The king receives 64 gold coins in taxes but has reason to believe

The king receives 64 gold coins in taxes but has reason to believe that one is counterfeit. He summons you to identify the fake coin. You have a balance that can hold coins on each side. How many times do you need to use the balance to find the lighter, fake coin?

Keep dividing the groups into 2.

Put 32 on each side, and get rid of the heavier group (because they're all real)

Put 16 on each side, and get rid of the heavier group.

.....and so on until you just have 2 coins, and then you know the lighter one is counterfeit.

I hope this information was very helpful.

Normally in these problems you don't know if the coin is heavier or lighter. Because 64 is 2^6 you obviously can narrow it down by first putting 32 coins on each side and so forth 6 times. However I would start with 21 on each side. If one side is lighter you can move on to putting 7 on each side from the lighter group which will tell you which of the groups of 7 is the lighter. For the third weighing put three on each side and if one is lighter, weigh any two to determine the lighter one. This is 4 weighings.
Going back to the first weighing, if they are the same then the lighter one is in the remaining 22. For the second weighing put 7 on each pan, and if they are different your third weighing will put 3 on each side which at worst will require one more weighing to find which of three is it. If the groups of 7 weigh the same, from the remaining 8 put three in each pan. If these are different you can finish in one more as above, and if they are the same one of the remaining 2 is lighter. Thus 4 weighings will do it no matter what.

7 times.

Split the 64 into 2, becomes 32. The side that is lighter becomes 16, 8, 4, 2, and 1.

do it in halves.
32 each side, throw out the light ones.
16 and so on

well there is a 1/64 chance of getting it right...