Thuppakki - A mathematical analysis.

In the recently released movie Thuppakki, there comes a scene.  It's an awesome movie in fact. But, let see the mathematical problem in this particular scene and let's analyse the solution for that.

The scene goes like this.  A member of a sleeping cell escapes captivity and plans to plant bombs at 12 different places using 11 other different members.  Immediately after escape, he devices a plan to contact the other 11 members and pass on the information. The protagonist, who wantedly released the terrorist so that he can find out the all 12 members, had to device a plan to eliminate all of them at once.  Missing one will lead to bombing and loss of lives.

There are two mathematical problems here.  For the terrorist, what's the fastest way of communication (without using cellphones). As shown in the movie, he chose a mathematical construct called binary tree. He goes and meets first person. Then the split into two ways. In the next level, each one meets one more person, and then split again into two ways. This is shown clearly in the picture on the right. So, roughly, the communication is cut short to O(log 12) roughly 3 hours. Otherwise, if one has to go and tell all 11 other person, it would have been 11 hours.





Ok, now, the real problem is how hero choses his solution. He simply follows the terrorist's approach. So far so good, but the problem is in the red nodes. Exactly at the red nodes, there are 3 policemen. They have to split into two. If they chose to split such that one person follows the left subree and other follows the right subtree as shown in the graph, it would be a success. But, what gaurentee that the left subtree meets the other two persons but the right one terminates. We would have left with 1 terrorist unmannaned. The probability of chosing the right path to send 2 policemen at this node or in any node is evenly distributed. i.e. 1/2. Unfortunately, this decision has to be made at 4 places where there are 3 policemen. So, choosing the right subtree 4 times again reduces the chance to eliminate all the terrorist. So, for these 4 independent events, the probabily reduces to 1 in 8 chances. (i.e. think of getting 4 times head in a row). So, probability of avoiding any single incident is very less here.

So instead of going with 12 people, he should have gone with at least 16 people for 100% success. Or the director should have selected 16 places instead :-).