How Many Squares?#1

Can you work out how many squares can be found in this diagram? Can you be sure you haven't missed any? Can you be sure you haven't counted any twice?

Show Some Clues

A Mathematics Lesson Starter Of The Day

Topics: Starter | Puzzles | Shape

  • Quinetta, NH
  • My fiance and I just did this puzzle; me on paper and him in head. I got 44 and he got 46. Very fun though.
  • Class 5G, Derby
  • We think that Mrs Green is the cleverest teacher in the world as she got all 42!! 5G are quite celever too... :).
  • Mrs Hill's Marvellous Maths Group - Form 5, Eversfield Preparatory School, Solihull
  • We found all 42 squares easily. We went on to look at the chess board and found lots of squares...204 in total. What's more, we noticed lots of other really interesting patterns. Square numbers are clever things...and so are we!
    Mrs Hill's last ever maths lesson at Eversfield. :-(.
  • Mr Cansdale, Wintringham Grimsby
  • The majority of 8X2 of Wintringham believe it is 44! (HA).
  • Mr Cansdale 8A4, Oasis Wintringham
  • 7 out of 21 think 34.
  • Mr Jing-Hao, Japan School
  • Extremely easy starter the pupil in my top set found 42 in a matter of seconds.
  • Purab, Noida
  • Quinetta can you tell me how you got 46 or 44 I only got 40.
  • Mary And Class 3, Class 3
  • Class 3 at Frettenham Primary (Year 4/5/6) all agree there are 42 squares.
  • Paulie, Cleveland, OH
  • There are:
    24 1x1 squares
    13 2x2 squares
    4 3x3 squares
    1 4x4 square
    for a total of 42 squares. Only 42!!!
    On the chessboard there are:
    64 1x1 squares
    49 2x2 squares
    36 3x3 squares
    25 4x4 squares
    16 5x5 squares
    9 6x6 squares
    4 7x7 squares
    1 8x8 square
    for a total of 204 squares.
  • Devon, UK
  • Can't wait to this with my class. We found 45.

How did you use this starter? Can you suggest how teachers could present or develop this resource? Do you have any comments? It is always useful to receive feedback and helps make this free resource even more useful for Maths teachers anywhere in the world.
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The important thing is not the answer but the method you used to arrive at the answer. Did you have a structured approach? Were you sure that you had the right answer? Can you make up a puzzle like this for others to solve?

Extension Challenge

How many squares are there on a chess board?

Chess Board

How many rectangles are there on a chess board?

Other shape counting starters:

How Many Squares 1? | How Many Squares 2?
How Many Triangles 1? | How Many Triangles 2? | How Many Triangles 3?
How Many Rectangles? | Rectangles Investigation | Icosahedron | Mystic Rose

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Laptops In Lessons

Teacher, do your students have access to computers such as tablets, iPads or Laptops?  This page was really designed for projection on a whiteboard but if you really want the students to have access to it here is a concise URL for a version of this page without the comments:

However it would be better to assign one of the student interactive activities below.

Laptops In Lessons

Here is the URL which will take them to another systematic listing activity.

Student Activity



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