In step 2a) please explain how you can split a group of 4 (Z) into 3 groups and make an accurate measurement. At some point surely you'll be weighing a group of 1 against a group of 2, and that won't tell you anything as the group of 2 will most likely always weigh more.
For instance, say you have the Z group: A, B, C, D.
You split it into Z1, Z2, and Z3 as follows:
Z1 = A
Z2 = B
Z3 = C, D
Let's suppose A is actually the odd coin, and it is heavier than the rest.
Weighing A against B (the second weighing) means that you don't get any information, since either A or B could be the odd coin (since the odd coin could also be lighter!). Weighing A against both C and D is a risk since it might not tell you anything. For example, assume the coin regular weight is 1, and A is a heavier coin at 1.5.
On the scale, the side with A on it would still go up when the other side has C and D (since the total weight is 2). Thus, you cannot say anything about the weight of coin A, since if A is of normal weight, the same tilt occurs.
Then...well, you don't have anymore weighings left.
For instance, say you have the Z group: A, B, C, D.
You split it into Z1, Z2, and Z3 as follows:
Z1 = A
Z2 = B
Z3 = C, D
Let's suppose A is actually the odd coin, and it is heavier than the rest.
Weighing A against B (the second weighing) means that you don't get any information, since either A or B could be the odd coin (since the odd coin could also be lighter!). Weighing A against both C and D is a risk since it might not tell you anything. For example, assume the coin regular weight is 1, and A is a heavier coin at 1.5.
On the scale, the side with A on it would still go up when the other side has C and D (since the total weight is 2). Thus, you cannot say anything about the weight of coin A, since if A is of normal weight, the same tilt occurs.
Then...well, you don't have anymore weighings left.
