159374

First, I thought of the numbers that are multiples of both 3 and 4, so I tried doing twelve.

However, if 3 coins were heads, which is essential because that is one quarter of 12, then if I turned two more heads, then it would make five coins heads which is not a third of 12.

Next, I tried doing 24. A quarter of 24 is 6 and 6 and 2 is 8. 8 is a third of 24, so that is how I got the answer. The methods that I used during this activity is logical thinking and guess, check and improve.