Heads or tails?

In the first post of this series on games and mathematics, I explained what brought me to thinking and writing on this topic. In subsequent posts, I analysed a game called Four Corners and explored the math behind the 100 m track and the 200 m/400 m track on my school playground.

From the title of this post, I guess that you can make an intelligent guess on what this post might be about! 😃

Let's be fair...

Think back to any team game or sport that you played as a child. In all likelihood, there would have been two teams, each led by a captain. When I was at school, such games included cricket, catches, basketball among others.

Now, let us look at how the game usually started. Two children would be chosen as captains - invariably the two considered to be the best at the game or the two bossiest kids in the group! 😅 Now, how did we decide which captain would get the first choice in the selection of his/her teammates?

Here is where an intuitive sense of fairness coupled with what we had observed sportsmen/women doing in such situations stepped in. We tossed either a Re. 1, Rs. 2 or Rs. 5 coin and one of the captains would call 'heads' or 'tails'. The coin was seen as a fair, unbiased tool that would not favour one captain over the other. It was pure chance (or luck) as to whether one would call correctly or not.

Source: http://technsofts.blogspot.com/2011/08/new-one-rupee-coin-2011.html

Mathematically, what makes this fair? Well, there are two possible outcomes - {heads, tails} - and two captains. So, whoever calls has a 1 in 2 chance of being right i.e. a 50% chance of being right. Consequently, the other captain also has a 50% chance of being right. This is as fair as it can get!

The simple explanation outlined in the previous paragraph contains key ideas that are foundational to probability - one of the chapters that I found most fascinating in high school. The notion of favourable outcomes and total outcomes, when combined with permuations and combinations (another core high school topic), leads to a host of enriching, stimulating problems that pushes one's logical reasoning skills to the limit.

When we didn't have a coin handy, we would resort to a third player picking up a tiny stone and holding it in his/her fist ('in') or leaving his/her fist empty ('out') without the two captains looking. Then, one of the captains would call 'in' or 'out'. Once again, his/her chances of being right was 50%.

After the teams were selected, we would repeat the process to determine court sides, which team would bat/bowl first etc.

In essence, we were using the ideas of probability and fairness much before formally learning the concepts in class!

Probability in individual games

One popular game that we boys used to play was Kings. All it required was a tennis/rubber ball and a reasonably flat area. If all the players were present, the game could literally begin in about 15 seconds - you can imagine how important every second of the games periods were to us! There was no limit on the number of players which further added to its appeal.

The rules were straightforward.
  • An unlucky person would be selected (called the 'den') and then the rest of us would run and disperse ourselves away from the den.
  • The den's objective was to hit a player on the leg below the knee with the ball.
  • He was allowed to run with the ball but, if he missed, he had to go and retrieve the ball which gave the other players time to reposition themselves further away from him.
  • When the den scored a successful hit, that person joined the den's team and now there would be two dens to run from. They could pass the ball to one another which added to the challenge for the rest of us.
  • This process continued until there was only one non-den player remaining; that player would be crowned the 'King'.
Now, the question arises - how did we kids choose the first den? There were no teams, no captains and no coins to help us out...

Well, here's what we did - we made a foot circle similar to the kind shown in the image below. We made the circle sufficiently big so that each of us could have the tip of one of our shoes touching the boundary of the circle and there was a large enough area for the ball to bounce within it. Then, one of the players would hold the ball at a height vertically above the centre of the circle and drop it. Due to the approximate nature of the vertical estimation above the centre and the uneven surface, the ball would naturally move in a random direction after its first bounce and eventually touch a particular player's foot first. That person would become the den! 😀

Source: https://www.nuffieldhealth.com/article/runners-feet-of-endurance

What is the hidden mathematical understanding (or assumption?) in the above exercise of choosing a den that made it seem inherently fair to us 10-12 year olds?

First, we realised (or intuitively sensed/guessed?) that, since most of our feet were of roughly similar sizes and that we all had to buy the same brand of shoes from a designated shoe shop, there was a symmetry to the foot circle that we were making. After all, a circle is the most symmetric two-dimensional shape in mathematics!

Second, due to the aforementioned symmetry, we gathered that each of us had a 1 in 'n' chance of becoming the den when we used the foot circle approach where 'n' represents the total number of players. This seemed reasonable to us and is, in fact, the fairest way of distributing the odds. Once again, we were using ideas of probability and fairness very early in our school lives without even realising it!

I'll end this post with a related example from basketball as I am an avid follower of the game. In the United States of America, the National Basketball Association (NBA) is the body responsible for the game. NBA games begin with what is called a 'jump ball' - the tallest player from each team stands around a circle (there's that wonderfully symmetric shape again!) and the referee tosses the ball vertically upwards (see the image below). Both players jump and aim to tip the ball to their teammates so that they can try and score on the opening possession of the game and gain an early advantage.

Source: https://www.zimbio.com/photos/Chris+Bosh/2014+NBA+Finals+Game+Five/CvZVaxAbjEh

While the NBA could have resorted to using a toss of a coin to determine sides and the opening possession, this is a far more exciting and electrifying way of starting the game while still being relatively fair to both teams! 🏀


  1. These comments are more for you than other readers.
    > Search for `fariness` on the webpage and you'll find a typo. I am sure you wanted `fairness` there.
    > I remember Kings was also played a lot using a ball made out of crushed aluminium foil found lying anywhere on the field :) I had forgotten the term `den` though!! We played a lot of fun games when we were kids. Imagine if all of us met now and tried to play Kings, oh what a disaster it would be :P
    > I wonder how fair the ball drop in Kings, or the Jumpball in basket ball is, considering there might be substantial human error because the person throwing the ball is not in-line with the direction of throw. In any case, I hope NBA referees train their vertical throwing skills well.
    Great post explaining probability. Cheers!!

    1. Good catch - I've corrected the typo.

      Yes, adults playing Kings might be either a disaster or a ton of unexpected fun!

      Neither the ball drop in Kings nor the jump ball in basketball are perfectly fair. These are approximations of fairness that are grounded in convenience (in the case of Kings) and trying to provide a more interesting experience for fans (in the case of the jump ball). There are numerous cases of NBA referees messing up the jump ball but, in those cases, players and fans look at it as "you win some, you lose some; it all evens out in the end!".

  2. Very interesting take on the passive role of probability in our childhood games. Indeed we were intuitively aware of the pressing need to be fair in our choices and we developed innocent methods like the circle of feet or the fist activity that you've mentioned. Thanks for revisiting these aspects of reasoning that now lie long-forgotten in a distant faction of our memory.

    1. Really glad you liked it! Personally, writing these takes me down memory lane too - all those games periods (and free periods!) at school were times that we looked forward to with so much anticipation!

  3. Hi shreyas. This post is a really fun and light read. And very nicely you have shown the co relation of symmetry in these games. And buying same shoes from same brand is a nice whacky idea

    1. The same shoes from the same brand was to make it truly a part of the school 'uniform'. It also reduced the chances of children discriminating against one another on the basis of the quality of their clothes or shoes. This could have happened because my school catered to children from a variety of economic backgrounds and cultures. :-)

  4. We used to play a variant of the Kings game you described. I would dread the game because, unfortunately, we didn't have a rule to hit only on the legs. As with most other outdoor games in my childhood this game also often resulted in silly fights.


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