by David Farhi and Casey McNamara

You have to figure out the consistent mapping of letters to numbers that makes each addition valid in a different base. As is usually the case for alphametic puzzles, each letter is mapped to a different number; furthermore, in this puzzle, the mapping is the same for all the alphametics. The solution is:

N=0
O=1
P=2
I=3
T=4
E=5
S=6
U=7
L=8
G=9
C=10
A=11
Z=12
K=13
Y=14
M=15
J=16
R=17
Q=18
F=19
X=20
D=21
H=22
W=23
B=24
V=25

The bases that the puzzles work in, then, are: 6, 25, 11, 8, 24, 11, 17, 21. Using the letters-to-numbers conversion above, that gives: SVALBARD.