1997 MIT Mystery Hunt Puzzle 33 Solution

Label the 20 pills A through T in order of their appearance in the diagram:

The clues can be rewritten as these equations:

28=B+C+D
33=B+D+E
22=F+G+H
26=F+H+I
43=H+I+J+K
21=I+J
41=J+L+M
45=M+N+O+P
39=P+Q+R+S
36=Q+S+T
22=C+G+Q
37=C+G+J+N
40=F+L+N+T
47=A+L+R+T
47=A+E+O+R
32=E+I+O
35=B+K+M
18=D+K+P
210=(sum of A through T, since they are all of 1 through 20 once each)

From 43=H+I+J+K and 21=I+J we know 22=H+K

From 28=B+C+D and 33=B+D+E we know E - C = 5

From 22=F+G+H and 26=F+H+I we know I - G = 4

From these last two results and 22=C+G+Q and 32=E+I+O we know 27=E+G+Q, 28=E+G+O, O - Q = 1

Also, from 37 = C+G+J+N and I - G = 4 and I + J = 21, we have C + N = 20

From 47=A+E+O+R, 35=B+K+M, 22=C+G+Q, 18=D+K+P, 40=F+L+N+T, 43=H+I+J+K, we know 205 = A+B+C+D+E+F+G+H+I+J+K+K+K+L+M+N+O+P+Q+R+T which yields 5 = S - 2K, or 2H+S=49.

Use these expressions to eliminate E, I, O, S, H, J, and N from the rest of the equations:

28=B+C+D
0=F+G-K
24=L+M-G
24=M+P+Q-C
34=P+Q+R+2K
31=Q+2K+T
22=C+G+Q
20=F+L+T-C
47=A+L+R+T
41=A+C+Q+R
35=B+K+M
18=D+K+P
E = C + 5
I = G + 4
O = Q + 1
S = 2K + 5
H = 22 - K
J = 17 - G
N = 20 - C

Then the following expressions can be arrived at successively with substitution after each step:

34=P+Q+R+2K - (24=M+P+Q-C) + 35=B+K+M - (28=B+C+D) + 18=D+K+P - (34=P+Q+R+2K) => 1 = 2K-Q
12=C+4K-F-L + 0=F+G-K - (23=C+G+2K) => 11 = L - K
0=F+G-K + 13=K+M-G => 13 = F + M
42=A+C+2K+R - (4=A+R-3K) => 38 = C + 5K

This lets everything be written in terms of K:

E = C + 5 = 43 - 5K
I = G + 4 = 3K - 11
O = Q + 1 = 2K
S = 2K + 5
H = 22 - K
J = 17 - G = 32 - 3K
N = 20 - C = 5K - 18
Q = 2K - 1
T = 32 - 4K
L = 11 + K
F = 13 - M = 15 - 2K
C = 38 - 5K
G = 3K - 15
M = 2K - 2
B = 37 - 3K
D = 8K - 47
P = 65 - 9K
R = 5K - 30
A = 34 - 2K

From T = 32 - K it is known that K is no more than 7, and from A = 34 - 2K, K is at least 7. Then:

A = 20
B = 16
C = 3
D = 9
E = 8
F = 1
G = 6
H = 15
I = 10
J = 11
K = 7
L = 18
M = 12
N = 17
O = 14
P = 2
Q = 13
R = 5
S = 19
T = 4

The center column contains AEHLORT, which translates to 20 8 15 18 14 5 4 by this solution, or (by index into the alphabet), THORNED.