Well, 16 is enough to make 4 bars and 16 times 16 is 20. That means that in the next bar, we’ll have 12. The next two, however, only add up to a few seconds and the fourth one just adds up to zero.

In other words, there is no way to make the numbers 16 × 20 × 4 as quickly as we do for 4 × 16 × 4.

And, in fact, you can only do this with a small amount of memory.

For example, if you wanted to make 16 × 2, you would need to read out the previous two numbers 4 × 16 to make 16 × 2. You could then just do:

16 × 2 + 4 × 16 + 16 × 2 = 24

But, you can’t read out the 16 × 2 numbers with 2 × 4. You can, however, add up the 8 numbers and write out the result for each, like so:

8 × 4 + 16 × 2 = 32

So, we know that in order to make 32 or 64 squares in each row, we need 16 × 2 and 4 × 16 or 32×4 and 16 × 2.

If you do an experiment that computes the time to get 16 x 16, for example, this is what we get:

I have written out 32 + 64 × 2 = 768 bytes for you. This number is equal to 64 bits, which is the size of memory. Now let’s see what the time for a 1-minute time window will be, which is 4×32=256 bytes.

Time = 24 × 1024 / 256 = 240 seconds.

Again, this number is the size of memory.

Now, there is a way to make squares in a row that are 16 x 16. But, what would be the fastest way?

Using the same principle, we can make squares in each other row as well. If we use 8 x 4 to make 16 × 15, we get 160 × 4 = 1,024 bytes from the second row. In other words, 16 × 4 × 8 = 2,048 bytes.

We can do this by writing out 2 × 8 and that gives 2,048 bytes of memory and you’ll still end up with the same time as before. In this case, the 8 × 4 = 16×16 algorithm is quicker than the 4 × 16 algorithm. It’s simply using more bits and bytes.

The math

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