death; he does not appear on the graph. Hill has the vertex degree of 5, but Schumacher won the championship by 1 point after their controversial collision at Adelaide.

## 1995 season

That’s pretty close to the wheel graph on six vertices – the only missing edge is Häkkinen-Coulthard. They would score a lot of 1-2 finishes for McLaren in the years to come, but at this time they were not teammates yet. At the center of the incomplete wheel, Schumacher won the championship by a wide margin.

## 1996 season

Another small component, another highly unusual race: wet Monaco Grand Prix, where only three cars made it to the finish and Panis scored the only victory of his career.

Hill won the championship in which no driver had vertex degree greater than 3, the only such season in our record.

## 1997 season

This season holds the record for the number of vertices (12). Two vertices have degree 6 (Villeneuve and Schumacher) but surprisingly, there is no edge between them. Although one of them was on the podium in every race except Italy, they were never on the podium together. Their infamous collision in the season finale at Jerez led to Schumacher being disqualified from the championship.

Villeneuve became the last non-European F1 champion to date.

## 1998 season

The small component is due to Carmageddon on the first lap of very wet Belgian Grand Prix.

This is where my decision to include only driver’s last names backfires: Ralf Schumacher gets to keep his initial. In other news, Williams suddenly faded from the picture and McLaren re-emerged with Häkkinen and Coulthard finishing 1-2 in five races. Häkkinen won the championship.

The seasons 1991 and 1998 is the only pair of isomorphic graphs in this collection. An isomorphism maps Schumacher and Häkkinen to Senna and Mansell.

## 1999 season

The small component is contributed by the partially wet Nürburgring race, where multiple retirements among the leaders left Herbert to score his last Grand Prix victory.

Schumacher’s injury at Silverstone took him out of contention. Still, the second championship of Häkkinen was a lot closer than the first one: he won by 2 points over Irvine.

## 2000 season

Finally, we get a complete subgraph on four vertices: the Ferrari and McLaren drivers. The sole appearance of a driver outside of these two teams was at Brazilian Grand Prix, where Fisichella finished 3rd but was promoted to 2nd after Coulthard’s disqualification. If not for this incident, we would have a regular graph in this collection, a rather unlikely event. Even so, this season set the record for fewest vertices (5). A closely fought championship ended with Schumacher collecting his third title.

## 2001 season

This was not close at all: the driver at the center of this diameter 2 graph won with a lot of room to spare.

## 2002 season

Another season of diameter 2. Schumacher finished every race in top two, except for the Malaysian Grand Prix, narrowly missing an opportunity to create a tree (a star graph). This season ties the fewest edges record (6) which was set in 1998.

## 2003 season

More vertices and larger diameter indicates a more interesting season. Schumacher won again, but by mere 2 points over Räikkönen.

## 2004 season

The final season of Schumacher/Ferrari dominance, in which Schumacher won 13 races and achieved the vertex degree of 6.

## 2005 season

This looks like it was between Alonso and Räikkönen – and it was, with Alonso becoming the youngest F1 champion yet.

## 2006 season

Button’s first career win (wet Hungarian Grand Prix) created the small component.

The large component has diameter 2, with Alonso (the champion) in its center. This is also the last graph in which Schumacher appears.

## 2007 season

As in 2000, Ferrari and McLaren combine to form a complete subgraph on four vertices. But this championship fight was as close as one could imagine, with three drivers finishing within one point: Räikkönen 110, Hamilton 109, Alonso 109. And this was Hamilton’s first season in F1.

## 2008 season

For the first time, we have a small component with more than two vertices. Kovalainen’s only F1 victory came in Hungary, where Glock took second place. More notable was Vettel’s first victory, which came in Monza and made him the youngest driver to win a F1 race [up to that time]. Even more notably, Hamilton won the championship by one point, at the end of the final lap of the final race, and became the youngest F1 champion at that time. Here is the Glock’s view of the action, his car slip-sliding on dry-weather tyres.

On the graph, “Jr.” is Piquet Jr. who took second place in Germany but his brief stint in Formula 1 would be remembered for an entirely different reason.

## 2009 season

The graph of largest diameter (6) captures a strange season after major rule changes. It is so close to being a complex tree, but the 3-cycle was completed at Istanbul, where the polesitter Vettel lost the lead on the first lap and then fell behind his Red Bull teammate Webber as well, finishing just 0.7 seconds behind in the 3rd place. If Vettel was first or second in Turkey, we would have a tree. Button won the championship on the strength of the first half of the season.

## 2010 season

The third time we see a ${K_4}$ subgraph, but the first time that it involves more than two teams: the vertices come from Red Bull (Vettel and Webber), McLaren (Hamilton), and Ferrari (Alonso). Although Vettel’s vertex degree is only 3, trailing Hamilton’s 4 and Alonso’s 5, he became the youngest F1 champion in history, a record he still holds.

## 2011 season

The season tied 2000 for the fewest vertices, with 5. The fewest edges record (6) is tied as well: it was McLaren in 1988 and Ferrari in 2002; this time it is Red Bull’s turn. Vettel won the championship by 122 points but it’s not all in the car; his teammate Webber finished only third.

## 2012 season

With 16 edges, this season beat the previous record set by 1997 season, even though there are fewer vertices here. The two degree-6 vertices led the way in the championship, with Vettel beating Alonso by 3 points. Was this the last great season to watch?

## 2013 season

Vettel over Alonso again, but by 155 points this time. This was the last season of V8 engines, and last season of Red Bull domination. Hamilton appears on the graph only because of his victory in Hungary, after which Vettel won the remaining 9 races. The season opener turned out to be the last race [at the time of writing] won by someone not driving Mercedes, Ferrari, or Red Bull:

## 2014 season

The beginning of a new era: V6 hybrid engine, Mercedes, and Hamilton. Also the last time we see a McLaren driver (Magnussen) on the graph: he appears because of the 2nd place in the dramatic season opener.

In a brief moment of Williams resurgence, Bottas took 2nd place in Britain and Germany, forming a cycle with the Mercedes drivers. If not for him, we would have a tree.

## 2015 season

Another 6-edge graph, another season without much competition. Vettel was the only driver to challenge Mercedes on occasions, thus contributing a cycle to the graph. The entire graph is formed by Mercedes, Ferrari, and Red Bull. Hamilton won the championship again.

## 2016 season

The first time we get a Hamiltonian cycle, for example: Hamilton, Vettel, Rosberg, Räikkönen, Verstappen, Ricciardo, and back to Hamilton. Another 6-vertex graph formed by Mercedes, Ferrari, and Red Bull exclusively. Among them, Mercedes and Red Bull drivers form a complete subgraph. With Ferrari fading to third, neither Vettel nor Räikkönen had enough success to extend ${K_4}$ to ${K_5}$ and thus create the first non-planar season. We would have ${K_5}$ if (a) Räikkönen overtook Verstappen in Austria (he was 0.3s behind), after Hamilton and Rosberg collided on the last lap:

and (b) Räikkönen finished 2nd instead of the 4th in Malaysia, where Hamilton’s engine went up in smoke, costing him the championship.

As it happened, we did not get ${K_5}$ and Hamilton did not get the championship, which went to Rosberg instead. But Verstappen got his first victory at Barcelona and still remains the youngest driver ever to win an F1 race.

## 2017 season

Once again, it is all about Mercedes, Ferrari, and Red Bull, with the Mercedes drivers enjoying higher vertex degree. But this time Ferrari drivers are connected by an edge. The last 1-2 finish of Ferrari to date was in Hungary, arguably their high point of the season.

It was all about Hamilton the rest of the season.

## 2018 season

Second time a Hamiltonian cycle appears, for example: Hamilton, Räikkönen, Verstappen, Vettel, Ricciardo, Bottas, and back to Hamilton. Fourth year in a row that only Mercedes, Ferrari, and Red Bull drivers appear on the graph. Second year in a row that Hamilton wins, and his fifth time overall.

## 2019 season

So close to a 5-clique, only one edge is missing: Bottas-Verstappen. It looked like they could finish 1-2 in Austin, but the Law of Planarity would not allow it, causing yellow flags that prevented Verstappen from an attempt at moving from 3rd to 2nd. Hamilton shared the maximal vertex degree with Verstappen, Leclerc, and Vettel, but was never threatened by any of them in the championship.