146 Combinatorial & Topological Problemsthe six points add up to 24. If for every number in its present position you
substitute its difference from 13 you will get another solution, its complemen-
tary, with the points adding up 54, which is 78 less 24. The two complementary
totals will always sum to 78.
I will give the total number of different solutions and point out some of the
pretty laws which govern the problem, but I will leave the reader this puzzle
to solve. There are six arrangements, and six only, in which all the lines of
four and the six points also add up to 26. Can you find one or all of them?- THE SEVEN-POINTED STAR
We have already dealt briefly with
stars of five and six points. The case
of the seven-pointed star is particu-larly interesting. All you have to do
is to place the numbers 1, 2, 3, up to
14 in the fourteen disks so that every
line of four disks shall add up to 30.
If you make a rough diagram and
use numbered counters, you will soon
find it difficult to break away from
the fascination of the thing. Possibly,
however, not a single reader will hit
upon a simple method of solution;
his answer, when found, will be ob-
tained by mere patience and luck. Yet,
like those of the large majority of the
puzzles given in these pages, the solu-
tion is subject to law, if you can un-
ravel it.- TWO EIGHT-POINTED STARS
The puzzles of stars with five, six, and seven points that I have given lead
us to the eight-pointed star. The star may be formed in two different ways, as
shown in our illustration, and the first example is a solution. The numbers 1
to 16 are so placed that every straight line of four adds up to 34. If you substi-
tute for every number its difference from 17 you will get the complementary
solution.