Divide 27 coins in 3 groups, each of 9 coins, say 9A, 9B, & 9C. W1: Compare 9A & 9B. Three possibilities are as under:

[a] 9A = 9B, OR [b] 9A > 9B, OR [c] 9A < 9B. CASE [a]: W1: 9A = 9B, (counterfeit coin is in group 9C).

Divide 9C in 3 groups, say 3C1, 3C2, & 3C3. W2: Comparing 3C1 & 3C2, again there are three possibilities as under:

[a1] 3C1 = 3C2,

[a2] 3C1 > 3C2,

[a3] 3C1 < 3C2,

CASE [a1]:

W1: 9A = 9B, (counterfeit coin is in 9C).

W2: 3C1 = 3C2, (counterfeit coin is in group 3C3). Name three coins in 3C3 as 1C31, 1C32, & 1C33.

W3: Compare 1C31 & 1C32. Again there are three possibilities as under:

[a11]: 1C31 = 1C32,

[a12]: 1C31 > 1C32,

[a13]: 1C31 < 1C32.

CASE [a11]:

W1: 9A = 9B, (counterfeit coin is in group 9C).

W2: 3C1 = 3C2, (counterfeit coin is in group 3C3).

W3: 1C31 = 1C32, (1C33 is COUNTERFEIT COIN).

W4: Compare 1C31 v/s 1C33, to know whether 1C33 is LIGHTER or HEAVIER than the normal coin 1C31.

CASE [a12]:

W1: 9A = 9B,

W2: 3C1 = 3C2, (counterfeit coin is in group 3C3).

W3: 1C31 > 1C32, (Either 1C31 or 1C32 is counterfeit coin).

W4: Compare 1C31 v/s 1C33,

If 1C31 = 1C33, then 1C32 is COUNTERFEIT COIN, and is LIGHTER.

If 1C31 > 1C33, then 1C31 is COUNTERFEIT COIN, and is HEAVIER.

If 1C31 < 1C33, then 1C31 is COUNTERFEIT COIN, and is LIGHTER.

CASE [a13]: Similar procedure as described in CASE [a12].

CASE [a2]:

W1: 9A = 9B, (counterfeit coin is in group 9C).

W2: 3C1 > 3C2, (counterfeit coin is either in group 3C1 OR in 3C2.)

W3: Compare 3C1 v/s 3C3, there are three possible results:

[a21]: 3C1 = 3C3,

[a22]: 3C1 > 3C3,

[a23]: 3C1 < 3C3,

CASE [a21]:

W1: 9A = 9B, (counterfeit coin is in group 9C).

W2: 3C1 > 3C2, (counterfeit coin is either in group 3C1 OR in 3C2.)

W3: 3C1 = 3C3, (counterfeit coin is in the group 3C2 and is LIGHTER).

Name three coins in 3C2 as 1C21, 1C22, & 1C23.

W4: Comparing 1C21 & 1C22, three possibilities are as under:

If 1C21 = 1C22, then 1C23 is COUNTERFEIT COIN, and is LIGHTER.

If 1C21 > 1C22, then 1C22 is COUNTERFEIT COIN, and is LIGHTER.

If 1C21 < 1C22, then 1C21 is COUNTERFEIT COIN, and is LIGHTER.

CASE [a22]:

W1: 9A = 9B, (counterfeit coin is in group 9C).

W2: 3C1 > 3C2, (counterfeit coin is either in group 3C1 OR 3C2.)

W3: 3C1 > 3C3, (counterfeit coin is in the group 3C1 and is HEAVIER). Name three coins in 3C1 as 1C11, 1C12, & 1C13.

W4: Comparing 1C11 & 1C12, three possibilities are as under:

If 1C11 = 1C12, then 1C13 is COUNTERFEIT COIN, and is HEAVIER.

If 1C11 > 1C12, then 1C11 is COUNTERFEIT COIN, and is HEAVIER.

If 1C11 < 1C12, then 1C12 is COUNTERFEIT COIN, and is HEAVIER.

CASE [a23]: This can also be solved similarly as above CASE [a22].

CASE [a3] can also be solved with the same procedure as in CASE [a2].

CASE [b]:

W1: 9A > 9B, (counterfeit coin is either in group 9A or in 9B).

W2: Compare 9A v/s 9C, three possibilities are as under:

[b1] 9A = 9C, (counterfeit coin is in 9B and is LIGHTER)

[b2] 9A > 9C, (counterfeit coin is in 9A and is HEAVIER)

[b3] 9A < 9C, (counterfeit coin is in 9A and is LIGHTER)

CASE [b1]:

W1: 9A > 9B,

W2: 9A = 9C, (counterfeit coin is in 9B and is LIGHTER) Divide 9B in three groups 3B1, 3B2, & 3B3. W3: Compare 3B1 & 3B2,. Again there are three possibilities as under:

[b11]: 3B1 = 3B2, (counterfeit coin is in 3B3 and is LIGHTER).

[b12]: 3B1 > 3B2, (counterfeit coin is in 3B2 and is LIGHTER).

[b13]: 3B1 < 3B2, (counterfeit coin is in 3B1 and is LIGHTER).

CASE [b11]:

W1: 9A > 9B, (counterfeit coin is either in group 9A or in 9B).

W2: 9A = 9C, (counterfeit coin is in 9B and is LIGHTER)

W3: 3B1 = 3B2, (counterfeit coin is in 3B3 and is LIGHTER).

Name three coins of 3B3 as 1B31, 1B32, 1B33.

W4: Compare 1B31 v/s 1B32,

If 1B31 = 1B32, then 1B33 is COUNTERFEIT COIN, and is LIGHTER.

If 1B31 > 1B32, then 1B32 is COUNTERFEIT COIN, and is LIGHTER.

If 1B31 < 1B32, then 1B31 is COUNTERFEIT COIN, and is LIGHTER.

CASE [b12]:

W1: 9A > 9B, (counterfeit coin is either in group 9A or in 9B).

W2: 9A = 9C, (counterfeit coin is LIGHTER and is in group 9B).

W3: 3B1 > 3B2, (counterfeit coin is in 3B2 and is LIGHTER).

Name three coins of 3B2 as 1B21, 1B22, 1B23.

W4: Compare 1B21 & 1B22,

If 1B21 = 1B22, then 1B23 is COUNTERFEIT COIN and LIGHTER.

If 1B21 > 1B22, then 1B22 is COUNTERFEIT COIN and LIGHTER.

If 1B21 < 1B22, then 1B21 is COUNTERFEIT COIN and LIGHTER.

CASE [b13] may be solved as above CASE [b12].

CASE [b2]:

W1: 9A > 9B, (counterfeit coin is either in group 9A or in 9B).

W2: 9A > 9C, (counterfeit coin is in 9A and is HEAVIER)

Divide 9A in three groups as 3A1, 3A2, 3A3.

W3: Compare 3A1 & 3A2, there are three possibilities as under:

[b21] 3A1 = 3A2, (3A3 contains counterfeit coin and is HEAVIER).

[b22] 3A1 > 3A2, (3A1 contains counterfeit coin and is HEAVIER).

[b23] 3A1 < 3A2, (3A2 contains counterfeit coin and is HEAVIER).

CASE [b21]:

W1: 9A > 9B, (counterfeit coin is either in group 9A or in 9B).

W2: 9A > 9C, (counterfeit coin is in 9A and is HEAVIER)

W3: 3A1 = 3A2, (3A3 contains counterfeit coin and is HEAVIER).

Name three coins of 3A3 as 1A31, 1A32, 1A33.

W4: Compare 1A31 & 1A32,

If 1A31 = 1A32, then 1A33 is the COUNTERFEIT COIN and is HEAVIER.

If 1A31 > 1A32, then 1A31 is the COUNTERFEIT COIN and is HEAVIER.

If 1A31 < 1A32, then 1A32 is the COUNTERFEIT COIN and is HEAVIER.

CASE [b22]:

W1: 9A > 9B, (counterfeit coin is either in group 9A or in 9B).

W2: 9A > 9C, (counterfeit coin is in 9A and is HEAVIER)

W3: 3A1 > 3A2, (3A1 contains counterfeit coin and is HEAVIER).

Name three coins of 3A1 as 1A11, 1A12, 1A13.

W4: Compare 1A11 & 1A12,

If 1A11 = 1A12, then 1A13 is the COUNTERFEIT COIN and is HEAVIER.

If 1A11 > 1A12, then 1A11 is the COUNTERFEIT COIN and is HEAVIER.

If 1A11 < 1A12, then 1A12 is the COUNTERFEIT COIN and is HEAVIER.

Similarly CASE [b23] may be solved.

CASE [c] may be solved similarly as CASE [b].

THUS WE SEE 4 WEIGHING ARE REQUIRED TO KNOW THE COUNTERFEIT COIN OUT OF 27 COINS.