Generative AI proves the Collatz Conjecture

 It was proudly reported earlier today that mathematics' most famous and challenging problem has been solved.  A team from the UK's University of Cambridge, using a combination of classical maths and Generative Artificial Intelligence (Gen AI) have demonstrated that the Collatz Conjecture (also known as the 3n+1 problem) has been proved once and for all, using a new dynamic algorithm, similar to the type used by the Chess-playing program, Alpha Zero.

The problem is easy to state, but has confounded mathematicians for almost 100 years:  take any number, and if it's odd, then multiply by three and add one.  If it's even, divide by two.  Take this number, and repeat the operation:  if odd, then multiply by three and add one; if even, divide by two.  Continue to repeat this operation, and eventually, you reach 1.  (1 *3) +1 = 4, 4 /2 = 2, 2/2 = 1.

Mathematically:

The question, which has previously remained unanswered, is: does this apply to ALL numbers?

The team from Cambridge's Department for Applied Mathematics and Theoretical Physics worked on devising an algorithm that was able to overcome the Collatz Conjecture's key challenges.  Instead of trying to unpick the chaotic nature of the Conjecture's sequence, they embraced this using their dynamic Gen AI model.  Previously, the challenge of the Collatz Conjecture lay in its number sequence, which can grow to immense sizes swiftly, only to diminish just as quickly.  However, when they programmed their AI algorithm to map every integer in a variable 1-4 dimensional space, and plot each term in all sequences in a four-dimensional matrix, they uncovered a spherical symmetry that they had not expected.  As all real numbers are contained within this four-dimensional hypersphere, the team were able to prove the Collatz Conjecture for all real positive integers.

A 3-D visualization of the 4-D Collatz Conjecture solution
University of Cambridge

As the conjecture’s proof is tied to other mathematical domains, such as number theory and dynamical systems, it is expected that proving it will have far-reaching consequences in these areas, necessitating a profound review of these disciplines.  

A second visualisation of the 4-D Collatz Conjecture solution showing a different 'shadow'
 University of Cambridge

The team have not yet shared full details of how the proof works, but they explained that they mapped all known sequences into the 1-4 dimensional space, and the AI algorithm then arranged them in a way that would maximise their spatial symmetry.  The next step was then to map all the odd, so-called 'April' numbers and connect them to the even numbers.  By demonstrating that any odd number would always eventually path to another point on or within the same hypersphere, they were able to prove that all numbers eventually path to 1.  The algorithm was able to plot 'shadows' of this in 3D, and the visualizations have been as beautiful as they have symmetrical.  

A close up of a 3-D shadow of the 4-D solution, showing the connections between the real integers in Collatz sequences
University of Cambridge

The team plan to publish full details of their findings and proof in an upcoming issue of the Journal of the European Mathematical Society, following a thorough peer review.  I will provide more updates as I find them; I have my own series of articles here on this blog on the Collatz Conjecture, and the variations  5n+1, 3n+3 (which is wild) and 3n+5 (which grows very rapidly).

Rollarama World Football Dice Game

Rollarama World Football is a "shootout dice and card game", and so when I saw it on the shelf of a charity shop recently, I decided to give it a try.

The game comes with full instructions on how to play it "properly", but I just liked the idea and the packaging which clearly showed a six-sided die, a 12-sided die and a 20-sided die.

The dice values aren't 1-6, 1-12 and 1-20; instead the values are the number of goals a particular team scores in a game.  Each country (there are 30 in total) has a card which shows if it's assigned the 6, 12 or 20 sided die and you roll that die for that country when it plays a match.  For example, Canada is a D6 team, while Portugal is a D20 team and Mexico is a D12 team.  There are rules included on how to play this as a two-player game, using the dice and the cards, but I found a more interesting game for one player using the same raw materials.

The questions I was most interested in answering are how fair the game is, and if the game could be won by one of the 'weaker' D6 teams.  To do this, I set up my own World Football championship, the 2023-4 Winter World Cup.  This comprises five groups of six countries (the 30 countries in total) who will play in their own mini-league.  The top team from each group goes forward, along with the top three performing runners-up, to give eight teams. These eight will play in quarter-finals, then semi-finals and a final.  The draw for the knockout stages will be entirely random, I'm not seeding any team.

Here's the starting roster and the groups, all drawn randomly from the deck of 30 cards.  As you can see, Brazil got a very good draw as the only D20 country in a group of D6s, while Group A has no D20s.

Group A:
D12: Serbia, Denmark, Mexico
D6: Saudi [Arabia],  USA, Cameroon, 

Group B:
D12: Croatia, Switzerland
D20:  Spain, England, Germany, France

Group C:
D6: Tunisia
D12:  S Korea, Senegal, Japan
D20:  Netherlands, Uruguay

Group D:
D6:  Scotland, Ghana, Canada, Wales, Morocco
D20:  Brazil

Group E:
D6:  Ecuador
D12: Iran, Poland
D20: Portugal, Argentina, Belgium

Each team in the group plays the other teams once, so in a league of six, there are 15 games altogether.  There are three points for a win, one for a draw and none for a loss.  In the event of a tie on points, then goal difference will be taken into account.

Here are the results:

GROUP A

Serbia	1	USA	3
Cameroon	2	Denmark	2
Mexico	2	Saudi	1
Serbia	3	Cameroon	2
USA	3	Mexico	1
Denmark	2	Saudi	1
Serbia	4	Denmark	3
USA	1	Saudi	0
Mexico	3	Cameroon	1
Serbia	4	Mexico	1
USA	1	Denmark	3
Saudi	0	Cameroon	3
Serbia	1	Saudi	1
Cameroon	3	USA	0
Denmark	2	Mexico	4

GROUP B

Spain	3	Croatia	1
Switzerland	3	England	3
Germany	1	France	3
Spain	5	Switzerland	2
Croatia	4	Germany	0
England	5	France	2
Spain	3	England	2
Croatia	0	France	2
Switzerland	4	Germany	3
Spain	1	Germany	1
Croatia	2	England	5
France	5	Switzerland	3
Spain	3	France	5
Croatia	2	Switzerland	1
England	5	Germany	0

GROUP C

South Korea	2	Senegal	3
Japan	4	Netherlands	2
Uruguay	4	Tunisia	3
South Korea	2	Japan	1
Senegal	4	Uruguay	1
Tunisia	0	Netherlands	5
South Korea	1	Netherlands	4
Japan	3	Uruguay	0
Senegal	1	Tunisia	2
South Korea	2	Uruguay	3
Japan	1	Tunisia	0
Netherlands	0	Senegal	2
South Korea	2	Tunisia	3
Senegal	4	Japan	0
Uruguay	2	Netherlands	4

 GROUP D

Brazil	0	Scotland	3
Morocco	2	Canada	2
Wales	1	Ghana	2
Brazil	1	Morocco	0
Scotland	2	Wales	2
Canada	0	Ghana	0
Brazil	2	Canada	2
Scotland	3	Ghana	3
Wales	0	Morocco	2
Brazil	0	Wales	1
Scotland	2	Canada	3
Ghana	2	Morocco	1
Brazil	0	Ghana	0
Scotland	0	Morocco	2
Wales	0	Canada	2

GROUP E

Iran	3	Portugal	3
Argentina	2	Belgium	5
Ecuador	1	Poland	4
Iran	3	Argentina	5
Portugal	3	Ecuador	1
Belgium	2	Poland	3
Iran	4	Belgium	2
Portugal	4	Poland	1
Ecuador	2	Argentina	1
Iran	3	Ecuador	0
Portugal	4	Belgium	3
Poland	2	Argentina	2
Iran	1	Poland	4
Portugal	5	Argentina	2
Ecuador	0	Belgium	3

Final tables (this game is a statistician's or an analyst's dream!)

GROUP A
Team	P	W	D	L	F	A	GD	Pts
Serbia	5	3	1	1	13	10	3	10
Mexico	5	3	0	2	11	11	0	9
USA	5	3	0	2	8	8	0	9
Cameroon	5	2	1	2	11	8	3	7
Denmark	5	2	1	2	12	12	0	7
Saudi	5	0	1	4	3	9	-6	1

GROUP B
Team	P	W	D	L	F	A	GD	Pts
France	5	4	0	1	17	12	5	12
England	5	3	1	1	20	10	10	10
Spain	5	3	1	1	15	11	4	10
Croatia	5	2	0	3	9	11	-2	6
Switzerland	5	1	1	3	13	18	-5	4
Germany	5	0	1	4	5	17	-12	1

GROUP C
Team	P	W	D	L	F	A	GD	Pts
Senegal	5	4	0	1	14	5	9	12
Netherlands	5	3	0	2	15	9	6	9
Japan	5	3	0	2	9	8	1	9
Tunisia	5	2	0	3	8	13	-5	6
Uruguay	5	2	0	3	10	16	-6	6
South Korea	5	1	0	4	9	14	-5	3

GROUP D
Team	P	W	D	L	F	A	GD	Pts
Canada	5	2	3	0	9	6	3	9
Ghana	5	2	3	0	7	5	2	9
Morocco	5	2	1	2	7	5	2	7
Scotland	5	1	2	2	10	10	0	5
Brazil	5	1	2	2	3	6	-3	5
Wales	5	1	1	3	4	8	-4	4

GROUP E
Team	P	W	D	L	F	A	GD	Pts
Portugal	5	4	1	0	19	10	9	13
Poland	5	3	1	1	14	10	4	10
Iran	5	2	1	2	14	14	0	7
Belgium	5	2	0	3	15	13	2	6
Argentina	5	1	1	3	12	17	-5	4
Ecuador	5	1	0	4	4	14	-10	3

Notes:

Serbia and Mexico were first and second in Group A, as would be expected as D12s.

Group B was won by France, and there were no major upsets.

Senegal won Group C, ahead of the two D20 teams Uruguay and Netherlands.  Senegal won four of their five games, including a 4-1 win over Uruguay and a 2-0 win against the Netherlands.

In a significant shock, Brazil game fifth in Group D.  As a D20 country playing D6s, Brazil were expected to come first, but only won one of their games.  It was very tight among the other teams, as they were all D6 teams and therefore had a high probability of drawing each other.  Six of the 15 games were drawn.

Portugal topped Group E easily, winning four of their five games and remaining unbeaten throughout.  Ecuador, the only D6 team in the group, came last, winning only one game.

The group winners were:
Serbia (D12), France (D20), Sengal (D6), Canada (D6) and Portugal (D20).

The best-performing runners-up were Mexico (A), England (B) and Netherlands (D).

The Quarter Finals, drawn from the eight qualifying teams:

Serbia vs Netherlands
France vs Senegal
Canada vs Portugal
Mexico vs Ghana

Serbia       2     Netherlands     2 (Netherlands 5-2 on penalties)
France       2    Senegal              1
Canada     1    Portugal             2
Mexico     1     Ghana                1 (Mexico 11-10 on penalties)

The Semi Finals, drawn from the four winners

Netherlands vs France
Portugal vs Mexico (who have done surprisingly well)

Netherlands 4 - France 5
Portugal 4 - Mexico 4 (Portugal win 3-2 on penalties)

The Final was unsurprisingly between two D20 teams, France and Portugal.

France 2 - Portugal 1

The dice:

Although I've described the teams as D6, D12 and D20, the dice aren't normal dice with values 1-6, 1-12 or 1-20.  Most football matches don't end with scores like Liverpool 18 - Man Utd 11.  The dice have duplicated numbers - here are the values for their sides:

D6:  0, 1, 1, 2, 2, 3 --> mean = 1.5
D12: 0, 1, 1, 1, 2, 2, 2, 3, 3, 4 --> mean = 1.58 (19/12)
D20: 0, 0, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5 --> mean = 2.7 (54/20)

In conclusion - this is a great game.  I didn't play it according to the instructions, but took the raw materials from the game and made it my own - and thoroughly enjoyed it.  I didn't analyse all the maths (what's the probability of a D6 team drawing against a D12 team? Or winning?) but rolled the dice, recorded the scores, and built and Excel spreadsheet to solve the league maths for me!  Was it fair?  Yes - the D6 teams have a chance of winning (and the multiple upsets during the championship show this) and the D20s have a chance of being knocked out (Brazil in a league of D6 teams).  The values on the dice make sure that every team has a chance of winning, even if it's slim.

In a future post, I'll play a cricket-by-dice game, and compare the results!