01/06/2026
𝐖𝐇𝐄𝐍 𝟑𝟏% 𝐁𝐄𝐀𝐓𝐒 𝐈𝐌𝐏𝐎𝐒𝐒𝐈𝐁𝐋𝐄: 𝟏𝟎𝟎 𝐏𝐑𝐈𝐒𝐎𝐍𝐄𝐑𝐒 𝐏𝐔𝐙𝐙𝐋𝐄
By 𝐒𝐘𝐄𝐃 𝐌𝐔𝐒𝐇𝐅𝐈𝐐𝐔𝐑 𝐑𝐀𝐇𝐌𝐀𝐍 (𝟏𝟐𝟕𝟏𝟔𝟏𝟎𝟓)
𝐈𝐧𝐭𝐫𝐨𝐝𝐮𝐜𝐭𝐢𝐨𝐧:
Imagine you are the warden of a prison which holds the world’s most cunning prisoners. One the eve of their ex*****on, you decided to give them a chance to survive. But for that they have to crack the puzzle you gave them. And from here starts our “𝟏𝟎𝟎 𝐏𝐫𝐢𝐬𝐨𝐧𝐞𝐫𝐬 𝐏𝐮𝐳𝐳𝐥𝐞”.
𝐓𝐡𝐞 𝐏𝐮𝐳𝐳𝐥𝐞:
You pick 100 prisoners from the prison labelling each with a number from 1 to 100. They have to find a paper corresponding to their number from 100 boxes inside a room. The boxes are also labelled from 1 to 100. The prisoners will enter one by one and search a maximum of 50 boxes. They have to leave immediately after they find their paper or search 50 boxes. They can’t talk during and after searching boxes. But they can plan before entering the room.
𝐀𝐧𝐚𝐥𝐲𝐬𝐢𝐬:
At first glance this situation seems hopeless. If each prisoner searches 50 boxes at random, the chance of their success is (0.5)¹⁰⁰. Can you even imagine how small the number is! For comparison, let’s say your friend took 100 grains of sand in his hand from any part of the world. It is quite easier to find that exact 100 grains of sand than to escape this prison. But there’s a strategy which increases the probability of escape to about 31%. Can you guess what it is?
𝐖𝐡𝐚𝐭’𝐬 𝐭𝐡𝐞 𝐬𝐭𝐫𝐚𝐭𝐞𝐠𝐲 𝐭𝐡𝐚𝐭 𝐛𝐞𝐚𝐭𝐬 𝐭𝐡𝐞 𝐢𝐦𝐩𝐨𝐬𝐬𝐢𝐛𝐥𝐞?
The strategy is simple. Each prisoner will have to open the box with their corresponding number. That box will contain a paper having a number. Then he has to open the box of that number. Thus, a loop will be created. Let’s see what will be the scenario for 𝐏𝐫𝐢𝐬𝐨𝐧𝐞𝐫 𝟐:
Box No. 2 5 43 16 7
Paper No. 5 43 16 7 2
Here you can see a loop of 5 boxes was created and 𝐏𝐫𝐢𝐬𝐨𝐧𝐞𝐫 𝟐 got his number after searching 5 boxes. This process will be applicable for all the 100 prisoners. But here’s the catch, the loop must not be longer than 50 boxes as the limit is up to 50 boxes. So, if there exists any loop longer than 50, that means it’s game over. 𝐒𝐨, 𝐥𝐞𝐭’𝐬 𝐜𝐚𝐥𝐜𝐮𝐥𝐚𝐭𝐞 𝐭𝐡𝐞 𝐩𝐫𝐨𝐛𝐚𝐛𝐢𝐥𝐢𝐭𝐲 𝐧𝐨𝐰.
In how many ways can you arrange 100 boxes? It’s 𝟏𝟎𝟎!. So there are 𝟏𝟎𝟎! different ways you can create a loop of 100 boxes. But these are not just lines of numbers, they are loops. For instance, you can create a loop like 𝟏 → 𝟐 → 𝟑 → 𝟒 → … → 𝟏𝟎𝟎. Again, you can make 𝟐 → 𝟑 → 𝟒 → … → 𝟏𝟎𝟎 → 𝟏. Even though in the number line they are different they are the same loop. So at the end the total number of different loops you can make is (100!/100). So, you got the total number of unique loops and the total possible loops of the length 100. The probability that a random arrangement of boxes will contain a loop of 100 is (100!/100)/100!. Which equals to 1/100. Similarly, the probability that you get a loop of 99 is 1/99, a loop of 98 is 1/98 and so on. So the probability of loops longer than 50 is (1/51)+(1/52)+….+(1/100)=0.69.
If the probability of loops longer than 50 is 0.69, then the probability of loops shorter than or equal to 50 is 0.31 or 31%. 𝐓𝐡𝐚𝐭 𝐢𝐬, 𝐭𝐡𝐞 𝐩𝐫𝐨𝐛𝐚𝐛𝐢𝐥𝐢𝐭𝐲 𝐨𝐟 𝐬𝐮𝐜𝐜𝐞𝐬𝐬 𝐢𝐬 𝟑𝟏%.
𝐂𝐨𝐧𝐜𝐥𝐮𝐬𝐢𝐨𝐧:
You made a puzzle that was quite impossible to solve, but some simple strategy made the impossible chance a solid 31% possibility. The mentioned strategy is the best strategy so far. But you, the warden of the prison, could have done something that would have made the puzzle still impossible even after applying the strategy? Well, you could have taken more than 100 prisoners. But you already gave them a 31% possibility and realized that your 𝐈𝐦𝐩𝐨𝐬𝐬𝐢𝐛𝐥𝐞 𝐰𝐚𝐬 𝐛𝐞𝐚𝐭𝐞𝐧 𝐛𝐲 𝟑𝟏%.
