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Tom started us off with the following idea. Thanks Tom.
In the square of dots you can imagine the"L" of any odd number and if you take the square for the odd number below it from the square of the odd number you get the number you stated with. To find the two squares you have to find half the odd number and round it up and down. So half of $13$ is $6.5$ so the two squares have sides $7$ and $6$.So, using the diagram you know that $$105= \frac{105-1}{2} + \frac{105+1}{2}$$
$$ 105 = 52 +53$$
$$ 105 = 53^2 - 52^2$$
You can always find one way using this method. So
$$1155= \frac{1155-1}{2} + \frac{1155+1}{2}$$
$$ 1155 = 577+578$$
$$ 105 = 578^2 - 577^2$$
Alex, from the Grammar School at Leeds extends the algebra building on the second visualisation: