Balance puzzles

Balance puzzles are fun little logic puzzles in which you have to figure out which item is fake out of a bunch of them, but by using a maximum amount of weighings. An example of this is as follows: The party's been given 15 bars of gold, but one of them is fake. The fake weighs less, but looks otherwise identical to all the other bars of gold. They only get to make 3 weighings to figure out which bar is false, or else they'll go home empty handed.

The solution is simple. Divide the stack into five stacks of three, then weigh two against each other. If either one is lighter you know the fake is in that pile of three, if not, you know all of these are real gold. If you haven't found the stack with the fake weigh 2 more stacks against each other, again if one is lighter you know the fake is in there, if not, you know the fake is in the last stack.
Once you've identified the stack with the fake bar weigh any 2 of them, if one of them is lighter you know it's the fake, otherwise the fake is the one you didn't weigh.

Variants

The puzzle can be made to be more challenging, or at the very least seem more challenging. If you increase the number to 81 you would need only 4 weighings, but it may appear like you'd need more at first glance, especially to those who overthink. If you don't see the solution: Split 81 into three stacks of 27, after weighing two split the one with the fake into three stacks of 9, weigh, then three stacks of 3, weigh, done.

You can make it more challenging by not stating whether the fake is lighter or heavier, and by changing the amount of coins. A simple version of this is being presented with 9 bars, one of which is fake, but you don't know which. Again, divide into three, if there's an imbalance you know the fake is among these two stacks, if there's a balance you know the fake is in the last.
In case of the first, weigh the stack you didn't weigh against any one of the two you did. Check for an imbalance again, if there is you know the stack you left on the scale contains a fake, and that it's heavier or lighter depending on which way the scale tipped. If there is a balance you know the stack you removed contains a fake, and is heavier or lighter depending on the way the scale tipped.
Either way you now know which stack contains the fake, and you know if this stack was heavier or lighter than the others based on how the scale moved. Weigh two of these 3 bars, if there's a balance the fake is the one you didn't weigh, if the scale tips you know the fake is the one that made the scale tip the same way as the stack did earlier.

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