Lets multiply, just for kicks…
Multiplying a number by itself is pretty easy, even for large numbers. Just as you can factor 365+365 into 2300 + 260 + 25 (or any other method described here) You can factor the product as well
365365
= (300+60+5)(300+60+5)
= 300300 + 6060 + 55 + 2*(30060) + 2(3005) + 2(60*5)
= 90000+3600+25+36000+3000+600=133225

Missed the Edit Window…
Let me explain more if its not clear how to do this in your head…
You are multiplying each digit of the first number by each digit in the second number, adding the appropriate number of zeros, then adding up the results. The first three terms are easy, because you are just squaring 3, 6, and 5. Then you double each combination possible, 36, 35, and 6*5. Keeping track of the zeros is the hard part, then you have to add 6 numbers together, but they are all nice, neat numbers with lots of zeros.

I use radcon math - I will round the initial numbers to something I find convenient to manipulate: In this case 375. Which works out to 300 + 300 + 150 = 750. Then just subtract the amount I rounded by - 20, to get 730.

Of course, that’s only because I needed an exact answer. For most uses, I’d find 750 useful enough for rough calculations.

I did something very similar to InvisibleWombat’s first solution, although I rounded each instance of 365 up to 400, added them together (or multiplied 400x2, which gets the same result when using numbers of equal value) to get 800, then doubled 35 (400-365) to get 70, which I then subtracted from 800 to get 730.

I’ve said it before (although not in this arena): you want to learn how to do math quickly in your head, you work at a high school concessions counter. There’s no pressure like the kind experienced when a high school football fan orders three hot dogs, two Butterfingers and a large Coke, and he wants exact change back from his $10 bill.