Challenge Level

Charlie has been playing with calculations again...

$2 \times 4 + 1 = 9$

$4 \times 6 + 1 = 25$

$5 \times 7 + 1 = 36$

$9 \times 11 + 1 = 100$

**What do you notice?**

Click below to see what Charlie said:

"If you multiply two numbers that differ by 2, and then add one, the answer is always the square of the number between them!"

**Can you explain what's happening?**

Click below to see Charlie's explanation:

$ n(n+2) +1 = n^2 +2n+1 = (n+1)^2$

or alternatively,

$(n-1)(n+1)+1 = n^2-n+n-1+1=n^2$

Alison drew a diagram to explain the results. Click below to see:

**Can you make sense of Charlie's method and Alison's diagrams?**

Here are some more number patterns to explore. Some have been expressed numerically, some in words, and some algebraically.

Can you represent each pattern in all four ways,

and

- $2 \times 3 + 3 = \, ?$

$5 \times 6 + 6 = \, ?$

$4 \times 5 + 5 = \, ?$

$9 \times 10 + 10 = \, ?$

What do you notice?

- Choose three consecutive numbers, square the middle one, and subtract the product of the other two.

Repeat with some other sets of numbers.

What do you notice?

- $3 \times 3 - 1 \times 1 = \, ?$

$8 \times 8 - 6 \times 6 = \, ?$

$7 \times 7 - 5 \times 5 = \, ?$

$10 \times 10 - 8 \times 8 = \, ?$

What do you notice?

- $n(n+1) - (n-1)(n+2) = \, ?$

$(n+1)(n+2) - n(n+3) = \, ?$

$(n-3)(n-2) - (n-4)(n-1) = \, ?$

What do you notice?

- $3\times 5 + 1= \, ?$

$5\times 7 + 1= \, ?$

$7\times 9 + 1= \, ?$

$9\times 11 + 1= \, ?$

What do you notice?

- Choose three consecutive numbers and add the product of the smallest two to the product of the greatest two.

Repeat with some other sets of numbers.

What do you notice?

*With thanks to Don Steward, whose ideas formed the basis of this problem.*

*You may be interested in the other problems in our Factorise This! Feature.*