Maths Olympiad Problems | Maths Contest Problems | PRMO Questions | NSEJS Maths Questions Important for ---- NTSE Stage 2, Maths Olympiads, PRMO, RMO, INMO
Joy is making a rectangular patchwork quilt from square pieces of fabric. All the square pieces are the same size. She
starts by joining two black squares into a rectangle. Then she adds a border of grey squares
Next, she adds a border of white squares.
a) How many white squares does she need?
Joy continues to add borders of grey and borders of white squares alternately.
b) How many grey squares and how many white squares does she need for the next two borders?
c) Joy notices a pattern in the number of squares used for each border. Describe this pattern and explain why it
always works.
d) There are 90 squares in the outside border of the completed quilt. How many borders have been added to Joy’s
starting black rectangle?
Answers:
a) The quilt after adding one grey and one white border:
The number of white squares is 18.
b) The quilt after adding another grey and another white border:
There are 26 extra grey squares and 34 extra white squares.
c) The table shows the number of squares in each border that we have counted so far.
Except for the corner squares, each square on the previous border corresponds to a square in the new border. Each
of the corner squares C on the previous border corresponds to a corner square C' in the new border. In addition,
there are two new squares next to each corner.
Thus the number of squares from one border to the next increases by 4 × 2 = 8.
d) Alternative i
The first border has 2 + 8 squares.
The second border has 2 + 8 + 8 squares.
The third border has 2 + 8 + 8 + 8 squares.
And so on.
We want 2 + 8 + 8 + ··· = 90.
Hence the number of 8s needed is 11.
So Joy’s completed quilt has 11 borders.
Alternative ii
From Part c we have the following table.
So Joy’s completed quilt has 11 borders.
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