To solve this equation, we need to follow the order of operations (PEMDAS).

Calculate 4 2/3 6:
4 2/3 = 14/3
14/3 6 = 28

Calculate 1 23/42 3 1/3:
1 23/42 = 55/42
3 1/3 = 10/3
55/42 10/3 = 550/126 = 25/42

Calculate 2 1/8 1 29/51:
2 1/8 = 17/8
1 29/51 = 80/51
17/8 80/51 = 1360/408 = 85/51

Now, we add and subtract the results:
28 - 25/42 + 85/51

To add and subtract fractions, we need to find a common denominator:
42 and 51 have a common multiple of 42*51 = 2142

Now convert the fractions to have a common denominator:
(28 51) / 51 - (25 42) / 42 + 85/51

Ḧ/2142 - 1050/2142 + 85/51

Combine the numerators:
(28*51 - 1050 + 85)/2142

Simplify the numerator:
(1428 - 1050 + 85)/2142
(463)/2142

Final answer: 463/2142