Where is the Gold?

All three of the labels are incorrectly placed.

$\implies$ Chest $1$ does not contain $100$ gold coins.

$\implies$ We need to check only $\boxed{1}$ other chest,i.e., Chest $2$ to be guaranteed to know which chest contains gold!

After opening 1 chest you know what is in it.

You know two remaining option and they are at wrong place so as THE BRILLIANT SAYS "A known lie is just as useful as a known truth."

If you open one chest, whether it is just silver, gold and silver, or just gold, you can tell where the gold is. For example, if you open up the chest with all gold, you obviously know it has all gold. Also, if you open up the chest with some gold, you know that that chest contains gold. If you open up the chest that contains silver, you know that the other two chests both contain gold.

If all labels are incorrect, then:

• In Chest1 there is 50 golds and 50 silvers or 100 silvers.

• In Chest2 there is 100 golds or 100 silvers.

• In Chest3 there is 50 golds and 50 silvers or 100 golds.

So 100 golds are in Chest2 or Chest3.

But if we look at Chest3 and take a coin, it could be silver, and 100 golds are in Chest2, and if it's gold, it can be from 50 GOLD and 50 silver or 100 GOLD.

It doesn't make sense to look at Chest3 or Chest1.

But if we look at Chest2, there are 100 golds or 100 silvers. If I take a coin and it's gold, it's the 100 golds chest and if it's silver, than 100 golds are in Chest3.

100 coin chest can't have 100 coins in it because they are are wrong. That leaves two chests. if its not in one, its in the other. ( I like science not language so dont judge my spelling and grammar)

Chest 1 can't contain 100 gold coins , but it could contain either 50 gold and 50 silver, or 100 silver , meaning that chest 1 wouldn't tell you what its contents are, assuming that you can only take out 1 coin. same with chest 3. However, chest 2 cannot have 50 gold and 50 silver , meaning that it can either contain 100 gold or 100 silver:

• If it has a silver coin - then chest 1 must be the one with 50 gold and 50 silver (because chest 1 cannot be the one with the gold) and therefore chest 3 is the one with the gold, and vice versa

As all the labels are incorrect

$\therefore$ Chest 1 can't have gold coins

Now if you open chest 2 or chest 3 either you will get gold coins in it or not

If you open one of them and gold is not there, this means that gold is in the chest left

Else you may even find gold in it

In both the cases you only need to open only $\boxed{1}$ chest

If chest one is not gold but 100 silver - we know 2 facts

chest 1 - silver hence chest 3 label comes to chest 1

chest 2 label belongs to chest 3

And automatically chest 1 label goes to chest 2

It's one. If you pick from chest 2, you know that it is ether 100 gold coins or 100 silver coins because the boxes are labeled incorrectly. So, if you pick a gold coin, then you know that the one labeled 100 silver coins isn't 100 silver coins and now you also know that can't be 100 gold coins, because that is what chest 2 is. Leaving chest 1 the only one you don't know, and it has to be 50 gold and 50 silver because that is the only option left. If you pick a silver coin at first, you go through the same steps.

all labels are false thus, we open the chest that has uncertainty which is no.2 (it could either have all gold or all silver)

• if it has gold then we are done we found the chest

• if it has silver then the chest labeled "gold"(1st chest) can't have silver in it because the silver is in chest no.2

• and therefore the gold is in the chest labeled "silver" (3rd chest) and thus we have found the chest

in both cases, we deduced the location of the gold by opening one chest noting that we could take only one coin of chest no.2 and similarly end up with the same solution.

NOTE: the logic I made was true if and only if every label belonged to a unique chest of the three chests and every chest had a unique label in other words, no chest has more than one label (contradiction) and no one label belongs to two chests (which is trivial you can't duplicate labels).

Think of this like a sudoku

Make a matrix. The columns are G, GS, and S The rows are chests 1,2,3

An entry of the matrix is 1 if chest I contains treasure J

We know the labels are incorrectly placed. Thus we must have 0s along the diagonal. If we open a single chest and find a treasure, we mark that (chest, treasure) element with a 1, and fill in the rest of the column and rows with 0s. Because there must be at least one 1 in each row and column (since this is an assignment, each box has a treasure and each treasure has a box), as soon as we update the matrix from one box, we can complete the matrix.

We then read off the assignments from the matrix.