Answers 351- THE SEVEN-POINTED STAR
Place 5 at the top point, as indicated
in diagram. Then let the four numbers
in the horizontal line (7, 11,9,3) be
such that the two outside numbers
shall sum to 10 and the inner numbers
to 20, and that the difference between
the two outer numbers shall be twice
the difference between the two inner
numbers. Then their complementaries
to 15 are placed in the relative posi-
tions shown by the dotted lines. The
remaining four numbers (13, 2, 14, I)
are easily adjusted. From this funda-
mental arrangement we can get three
others. (I) Change the 13 with the I
and the 14 with the 2. (2 and 3) Sub-
stitute for every number in the two
arrangements already found its differ-
ence from IS. Thus, 10 for 5, 8 for 7,
4 for II, and so on. Now, the reader
should be able to construct a second
group of four solutions for himself,
by following the rules.
The general solution is too lengthy
to be given here in full, but there are,
in all, 56 different arrangements,
counting complementaries. I divide
them into three classes. Class I in-
cludes all cases like the above ex-
ample, where the pairs in the positions
of7-8, 13-2,3-12, 14-1 all sum to IS,
and there are 20 such cases. Class II
includes cases where the pairs in the
positions of 7-2, 8-13, 3-1,12-14 all
sum to IS. There are, again, 20 such
cases. Class III includes cases where
the pairs in the positions of 7-8, 13-2,
3-1,12-14 all sum to IS. There are 16
such cases. Thus we get 56 in all.
[Dudeney erred in his enumera-
tion of solutions for both the seven-
pointed and the six-pointed stars.
When I published his results in a
column on magic stars (Scientific
American, December 1965), two
readers-E. J. Ulrich, Enid, Okla-
homa, and A. Domergue, Paris-
independently confirmed that there
are 80 patterns for the six-pointed
star, or 6 more than listed by Dudeney.
That the seven-pointed star has 72
distinct solutions (as against Du-
deney's figure of 56) was first reported
to me by Mrs. Peter W. Montgomery,
North Saint Paul, Minn. This was
confirmed by the independent work of
Ulrich and Domergue. In 1966 Alan
Moldon, Toronto, using a computer
at the University of Waterloo, also
obtained 72 solutions for the seven-
pointed star, so there seems little
doubt that this is correct.-M. G.]