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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 - if you haven't read it yet, you might enjoy the section in 𝑆𝑢𝑟𝑒𝑙𝑦 𝑌𝑜𝑢 𝑀𝑢𝑠𝑡 𝐵𝑒 𝐽𝑜𝑘𝑖𝑛𝑔, 𝑀𝑟. 𝐹𝑒𝑦𝑛𝑚𝑎𝑛 about Feynman's calculational methods - and how Hans Bethe was better.
@johncarlosbaez Thanks for the memory!
Yes, I know "Surely You're Joking, Mr. Feynman" *very* well, and I'm familiar both with his techniques, and the claim that Bethe was better.
Great reference.
@ColinTheMathmo
I didn't know about that formula, where does b comes from?
If I want to calculate another squared number how do I get b?
@Andres So the starting point is the "Difference of Two Squares" formula. So we have:
a²-b² = (a-b)(a+b)
You can go ahead and check that to make sure it's valid.
Then we can switch is around to get:
a² = (a-b)(a+b) + b²
Now when we want to square a number (call it "a") we have a free choice of "b" to see if we can make it easy.
Take 62² for example. Then choosing b=2 gives us:
62² = (62-2)(62+2) + 2²
That's (60 times 64) plus 4.
Now, 60 times 64 is 60 times 60 (which is 3600) plus 4 times 60 (which is 240) so we get:
62² = (60 times 64) plus 4
62² = (3600 + 240) + 4
62² = 3844 ... and we're done.
But ...
1/n
@ColinTheMathmo I remember learing that (n5)^2 = n(n+1)25 (sorry for the atrocious notation mixing multiplication with concatenation), when I was very young, as a 'magic formula'. One of those things that just lived in my head without further thought - until now!
@RobJLow So there you are !!
For the square of a number ending on 5, there is another (and quicker) rule: it always ends on 25 and you need to put n * (n+1) in front. Hence 55 squared: 5 times 6 equals 30 and write 25 next to it; so 3025. So 75 squared would be 7 times 8 followed by 25, or 5625. The proof is very straightforward, but I’ll leave it to the reader 
@ColinTheMathmo if you like these mental calculations, I highly recommend “Dead Reckoning - Calculating without Instruments” by Ronald W. Doerfler.
@ColinTheMathmo I once shared this simple trick to a friend who is a professor at a university college. He told me a few days later that he used this trick during a lecture for chemistry students while calculating the results of an experiment on the blackboard. It seems they were very impressed that the could come up with the result (squaring a number ending in 5) in a split second, without the use of a calculator! It’s just a matter of seeing/knowing some simple patterns and a bit of practice.
@wernerkuper I know it well. I've not used it for a while so it would probably take a few minutes to find my copy.
@wernerkuper Indeed, and if you look elsewhere in this thread you will find that I have explained the proof.
I teach these sorts of things, and how they work, and much more besides, in math masterclasses.
@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.