Kanga wants to arrange the twelve numbers from 1 t0 12 in a circle in such
way that any neighboring numbers always differ by either 1 or 2.
That is to say that the "gaps" between each pair of
neighboring numbers must be of size 1 or 2.
The total "gap" between the smallest, 1, and the largest, 12,
is of size 12-1=11.
The only way "gaps" between neighbors of size 2 and 1 can add up to
a "gap" of size 11 is to have 5 "gaps" between neighbors of size
2 and 1 "gap" of 1. This must be the case whether we go from 1
to 12 clockwise or counter-clockwise.
The other numbers besides 1 and 12 are the 5 even numbers
2,4,6,8,10, and the 5 odd numbers 1,3,5,7,9. The "gaps"
between successive even or odd numbers is 2. This gives
us our 5 "gaps".
So we put the smallest number, 1, at the top and the
largest number, 12, at the bottom.
Then we put the successive odd numbers on one side and
the successive even numbers on the other side. So there
are two ways Kanga can arrange the twelve numbers from
1 to 12 in a circle in such a way that any neighboring
numbers always differ by either 1 or 2. Here they are:
Edwin
