Claps!

In an earlier post, I wrote about how children intuitively make use of probability in games. Here, I further explore that premise by taking a specific example called 'Claps!' which my friends and I frequently used to make decisions in our games. I would recommend reading the earlier post since I make references to it in this post.

I was reminded of the claps method by a reader of the blog who recalled using it as a child! 😃 It looked something like this...

Source: https://www.istockphoto.com/in/video/multicultural-kids-group-hands-on-top-of-each-other-outside-gm819959134-135028361

The idea was simple - players would come together and each would put out one of their hands towards the centre of the group until we ended up with a pile of hands. On the count of three, we would all raise our hands and bring them back down away from the centre in one of two ways - palm facing up or palm facing down.

     

Based on the distribution of the two different orientations in the group, we would either pick an odd one out or make teams of players with the same orientation depending on the need.

Here, I will explore the case of picking an odd one out...

To pick an odd one out (three players)

The possible outcomes when three players are involved is shown in Figure 1. The rows boxed in green indicate the cases where it is possible to pick an odd one out. In each green row, the odd one out is marked with a red dot.

Figure 1

Evidently, in 6 out of 8 cases (or 75% of the time), the method of claps is successful in picking an odd one out. Perhaps, this is why we gravitated towards using it when we had three players among which we needed to pick/exclude one.

Also, note the symmetry and fairness - the 6 cases are equally distributed among the 3 players.

The logical next question - will the claps method to pick an odd one out work equally well (or better? or worse?) for 4 players? How about for 5 players? 6 players? What happens as the number of players increases?

To pick an odd one out (four players)

The possible outcomes when four players are involved is shown in Figure 2. The green boxes and red dots are used in the same sense again. Here, the 'P' denotes 'player'.

Figure 2

We can see that, in 8 out of 16 cases (or 50% of the time), the method of claps is successful in picking an odd one out. Decent, but not as good as when there were only three players.

Again, note the symmetry and fairness - the 8 cases are equally distributed among the 4 players.

To pick an odd one out (five or more players)

Following a similar logic:

• if there were 5 players, the method of claps would be successful in 10 out of 32 cases (or ~ 31% of the time). For simplicity, only the 10 successful cases are shown in Figure 3 with the odd one out indicated by a red dot. In the remaining 22 cases, either all 5 hands are in the same orientation OR 3 hands are in one orientation and 2 hands are in the other orientation - either way, picking an odd one out directly was not possible in those cases.

Figure 3

• if there were 6 players, it would work in 12 out of 64 cases (or ~19% of the time)

• if there were 7 players, it would only work in 14 out of 128 cases (or ~11% of the time)

Are you seeing a pattern emerging?

Is there a link between the number of successful cases and the number of players?

What about the number of players and the total number of cases - is there a link there?

A key point here is that there is a steep decline in the effectiveness of the claps method for picking an odd one out as the number of players increases.

We children had an intuitive sense of this - I cannot recall a single scenario in which we tried to use the claps method for picking an odd one out when there were more than 4 players. We resorted to the strategy we used in Kings (see this post) or picked a person arbitrarily so that we could get the game started!

A hidden social element

I will conclude by highlighting something I found interesting and common to both the method used to start Kings and the claps method described above. In both cases, players come together to get the game started before dispersing when the den or the odd one is picked out. I found this act subtle but possibly integral to the inherent social element present in many games.

Standing around in a circle in close proximity or having one's hand a part of a pile of hands perhaps, subconsciously, makes each player feel a part of a group that is invested, for however short a time, in making the game a fun experience for all players. 😀