TIM WEBB wrote: No idea what the purchase is (math has never been my strong point!), but I added a cascade:

Each layer of cascade multiplies the purchase with a 2:1.
For the rest, in your picture, you can clearly see the 4 parts. Anytime you pull, all 4 parts get shorter by the reduced distance between the block. That means, you are pulling out line at 4 times the rate the blocks move, so you have a 4:1.
With one cascade you have an 8:1.
With the same blocks you can rig a 5:1 (or 10:1 with the cascade). You'd just have to turn the entire vang around, and, instead of pulling sideways, pull down (that is, away from the end of the purchase that moves). By pulling away from the moving end you'd effectively have 5 parts that get shorter (as the boom moves down, the distance between the top block and your pulling hand also gets shorter). That configuration is known as "rigged to advantage", and turns that 4:1 into a 5:1 purchase.
So, the "math" is really not very deep. Just count the number of parts in your purchase and you get "parts":1 as the ratio. If you've rigged things "to advantage" you add 1 to the "parts", because now you count the free end of the line as one of the parts.
For each cascade, multiply by two, that is, for a triple cascade you get a factor 8.
That's the theory. In practice, you loose a bit to friction. If each block looses 10%, after two blocks, you have only 81% left, after four, it's around 65%. Good blocks matter. Cascades help, because they add such a high ratio per number of additional blocks. Their downside is that they reduce the range of possible adjustment. That's one of the reason that you find them paired with a purchase, as in your setup.
If the losses in your blocks were only 5%, with five blocks you'd be at 77% remaining. If you had done an 8:1 purchase, you'd be at 66%. (Now, that math is one I use the calculator for, 95% to the 8th power is not something I do in my head

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