Enrichment in Mathematics – Algebra
Diagonal Sums
Teachers Notes
This investigation can be used as an extended activity to cover several lessons.
The solution is open-ended, so pupils can pose their own questions and follow
their own lines of investigation.
It is an excellent opportunity for pupils to:
Solve problems
Represent a problem using symbols, words, diagrams and tables.
Use algebra to generalise.
Solutions
Green Square
The sum of the numbers in opposite corners is 15.
Pink Square
The sum of the numbers in opposite corners is 47.
In both squares the sum is the same for both diagonals.
Yellow Square
The sum is 161 for both diagonals.
Other 2 × 2 Squares
Trying other squares with four numbers in them, this rule always seems to
apply.
Why does it happen?
If we give the number in the top left hand square the letter n, the rest look like
this:
n
n+1
n + 10
n + 11
So the sum of the two diagonal squares is:
n + n + 11 = 2n + 11
and
n + 1 + n + 10 = 2n + 11
Hence the reason why the two sums are always the same.
Red 3 × 3 Square
Moving on to the red square, which is a 3 × 3, we can represent this in a
similar way:
n
n+2
n + 20
n + 22
The sum of the opposite corners is:
n + n + 22 = 2n + 22
n + 2 + n + 20 = 2n + 22
Hence the same result occurs again, both totals are the same.
Worcestershire Numeracy Team
Enrichment Activities
Enrichment in Mathematics – Algebra
Pupils may now want to summarise the results so far, in a table:
Size of square
Sum of
diagonals
Algebraic sum
of diagonals
2×2
same
2n + 11
3×3
same
2n + 22
Different Sized Squares
Pupils can then make a prediction for the sum of the diagonals in a 4 × 4
square. For example:
 They will be the same for both diagonals
 That the generalisation might be 2n + 33.
Test this out by trying some specific squares on the 100 grid. The two
diagonals always sum to the same total. First prediction seems to be true.
Generalising, pupils get:
n
n+3
n + 30
n + 33
This gives the diagonals a total of:
n + n + 33 = 2n + 33
n + 3 = n + 30 = 2n + 33
proved!
Can we predict the sum of the diagonals in any sized square?
Let’s call the square width m.
The total of the diagonals will be:
2 times the left hand starting number (n) + 11 times the square width (m) – 1
2n + 11(m - 1)
So for a 10 × 10 square, the total of the diagonals will be:
2n + 11 × 9
2n + 99
This



investigation can be extended in many ways. For example:
What about a negative starting number?
What if I took the product of the diagonals, instead of the sum?
What about a rectangle? Would the same rule still apply?
Worcestershire Numeracy Team
Enrichment Activities
Enrichment in Mathematics – Algebra
Diagonal Sums
Look at the green square which contains the numbers 2, 3, 12 and 13.
What is the sum of the numbers that are diagonally opposite each other?
Do you notice anything?
Look at the pink square.
Does the same thing happen?
What about the yellow square?
You could try with other squares which have four numbers in them.
Why does this happen?
Look at the squares shaded red. They form the corners of a large 3 by 3
square. If you add the numbers diagonally opposite each other, what
happens? Why?
What happens for squares of different sizes?
Taken from nrich.maths.org, where there are templates of 100 squares for reproduction.
Worcestershire Numeracy Team
Enrichment Activities