If you pick any of the square numbers on the grid and then subtract the odd numbers starting with 1, you get the numbers on the diagonal going the other way.
So if we start with 36, subtract 1 to get 35, then subtract 3 to get 32, then subtract 5 to get 27.
(If you compare this diagram to the tables grid, you’ll see how the numbers all fit in.)
You can fill in the rest of the grid in exactly the same way, just using the even numbers (2,4,6,8…). Look at the diagonal running underneath the squares which goes 2,6,12, 20: you can fill this in by starting with 2, then adding 4, then 6, then 8 and so on. And then if you pick any of these answers (e.g. 20), you can subtract 2, then 4, then 6 to get the diagonal going the other way (e.g. 20 – 2 = 18 then 18 – 4 = 14 and 14 – 6 =8).
Using these sequences of odd and even numbers, you can go on to create a set of times tables as big as you like, and you don’t need to do any multiplying!