Answers 349Let us confine our attention to solutions with ten different positive whole
numbers. Then 24 is the smallest possible constant. A solution for any higher
constant can be derived from it. Thus, if we want 26, add I at each of the
points; if we want 28, add 2 at every point or I at every place in both points
and pentagon. Odd constants are impossible unless we use fractions. Every
solution can be "turned inside out." Thus, Figure IV is simply a different
arrangement of Figure II. Also the four numbers in G, K, D, ] may always be
changed, if repetitions do not occur. For example, in Figure II substitute 13,
7, 6, 5 for 12, 8, 5, 6 respectively. Finally, in any solution the constant will be
two-fifths of the sum of all the ten numbers. So, if we are given a particular
set of numbers we at once know the constimt, and for any constant we can
determine the sum of the numbers to be used.
394. THE SIX-POINTED STAR
I have insufficient space to explain fully the solution to this interesting
problem, but I will give the reader the main points.
(I) In every solution the sum of the numbers in the triangle ABC (Figure I)