Pulley Logic by Fernando da Vinci

Logic Gates.

As we can see below, the default state (0 input) of any logic gate can be either $\color{#20A900}On$ ( $\color{#20A900}1$ ) or $\color{#D61F06}Off$ ( $\color{#D61F06}0$ )

The set of unknown mechanisms that are $\color{#20A900}On$ (down) by default is (2,3,4,5).

The set of unknown mechanisms that are $\color{#D61F06}Off$ (up) by default is (1,6,7)

The following table identifies the Logic Gates in Order, and lists the possible Mechanisms each logic gate may have at each stage of deduction.

A $\color{#20A900}NOT$ gate only has one input, so that must be mechanism 5.

Mechanisms 1 and 2 go through two state changes between incrementing inputs:

• Zero input = default state
• One (either) input = state change
• Two (both) inputs = state change

This is because the only comparable spring cages are at the top of the mechanism (1,0,2) and (1,1,3), and affect the height of the '-' pulleys. The relative distance between the '-' pulleys dictates the 'out' state.

One input changes the relative distance between the '-' pulleys, and then both inputs brings them back to the original relative distance apart.

The $\color{#D61F06}XOR$ and $\color{#20A900}XNOR$ logic gates match these mechanisms. As they have different default states, we just need to assign the mechanisms based on their starting position.

• Mechanism 1's default position = $\color{#D61F06}Off$
• Mechanism 2's default position = $\color{#20A900}On$

The remaining logic gates in both sets have the same differences - One gate will have a state change with at least one input, and the other gate will only have a state change with two inputs. So we need to put some thought into how the mechanisms work to make the distinction.

The rule I used was (with respect to incrementing inputs):

$\small \color{#3D99F6}\text{The rope in the spring cage with the greatest initial directional force (compared to its opposing end) does one of two things;}$

• $\small \color{#3D99F6}\text{Moves in the direction of its force first,}$

• $\small \color{#3D99F6}\text{Or moves away from the direction of its force last.}$

• Moving first equates to one input (either).
• Moving last equates to two inputs (both).

(I could have been using this rule from the outset, but I didn't need to until now.)

With this logic, we can determine the following:

Mechanisms 3 and 4

• Mechanism 3
  • Default position = $\color{#20A900}On$
  • The 'out' spring cage (1, 1 ,0) has a force greater than its opposing end (1, e ,0).
  • The rope in the 'out' spring cage can only move in the direction of its force, so it will move first.
  • So a state change will occur with at least one input into the system.
  • $\color{#20A900}NOR$ matches this logical system.
• Mechanism 4
  • Default position = $\color{#20A900}On$
  • The 'out' spring cage (1, e ,0) has a force less then its opposing end (1, 1 ,0).
  • The rope in the 'out' spring cage can only move in the direction of its force, so it will move last.
  • So a state change will only occur with two inputs into the system.
  • $\color{#20A900}NAND$ matches this logical system.

Mechanisms 6 and 7

• Mechanism 6
  • Default position = $\color{#D61F06}Off$
  • The 'out' spring cage (1. 0 .1) has a force less than its opposing end (1, 1 ,2).
  • The rope in the 'out' spring cage can only be pulled away from the direction its force, so will move first.
  • So a state change will occur with at least one input into the system.
  • $\color{#D61F06}OR$ matches this logical system.
• Mechanism 7
  • Default position = $\color{#D61F06}Off$
  • The 'out' spring cage (1, 1 ,2) has a force greater than its opposing end (1, 0 ,1).
  • The rope in the 'out' spring cage can only be pulled away from the direction its force, so will move last.
  • So a state change will only occur with two inputs into the system.
  • $\color{#D61F06}AND$ matches this logical system.

$\large \boxed{5761432}$

I don’t know what the question is. I just randomly thought some number and typed it. The answer is exactly correct

What a great problem, thanks for contributing! Nice to learn about Fernando da Vinci and that he too was centuries ahead of his time!

(Figure 1) pulling either input string with force F pulls the lower two pulleys down with force 2F. Each of the upper pulleys then is pulled down with force F. They each may move down over a distance 1, against a spring force. The rightmost (1,0,2) spring has no initial tension, so it will move down first, tightening the output rope. When F=1, that spring box is halfway, the (1,1,3) box is still up, and the output box is already fully down. When both inputs are down however, both pulleys will be down and the output string is relaxed again. The truth table is that of XOR:

(Figure 2) The only difference with (1.) are the spring cages at the top. The (1,0,2) box now is connected to the leftmost pulley, so that the output rope is not tightened, but loosened when either input rope is pulled. Output is initially down, corresponding to a NXOR:

(Figure 3)  Initially, the output string tension is 1, the big pulley is pulled down by twice that, up by the upper left spring box (also 2). When pulling either input string, the top spring will yield so that the big pulley lowers, relaxing the output string. The output spring (1,1,0) pulls harder on it than the other end (1,e,0), so it will take its chance to go up (0). When both inputs are down, the output string is relaxed even more, which does not alter the output. This is a NOR gate.

(Figure 4) Similar, but here the output and lower left spring boxes have swapped characteristics. This is a NAND gate.

(Figure 5) Pulling the input with force F causes 2F at the left top box. So that box is fully down when 2F=1. The fixed pulley inverts the input, so this is a NOT gate.

(Figure 6) If one of the inputs is down, the pulleys are halfway down. The output is the first to go down, this is an OR gate.

(Figure 7) If one of the inputs is down, the pulleys are halfway down. The output is the last to give in, this is an AND gate.

Listing the figure numbers in the given order:

1)  NOT -> 5

2)  AND -> 7

3)  OR   -> 6

4)  XOR   -> 1

5)  NAND   -> 4

6)  NOR   -> 3

7)  XNOR  -> 2

The final answer then is $\boxed{5761432}$