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The grid multiplication method works because it is splitting up the numbers (e.g. 23 into 20 and 3) to times them together easier (as it is much easier trying to do 20x20 than 23x21). After you times every figure in the first number by every figure in the second number, you simply add them all up.
Column multiplication works in a similar way to the grid multiplication method, as the column method splits the numbers up in a similar fashion. With the sum 23x21, because you do 1x3, then put the answer in the units column, then 1x2, but put the answer in the tens column, you are basically doing 1x20 and carrying over from the units to the tens. For the next line, as you are really doing 20x3, you put a 0 as the number in the units column, because you act like you are carrying the 20x3 (60) across to the tens once again.
The line method works so that after you have drawn all your lines and added them up, they are split into units, tens, hundreds, and so on. This is done by multiplying the units by units (the only way to get units), the tens by units and units by units (the only ways to get tens), the units by hundreds and the tens by tens (the only ways to get hundreds), and so forth for any other columns. Basically, this method is quite simple when you really understand it, as the same effect is achieved if you do a sum so that units x units go into the units column, the tens x units or units x units go into the tens and hundreds x units or tens x tens go into the hundreds. For example (without using lines, but using the theory behind it), 104x52 would be 8 in the units (4x2), 20 in the tens (carried over to the hundreds (0x2 and 4x5)) and 2 in the hundreds (plus the carried 2 from the tens column (1x2 and 0x5).
The Gelosia method still works with the units x units, tens x units, etc. system explained for the line multiplication. Basically, the reason the squares are halved diagonally is that it is so that is the number was to multiply to over 9 (therefore needing to be carried to the next column), it would simply be added in with the next column, instead of you having to carry it over yourself. The way it is diagonally crossed of means that instead of doing 20x20, you can just do 2x2, because the 2s are boosted up into the hundreds column. The only number that goes into the units column is units x units, and the tens column is units x units and tens x units. However, because the numbers are already in their columns, when you add up the numbers in the end, they will appear in sequence, like 4-8-3 for 23x21, which just goes to 483.
The methods all have in common the fact that they split numbers down into their columns, as this is necessary for multiplication.
The advantages of the grid are that it splits it down into simple numbers, like 20 and 3, or 20 and 1. The disadvantages are that this method might get very complex if a lot of numbers are introduced, like 69283x24846.
The advantages to the column method are that this method is quite quick to do, and takes up minimal space, but it is also fairly easy to make a mistake, mostly because there are quite a lot of numbers around.
The advantages to the line method is that it would really help people who liked to visualise things, but the bad things are that it takes up a lot of space and can get very confusing, for example if it is 102x7, because of the 0 in the middle of 102. If this happened, you would have to leave a space, but the best way is to draw a dotted line across so you can figure out where it would be.
The advantages to the Gelosia method are that it is very helpful as it carries large numbers across for you, but it would be quite confusing as you may occasionally mess up on the adding up diagonally. This method would take some time, as it takes a little while to draw the grid for the multiplication.