# Math and Chess – Similarities and Differences By FM Armin Mušović

From a perspective of an average person, math and chess are somehow similar – the word that connects them would be ‘’thinking’’, or even ‘’overthinking’’. And indeed, you must be a good thinker if you want to be successful either in math or in chess. But what surprised me, even from teenage days, is that mathematicians do not often find interests in chess, and vice versa – chess players that I’ve known were not that good in math! How is that possible? Let’s try to speak about similarities between these two areas, about their differences, and at the end about some possible psychological explanations for this phenomenon.

Here are some similarities that I’ve found:

• Both areas are based on scientific principles. Rules, exceptions, axioms, consistency and checking in practice – these are all scientific principles that you can also find in chess, and without which you cannot be a good one;
• Both areas explore numbers and geometry. For math we already know, but in chess, numbers are very important. Pieces value are counted as number of pawns (bishop equals 3 pawns, knight as well, rook equals 5 pawns and so on), and using this method, chess players always evaluate particular position; while calculating, sometimes chess player must compare the number of advanced moves with the number of the moves of the opponent). There are also other calculations of moves when players do not want to lose tempo move (or move which will lose, in the chess meaning, time for something else). When we’re speaking about geometry, in chess we have an artificial one, and let’s say that it’s always in 2D format (maybe pieces are in 3D form, but the patterns are actually in 2 dimension). Let’s try to see one interesting example. In math we have this:

We know Pythagorean Theorem: one leg of triangle squared plus another leg of triangle squared equals the hypotenuse squared (or a2 + b2 = c2). In chess, however, we have something completely different:

If we look at this triangle, and if we count the number of the moves for the king to reach the square h2 from h8, we see that it doesn’t really matter if we go straight or if we go by using two legs of the triangle – it will be again 6. That is not possible in real life, because the shortest distance is always the straight one. From this example we can see that all other geometrical conclusions in chess are also very complex (similar to advanced mathematics).

• Both math and chess have their own ‘’calculator’’ to help solve a lot of tasks. We know that without a calculator we would need a lot of time to solve some tasks with numbers. In chess, engines (or ‘’programs’’) can give you an evaluation of the particular position, way better and faster than even a grandmaster can give. These engines are now so sophisticated that humans literally have no chance to win the game; there are some differences between ratings for humans and engines, but let’s try to give context to understand this better:
1) to become a grandmaster, you need to achieve 2500 rating points (in slow chess)
2) the highest rating in (slow) chess ever was 2882, achieved by the current World champion Magnus Carlsen (in 2014)
3) some engines cannot be even measured, and it seemed that AlphaZero (in 2017) had the rating between 3575 and 3725 points!
• Both math and chess explore possibilities. This is maybe even the biggest connection between these two areas. In math, we know that using the probability theory we can measure percentages and make conclusions, which is also definitely a part of everyday life. In chess, we have the same thing: players are always thinking and calculating about different possibilities. It will be more understandable if we say that the number of possible variations in chess in (only!) the first 40 moves is around 10120! To understand this number better, they say that the number of atoms of the known universe is ‘’just’’ around 1080!

Now let’s try to see what are the differences between these two areas:

• In chess, fighting spirit is an unavoidable part of the game; in math, competitions are present to a much lesser extent.
• In math we cooperate much more with constants and invariant values, while in chess it is opposite – in almost every moment it is possible for the pieces to change their own value (the only ‘’sure’’ thing is that the king is the most important piece and that it cannot be taken!)
• Because chess is a game between two people, there has to be some kind of psychology. And indeed, in real tournaments you have great preparations before games (what opponent is typically playing, how to find some of their weaknesses in the game using chess base and engines, what kind of openings the opponent doesn’t like…), but some players go even further (finding ways to distract opponent from concentrated thinking during the game). In math, we simply do not have anything similar to this.  The opponent is static, simply theoretical.

Now when we know some similarities and differences, let’s try to answer the question: why the representation of people in both areas is not so frequent? Possible conclusion might be that the presence of psychology, sports and fighting spirit (and thus the search for more adrenaline) and choosing greater social cohesion leads certain people to devote their abstract thought to chess, rather than mathematics. It seems that in chess, unlike mathematics, it is not enough just to understand something, but it is necessary to prove the same thing – first to yourself, and then to others. Also, it is easier to create new opening variation or new approach in chess than to find out new principle or axiom in the mathematics, so it is possible that innovative and creative people choose more chess than mathematics to express their own thoughts and feelings. Like we said at the very beginning, it is true – both of them must be good thinkers above and before all.
What are your conclusions?  Is it in fact easier to be creative in chess?  Does serious mathematics lack the psychological aspect competition against an opponent brings?