174 Combinatorial & Topological Problems454. THE SIX COTTAGES
A circular road, twenty-seven miles
long, surrounds a tract of wild and
desolate country, and on this roadare six cottages so placed that one
cottage or another is at a distance of
one, two, three up to twenty-six miles
inclusive from some other cottage.
Thus, Brown may be a mile from
Stiggins, Jones two miles from Rogers,
Wilson three miles from Jones, and
so on. Of course, they can walk in
either direction as required.
Can you place the cottages at dis-
tances that will fulfill these condi-
tions? The illustration is intended to
give no clue as to the relative dis-
tances.- FOUR IN LINE
Here we have a board of thirty-six squares, and four counters are so placed
in a straight line that every square of the board is in line horizontally, verti-
cally, or diagonally with at least one counter. In other words, if you regard
them as chess queens, every square on the board is attacked by at least one
queen. The puzzle is to find in how many different ways the four counters
may be placed in a straight line so that every square shall thus be in line with
a counter.
Every arrangement in which the
counters occupy a different set offour
squares is a different arrangement.
Thus, in the case of the example given,
they can be moved to the next column
to the right with equal effect, or they
may be transferred to either of the
two central rows of the board. This
arrangement, therefore, produces four
solutions by what we call reversals or
reflections of the board. Remember