Nick Lord says "This problem encapsulates for me the best features
of the NRICH collection."
What fractions can you divide the diagonal of a square into by
The diagonal of a square intersects the line joining one of the unused corners to the midpoint of the opposite side. What do you notice about the line segments produced?
Cut off three right angled isosceles triangles to produce a
pentagon. With two lines, cut the pentagon into three parts which
can be rearranged into another square.
A square of area 40 square cms is inscribed in a semicircle. Find
the area of the square that could be inscribed in a circle of the
The largest square which fits into a circle is ABCD and EFGH is a square with G and H on the line CD and E and F on the circumference of the circle. Show that AB = 5EF.
Similarly the largest. . . .
Change the squares in this diagram and spot the property that stays
the same for the triangles. Explain...
It is obvious that we can fit four circles of diameter 1 unit in a square of side 2 without overlapping. What is the smallest square into which we can fit 3 circles of diameter 1 unit?
What can you see? What do you notice? What questions can you ask?
Given that ABCD is a square, M is the mid point of AD and CP is
perpendicular to MB with P on MB, prove DP = DC.
Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square.
A square of area 3 square units cannot be drawn on a 2D grid so that each of its vertices have integer coordinates, but can it be drawn on a 3D grid? Investigate squares that can be drawn.
ABCD is a square. P is the midpoint of AB and is joined to C. A line from D perpendicular to PC meets the line at the point Q. Prove AQ = AD.
Creating designs with squares - using the REPEAT command in LOGO.
This requires some careful thought on angles
This LOGO Challenge emphasises the idea of breaking down a problem
into smaller manageable parts. Working on squares and angles.
If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.
A and C are the opposite vertices of a square ABCD, and have
coordinates (a,b) and (c,d), respectively. What are the coordinates
of the vertices B and D? What is the area of the square?
What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?
Just four procedures were used to produce a design. How was it
done? Can you be systematic and elegant so that someone can follow
Four identical right angled triangles are drawn on the sides of a
square. Two face out, two face in. Why do the four vertices marked
with dots lie on one line?
The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
Using LOGO, can you construct elegant procedures that will draw
this family of 'floor coverings'?
Look at how the pattern is built up - in that way you will know how
to break the final pattern down into more manageable pieces.
Charlie likes to go for walks around a square park, while Alison likes to cut across diagonally. Can you find relationships between the vectors they walk along?
A Short introduction to using Logo. This is the first in a twelve part series.
Which has the greatest area, a circle or a square inscribed in an
isosceles, right angle triangle?
Can you use LOGO to create a systematic reproduction of a basic
design? An introduction to variables in a familiar setting.
Can you use LOGO to create this star pattern made from squares.
Only basic LOGO knowledge needed.