The claims made on that web page are true. There have been several attempts to rebuild one of his machines. One of them used to be shown in the Technikmuseum in Berlin, but apparently it has been moved to the Konrad Zuse Museum in Hünich.
The machine used binary arithmetic and could do floating point addition (0.8 seconds) and multiplication (3 seconds) and had 64 22-bit words of memory. It's actually quite a technological marvel considering it was built in the war and he had trouble getting the materials to build it as military projects were prioritized and the Nazis didn't consider his machine important for the war effort.
If you ever visit Berlin, especially with kids, the city has a lovely Technology Museum with an computer section that also features Zuse replicas and gear, including a replica of the still-mechanical Z1. I think Zuse himself made and donated it to the museum.
I absolutely love the giant honking hand-crank you can manually turn to make it go.
As a child seeing the mechanical Z1 really drove home for me that computers are understandable and not magic, literally breaking what they do down into little managable parts and steps and the art of making a computer is in sizing (how many adders do you need?) and arrangement (how can you make it produce results faster?).
With some erudite commentary and informed pointing-at-things I think you have a good chance to recreate that revelation for your kids too.
One thing that struck me about the Z1 was how closely it paralleled things I'd expect to see in a silicon implementation of a CPU:
I recall seeing: different physical blocks for different logic, a big regular blog for the memory and other blocks for ALU, etc, addressing handled by column and row selection lines, evaluating bit values by putting a signal onto the data line and seeing whether the value changes...
But all implemented with mechanical linkages (so a signal is a pull of a rod, a connection is a hooking of a rod onto some other linkage, etc.
The claim that the Z3 computer was Turing-complete is not true. There is a paper arguing for it, but a detailed reading of it shows that this is an extremely far-fetched and somewhat disingenuous stretch. (The disingenuous part is because any fixed-function calculator could then be claimed to be "Turing-complete", not just the Z3). The central point of the Church-Turing thesis is that a finite set of instructions, given an unlimited memory to work with, can perform any calculation we can imagine (where the "can imagine" part makes the thesis philosophical). The "finite set of instructions" is indispensable, however, since if the instructions are unlimited, you can simply encode any answer you want into them. The "Turing" mode of Z3, which was of course never used, involves a program which essentially scales in length with the total number of calculations it will perform - or even the exponential of that number, if there are many branches - which is not a good model of a Turing machine.
Of course, no computer is a true Turing machine, since the memory is always limited, but our computers are a useful physical approximation of a Turing machine because a small program can compute using a large memory. The Z3 is not that type of a device at all.
If a computer can emulate another computer that is known to be Turing complete then it must itself be considered Turing complete. One thing we must decide is if we allow the addition of memory that the first machine didn't have originally. For example, an Apple ][ can emulate a modern PC (at a tiny fraction of the speed) if we can add a card with a few GB of RAM to it.
A very simple Von Neumann style computer is the ByteByteJump. It has a single instruction (so no op code) with 3 address fields and it copies a single byte from the first address to the second address and then always jumps to the third address. If the addresses are 3 bytes long then every instruction takes up 9 bytes in memory. You can do math and logic operations by setting up 16KB tables in memory and then patching an instruction so the two operands are the bottom two bytes of the first address. To subtract the byte in address 0x001234 from the byte in address 0x00C0F0 and store the result in address 0x003333 on a little endian BBJ with a subtraction table at 0xD00000 you could use this sequence:
There are several ways of implementing conditional jumps by patching the value of the third address.
Could the Z3 emulate this machine if given a 16MB memory it could interface to? Its instruction tape could be made into an infinite loop, which is good enough for this application. There are no other jumps or conditional execution in the emulator.
If you do need conditional execution to emulate a Turing complete machine (the Game Of Life, for example) then you might get by with conditional assignment instead. If the value of B is either 0 or 1, then Z:=A*B+C(1-B) will assign either A or C to Z. I am not familiar enough with the Z3 but would be surprised if it can't even do that.
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The machine used binary arithmetic and could do floating point addition (0.8 seconds) and multiplication (3 seconds) and had 64 22-bit words of memory. It's actually quite a technological marvel considering it was built in the war and he had trouble getting the materials to build it as military projects were prioritized and the Nazis didn't consider his machine important for the war effort.