Pears Make Squares

A Maths Starter Of The Day

Pears Make Squares


Let a, b and c be the pear numbers.

a + b is a square number

b + c is a square number

c + a is a square number


Topics: Starter | Arithmetic | Number

  • COLA, London Y9 Set 1
  • 0.5, 0.5, 0.5.
  • Matthew, 3D, Craigslea State School
  • I figured out another solution. All the pears' numbers are 0!
  • R G, City Of Leeds
  • I believe it works for all values to equal a square number divided by 2. eg 2, 2, 2 or 8, 8, 8.
  • Transum,
  • As you can see there are many answers below. It may be interesting to group the answers to help understand the patterns. There are some answers where all three numbers are the same. How would you describe these solutions? The comment above suggests the rule for even square numbers. What about odd square numbers? Must the pears be thinking of integers or could they have mixed numbers in their fruity minds (as proposed in the first comment above)?
  • Y4GB, Leigh, England
  • My Year 4 primary top set found 1 , 120 and 24 to work. Mr B.
  • Wendover C.E Junior 6J,
  • Freddy found 2,2,14 and 0, 16, 9! Connor found 2,34,2!
  • 4W, Wendover C E Junior School
  • Dylan found 20, 5 and 44.
  • Sam, Carnoustie
  • - half a square number, write it twice subtract that from any other square to find the third number eg. 2,2,7 2,2,14 2,2,23, 2,2,34... 8,8,1 8,8,17 8,8,28 8,8,41... 12.5,12.5,3.5
    if you want three different numbers
    - find two numbers with an odd difference which add up to an odd square: write them down
    - find the two squares with that same odd difference
    - subtract the numbers you wrote down from the two squares you just found: you should get the same answer twice. write it down.
    eg. 1, 24, (with a difference of 23: just like 121 and 144) 120
    2, 23 (with a difference of 21: just like 100 and 121) 98
    3, 22 (with a difference of 19: just like 81 and 100) 78
    4, 21 (with a difference of 17: just like 64 and 81) 60
    5,20,44 6,19,30 7,18,18 8,17,8 9,16,0 10,15,-6 11,14,-10 12,13,-12
    notice the pattern: first number increases by 1 second number decreases by 1 third number decreases by previous even number
    1,48,528 2,47,482 3,46,438 4,45,396 5,44,356 6,43,318 7,42,282 8,41,248 9,40,216 10,39,186 11,38,158 12,37,132 13,36,108 14,35,86 15,34,66 16,33,48 17,32,32 18,31,18 19,30,6 20,29,-4 21,28,-12 22,27,-18 23,36,-22 24,25,-24
    the solutions are infinite!
  • Class 4 Highfields, Twitter

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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 a similar activity.

Here is the URL which will take them to another activity involving square numbers.

Student Activity


Curriculum Reference

See the National Curriculum page for links to related online activities and resources.


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