Not sure if this is allowed here, but it would be nice to have a collection of tips to speed up one's mental arithmetic (which will help when it comes to doing Maths Challenges, non-calculator exam papers etc.).

One of my favourites is this trick which is used to quickly find out the square of any two-digit number much more quickly than column multiplication:

To square any two-digit number, decide what number that needs to be added or subtracted to it to get it to the nearest multiple of 10. Then do the inverse on the second number (i.e. if you added 3 to 67 to get 70, take away 3 from the other 67 - since we're multiplying 67 by itself - to get 64) and then - after multiplying the two manipulated numbers (70 * 64) add the square of the number you used to manipulate the numbers (in this case, 3^2, which equals 9). (So, 70 * 64 + 3² = 4489 = 67²⁾ This is a lot quicker than column multiplication since multiplying 70 by 64 is not too hard of a sum for the seasoned mathematician. Bit of a mouthful at first, but it does - in my opinion - pay off when this is mastered.

Here's a proof for why it works:

67 * 67 = (67 +3)(67-3) + 3²

The reason why we need to add the square of the number used to the end of '(67 +3)(67-3)' (i.e. the two manipulated numbers multiplied by each other) is because (67 +3)(67-3) = 67² - 9 via FOIL.

Another useful trick - albeit slightly situational and specific - is the 'percentage swap trick:'

i.e. 36% of 25 = 25% of 36 (the latter being much easier to work out in one's head).

Why? Well, 36% of 25 = 36/100 x 25 = 36 x 1/100 x 25.

Due to the associative law of multiplication, we can multiply in any order we want so therefore (36 x 1/100) x 25 = 36 x (1/100 x 25).

I understand that these might already be known by most of you, but as someone who has any just discovered a passion for mathematics, I would love to broaden my knowledge of these arithmetic tricks, and maybe you guys could learn something new from each others' comments!

Thanks for your time. :)

Edit: Here's another trick which helps me to multiply fiddly non-whole numbers together (it's quite simple, but it's indispensable knowledge and will help those who don't know it):

The product of '1.03 x 2.5' is quite fiddly and not a nice sum to work out in your head. However, we could do this by ignoring the decimal points and therefore the sum becomes '103 x 25' which is a lot less fiddly. So, 103 x 25 = 2575. In order to work out the answer to the original question, we place a decimal point in such a way that it creates n decimal places where n is equal to the number of decimal places in the original question. In this case, there are 3 decimal places (.03 and .5) so therefore we place a decimal point in such a way that it creates 3 decimal places in the answer we got when we worked out the manipulated multiplication question (so therefore the answer to the original question is '2.575'.)

Why does this work? Well, what we are doing in the original question is multiplying '1.03' by 100 and '2.5' by 10 to get rid of the decimal places; therefore the manipulated sum is '1.03 x 100 x 2.5 x 10' which, due to the associative law of multiplication, is equal to '1.03 x 2.5 x 1000.'

1.03 x 2.5 x 1000 = 2575, so in order to get the answer to the original question, we simply divide both sides by 1000 (which basically means to create 3 decimal places in the answer with a decimal point) to get '2.575'

Apologies if this tip is a bit long winded and verbose, but it is a lot quicker to do than explain and I hope it helps! :)

Edit 2: Used bold text to help separate the tips.