Find 30 fake coins out of 99 coins with minimum amount of weighting?

Question

You are given 99 coins which consists of 30 fake ones. You also have a digital balance scale with perfect precision that shows how much difference between weighs you put on. For example, if you put 10 g on the left side and 20 g on the other side, it will show -10, otherwise +10.

You are asked to find a fake coin among given 99 coins:

• You know that all genuine coins have the same weight but you do not know their weights.
• You also know that every fake coin is heavier or lighter by 1 gram than any genuine coin.

So, what is the minimum amount of weighing which guarantees to find any fake coin you are looking for? (The fake coin you are going to find might be heavier or lighter, it does not matter, you just need to find any fake one.)

Answer 1

If the goal here is to just find a single fake coin then from the information given there are 99 - 30 = 69 original coins. So by seperating 70 coins from the rest we are assuring that atleast one fake coins enters the 70 coin pile.
Now we must weigh 2 coins at a time. Whenever there are imbalances the coins should be marked '+' if it's heavier '-' if it's lighter. Whenever the coins balance each other the coins are both original or fake which can be used to inspect + and - coins later on.

Now coming to determining the minimum steps required an extreme case of 40 original and 30 fake coin scenario can be helpful. Here 30 original coins could possibly be paired with 30 fake while weighing ( another worst case) giving us + and - coins, which leaves us with 30 weighs done. Now the remaining 5 pairs are going to balance with each other ( again a sample case) and here 35 weighs are done. Now weighing + coins one at a time with all the coins that balanced ( again one at a time ) will help decide which coin is fake. If all the balanced coins balanced with + coins then - is the fake coin else + is fake. But we have to make sure that all the balanced sets are original (and in case both + and - coins show imbalance). To do that 2 balanced pairs are taken. One of the coins of a pair is placed on one side of a pan and and a coin from another pair on the other. If it's balanced then both pairs are original if not then one of the 2 pairs have a fake set in it. So these pairs can be compared with tested original pairs to determine the fake
So the final tally of weighings is 43 weighs.

[ P.S: I could easily be wrong here as well LOL ! ]