Updated on March 8, 2016

## The 100 Coin Cluster

Here is my favorite riddle of all times. Think of the following scenario. You are blindfolded sitting at a table with 100 coins on it. You know that 10 of these show heads and 90 show tails. However, you don't know which of the coins show heads since you are blindfolded (and not a cheater). We will come to the task in a second but first lets talk about the options you have to complete this task.

You can create piles of coins by counting them with your hands. You are also allowed to flip as many coins and as often as you want. Now here is the tricky objective. You have to form two piles of coins which have the same number of coins showing heads in them. That does not necessarily mean that both piles have to be the same size. It also does not mean that both piles need to have 5 coins showing heads ( remember, you can flip coins and that affects the number of coins showing heads or trails). Both piles just need to have the same number of coins with heads up. Here is simple an example where one pile consists of 10 and the other pile of 90 coins. Since both piles comprise 3 coins showing heads, this would be a valid solution to the riddle.

## Finding The Right Path

You are running down a path in a jungle trying to find your way out. Before your escape, a friend has warned you of a dangerous bridge that lies ahead of you. It might collapse when you try to cross. Unfortunately, he only remembered that this bridge came right after a fork but not on what side of the fork the dangerous bridge is. The other side of the fork leads you to a bridge that is safe to cross.

Your friend told you that at each side of the fork sits an oracle, and you can ask each oracle one question. He knows for sure that one oracle always speaks the truth, the other one always lies, and each oracle knows if the other oracle lies or not.

You arrive at the fork and spot the two oracles immediately. At each side of the fork you can see a bridge in the distance, and you have no idea which is the safe one. Assuming you hold your life dear, what question do you have to ask the oracles to be absolutely sure which is the safe route?

## The Three Door Problem

You are in a game show where the moderator makes you one of three closed doors. You know that there is a price hidden behind one door and the other two are blanks. Of course, you don't know which door holds the price, so you pick the first door randomly. The moderator opens now one of the two doors you haven't picked and hides a blank.

Apparently, the price has to be either behind the door you initially selected or behind the other door. Now it is getting interesting. The moderator offers you the option to switch doors and go with the door you have ignored so far. The key question now is, do your chances of winning the price increase if you switch doors or not? To ask a bit more specific, does each door have a 50% probability to hold the price?

A little hint: Intuition might not be very helpful to solve this riddle

## SOLUTION: The 100 Coin Cluster

I am sure you will find the solution ridiculously simple. You create two piles, one with 10 coins and the second pile with the remaining coins. You flip all 10 coins in the smaller pile and you have the same number of heads in both piles. That is all there is to it.

Lets look at an example. Assume 2 coins are in the small and 8 coins are in the large pile. After you flipped all coins in the smaller pile, you also have 8 coins showing heads. That is the same number as in the large pile. This works for any scenario you can think of ranging from zero to ten coins showing heads up after the flipping.

## SOLUTION: Finding the Right Path

Actually one question will do the trick. You ask one of the oracles "what would the other oracle tell me, what the dangerous path is". The path the oracle tells you is actually the safe route out. Let us think about the two possible scenarios for a moment.

Maybe you address the honest oracle. Since it knows the other oracle would lie, it is going to tells us that very lie.

On the other hand, if you ask the liar what answer the honest oracle would give, it will falsify the honest answer of the other oracle. Instead of getting the dangerous path, you end up with the right path in both cases. Essentially by linking the behavior of both oracles, you are always lied to which allows you to determine the safe route.

## SOLUTION: The Three Door Problem

Your chances of winning improve from 33% to 66% if you switch to the door you did not pick initially. Hard to believe? Lets play with three different scenarios and look at the outcomes if you switch doors. For simplicity, I am going to enumerate the doors with the letters A, B, and C and assume the price is hidden behind door A. The three rows show the outcome (last column "Final Door") if you pick initially door A, B and C, respectively.

Initial Door
Moderators Door
Final Door
Door A
Door B
Door C
Door B
Door C
Door A
Door C
Door B
Door A

Looking at the last column of the table above, we can see that in 2 of 3 cases, we ended up with Door A and won the price by switching doors. Apparently, switching doors improves our chances of winning to 66%