Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP
have equal areas. Prove X and Y divide the sides of PQRS in the
Straight lines are drawn from each corner of a square to the mid
points of the opposite sides. Express the area of the octagon that
is formed at the centre as a fraction of the area of the square.
Find the ratio of the outer shaded area to the inner area for a six
pointed star and an eight pointed star.
The special clue-numbers in this Sudoku variant are fractions or
ratios in the lowest terms, hence the name for the puzzle.
The clue-numbers are always placed on the border lines between
selected pairs of neighbouring cells of the grid.
Each clue-number is the fraction of the two numbers in adjacent
cells (to the left and right). Each fraction is written in its
lowest terms, with the smaller number denoted as the numerator.
Thus $1/2$ can stand for the following combinations of numbers in
the two adjacent cells: $1$ and $2$, $2$ and $1$, $2$ and $4$, $4$
and $2$, $3$ and $6$, $6$ and $3$, $4$ and $8$, $8$ and $4$.
Suppose the answers in the two adjacent cells of a puzzle are
actually $7$ and $5$, the clue-number will be written in the form
$5/7$ instead of $7/5$, otherwise this would give away the answer
and the puzzle would be too easy to solve!
When fractions are placed on consecutive border lines, a number
of combinations are possible. For example:
if two fractions, $2/3$ and $1/4$, are placed consecutively from
left to right, the possible combinations of answers in the three
neighbouring cells would be the following two sets of numbers $3$,
$2$ and $8$, and $6$, $4$ and $1$;
if three fractions are placed on the consecutive border lines in
the following order $2/3, 1/4$ and $1/2$, the combinations of
answers in the four neighbouring cells would be $3, 2, 8$ and $4$,
and $6, 4, 1$ and $2$.
The remaining rules are as in a "standard" Sudoku, and the
object of the puzzle is to fill in the whole $9 \times 9$ grid with
numbers $1$ through $9$ (one number per cell) so that each
horizontal line, each vertical line, and each of the nine $3 \times
3$ squares (outlined with the bold lines) must contain all the nine
different numbers $1$ through $9$.