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Problem of the Week Archive
Fence Me In – October 3, 2016
Problems & Solutions
Zaria and Maisie joined the National Math Club, and during the first club meeting they played Fence
Me In. In the game, Zaria and Maisie were given two six-sided dice and a game board that was a
20­unit by 20-unit grid. During each turn, one player rolled the dice and calculated the product of
the two numbers that were rolled. This product could be considered either the area of a rectangle, in
units squared, or the perimeter of a rectangle, in units. The player then had to draw a rectangle with
either that area or that perimeter and with integer dimensions (not necessarily the same as the two
numbers rolled) on the game board. Zaria and Maisie took turns rolling the dice and then drawing a
rectangle. If either was unable to draw a rectangle in the remaining area, the player would lose a
turn. Play continued until the entire board was completely filled with non-overlapping rectangles.
How many possible products can Zaria and Maisie roll?
The smallest possible product that can be rolled is 1 × 1 = 1 and the
largest is 6 × 6 = 36, but not all the integers between 1 and 36 can be
created as a product of two numbers using only the integers 1 through
6. The products that can be rolled are 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15,
16, 18, 20, 24, 25, 30 and 36. A total of 18 possible products.
What is the area of the largest possible rectangle that
can be drawn during a turn in this game?
The largest product that can be rolled is 36. This can be an area of
36 units2, but if we use 36 units as the perimeter, it is possible to get a
larger area. The dimensions of possible rectangles with perimeter
36 units are 1×17, 2×16, 3×15, 4×14, 5×13, 6×12, 7×11, 8×10
and 9×9. The largest area is created by a 9×9 rectangle and is
81 units2.
If Zaria rolled a 3 and a 6 on her first turn, how many different sized rectangles could she draw?
The product is 18. A rectangle of area 18 units2 can be made 3 ways, which are 1×18, 2×9 or 3×6. A rectangle with
perimeter 18 units can be made 4 ways, which are 1×8, 2×7, 3×6 and 4×5. We see that a 3×6 rectangle is counted twice
because its area, 18 units2, and perimeter, 18 units, are numerically equal. In total, there are 3 + 4 – 1 = 6 different sized
rectangles.
What is the minimum number of rectangles it could possibly take for Zaria and Maisie to completely
fill the game board?
If we use 4 rectangles of dimensions 8×10, with A = 80 units2 and P = 36 units, 1 rectangle of dimensions 4×14, with
A = 56 units2 and P = 36 units, and 1 rectangle of dimensions 4×6, with A = 24 units2 and P = 20 units, then we can fill the
board completely with 4 + 1 + 1 = 6 rectangles. There are other possible combinations but in any of these, the minimum
number of rectangles that can be used to completely fill the board is 6 rectangles.
Problem of the Week Archive
Fence Me In – October 3, 2016
Problems
Zaria and Maisie joined the National Math Club, and during the first club meeting they played Fence
Me In. In the game, Zaria and Maisie were given two six-sided dice and a game board that was a
20­unit by 20-unit grid. During each turn, one player rolled the dice and calculated the product of
the two numbers that were rolled. This product could be considered either the area of a rectangle, in
units squared, or the perimeter of a rectangle, in units. The player then had to draw a rectangle with
either that area or that perimeter and with integer dimensions (not necessarily the same as the two
numbers rolled) on the game board. Zaria and Maisie took turns rolling the dice and then drawing a
rectangle. If either was unable to draw a rectangle in the remaining area, the player would lose a
turn. Play continued until the entire board was completely filled with non-overlapping rectangles.
How many possible products can Zaria and Maisie roll?
What is the area of the largest possible rectangle that
can be drawn during a turn in this game?
If Zaria rolled a 3 and a 6 on her first turn, how many
different sized rectangles could she draw?
What is the minimum number of rectangles it could
possibly take for Zaria and Maisie to completely fill the
game board?