Larry had 6 singles and 28 doubles.

Curly had 27 singles and 7 doubles.

Moe’s singles = 18 = 3 x 6 = 3 x Larry’s singles

Larry’s doubles = 28 = 4 x 7 = 4 x Curly’s doubles

Moe’s hits = 18 + 16 = 34

Larry’s hits = 6 + 28 = 34

Curly’s hits = 27 + 7 = 34

Total singles = 18 + 6 + 27 = 51

Total doubles = 16 + 28 + 7 = 51

Total hits = 51 + 51 = 102

If MS and MD represent the number of Moe’s singles and doubles, respectively, and LS and LD are the numbers of Larry’s singles and doubles, and CS and CD are Curly’s singles and doubles, then the facts that were stated lead to the following five equations:

1) MS = 3 LS

2) LD = 4 CD

3) MS + MD = LS + LD

4) MS + MD = CS + CD

5) MS + LS + CS = MD + LD + CD

Also, MS + LS + CS + MD + LD + CD < 200 If you plug 1) and 2) into 3), 4), and 5), you get 3') 3 LS + MD = LS + 4 CD, or MD = 4 CD - 2 LS 4') 3 LS + MD = CS + CD 5') 3 LS + CS + LS = MD + 4 CD + CD, or 4 LS + CS = MD + 5 CD Plug 3') into 4') and 5') and get 4'') 3 LS + 4 CD - 2 LS = CS + CD, or LS + 3 CD = CS 5'') 4 LS + CS = 4 CD - 2 LS + 5 CD, or CS = 9 CD - 6 LS Then LS + 3 CD = CS = 9 CD - 6 LS, or 7 LS = 6 CD. The smallest positive whole number solutions for this is LS = 6 and CD = 7. This means that MS = 3 LS = 3 x 6 = 18 and LD = 4 CD = 4 x 7 = 28. Then LS + LD = 6 + 28 = 34, which is the total number of hits for each of them, so MD = 34 - MS = 34 - 18 = 16 and CS = 34 - CD = 34 - 7 = 27. Now you have all six values and you can just go back and verify that these values work for all of the equations