Ten hidden squares
These points all mark the vertices (corners) of ten hidden squares. Can you find the 10 hidden squares?
Problem
On the graph below there are $34$ marked points.
These points all mark the vertices (corners) of ten hidden squares. Each of the $6$ red points is a vertex shared by two squares. The other $28$ points are each a vertex of just one square. All of the squares share at least one vertex with another square.
All the squares are different sizes.
There are no marked points on the sides of any square, only at the vertices. (There are two near misses!)
Can you find the ten hidden squares?
You might like to play the game Seeing Squares which uses some similar ideas.
Getting Started
Start with one point and look for others that make a square with it.
Don't forget that some squares may be tilted!
Student Solutions
Harjun sent in this image to show us where the ten squares are:
There is just one square which you've mis-drawn a line in I think, Harjun. Can you see which one it is?Akalya and Michelle from Devonshire Primary School and Esther sent in the coordinates of the 10 squares so we can check that way, too:
1.(2,-3) (-1,-3) (-1,-6) (2,-6)
2.(0,0) (5,-3) (2,-8) (-3,-5)
3.(0,0) (2,2) (0,4) (-2,2)
4.(1,0) (4,1) (5,-2) (2,-3)
5.(2,10) (8,10) (8,4) (2,4)
6.(2,10) (7,8) (5,3) (0,5)
7.(-1,-1) (-1,-3) (-3,-3) (-3,-1)
8.(-4,2) (-9,2) (-9,-3) (-4,-3)
9.(-7,5) (-10,2) (-7,-1) (-4,2)
10.(-7,5) (-5,4) (-4,6) (-6,7)
Thank you to all of you - I thought this problem was more difficult than it sounded!
Teachers' Resources
Ten Hidden Squares
On the graph below there are $34$ marked points.
These points all mark the vertices (corners) of ten hidden squares. Each of the $6$ red points is a vertex shared by two squares. The other $28$ points are each a vertex of just one square. All of the squares share at least one vertex with another square.
All the squares are different sizes.
There are no marked points on the sides of any square, only at the vertices. (There are two near misses!)
Can you find the ten hidden squares?
You might like to play the game Square It which uses some similar ideas.
Why do this problem?
This problem has two aspects. A knowledge of coordinates in all four quadrants is necessary but also a secure understanding of the properties of squares. The activity challenges learners' misconceptions about how shapes "should be" oriented.
Possible approach
Key questions
Possible extension
Those who were able to do the problem quickly could plot more squares on this sheet, making sure that all the squares are still different sizes.
Possible support
Those who are experiencing difficulties could try this simpler version of the problem which only uses coordinates in the first quadrant.