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I don't know what 55² is, and I wouldn't bother to compute it by hand.
But I do know:
a²-b² = (a-b)(a+b)
a² = (a-b)(a+b)+b²
Setting a=55 and b=5 we get
55² = (55-5)(55+5) + 5²
Suddenly it's obvious, not because of the numbers, but because of the structure.
The facility with numbers is an epiphenomenon.
2/n, n=2
@ColinTheMathmo
My grandmother gave me this trick:
When you need to calculate the square of a number ending in 5 (a5), you write down "25" and put (a+1)a in front of it.
In your example: 25 with 30 in front: 3025.
Later, I updated the trick for a5 × b5 =
(The average of (a+1)b and (b+1)a) × 100 + 25.
It works, but it's still a trick...
@cvwillegen Yup, and all explained in other branches of this thread. At all comes back to the difference of two squares formula.
It's a lovely little thing.