I first started out with the three times time tables. The first number I decided to choose was 3 itself. So square the number on its sides, so 2= 4 and 4=16. Now I subtracted them, and I got 12. Next I used 6. So I squared 5 and 7 which gave me 25 and 49 and then I subtracted them, which gave me 24. The last number I used was 9. So I squared 8 and 10 and got 100 and 36. Subtract them, and their difference is 36. Now let's put the numbers in ascending order:
3=12
6=24
9=36
If we look at the numbers, we can see that the number from the three times table gets multiplied by four to get to the difference of the squares of the numbers in its sides.
The next part asks us to try with a number from the five times table. So I chose 5 first. So I squared 4 and 6 and got 16 and 36. Their difference gave me 20. Then I tried 10. So I squared 9 and 11 to give me 81 and 121, which share a difference of 40. Finally, I tried 15. So I squared 14 and 16 to get 196 and 256. Subtract them, and I got 60. Now let's do the same as before- arranging them in ascending order-
5=20
10=40
15=60
And the same thing happens as before. The number from the 5 times table is multiplied by 4 to get the difference of the squares of the numbers on its sides.
So if you want to find the difference of the squared numbers on either sides of a particular number, you just need to multiply it by four. It works with all the numbers.
For the extension, they ask us to choose a number and then find the difference of the squares of the numbers two sides up and two below. So I chose 4. That means I'll have to square 2 and 6 which gave me 4 and 36. Their difference is 32. Previously, you had to multiply the number by four to get the difference of the squares of the numbers on either of its sides. But this time, you have to multiply it by 8. This is because it asks us to square the numbers TWO sides above and below the number.
Let's take this a step further. I tried doing three steps up and below the number. So I chose 12. My prediction is the number 144, since you need to multiply it by 12 and since it's three sides up and below. So I squared 9 and 15 and got 81 and 225. Their difference is 144. So I was right!
This will work the same way for four places up and down, five and six, excpet that you need to multiply four by the number of places you are moving up. Then, the product you obtain from that is the number you need to multiply it with.
And that's how I solved it.