I don't have a definitive answer and it is late here, so sorry if that trails off.
Given the [Amazon description](https://www.amazon.com/dp/B000BFORIK
), I'd say you need to place this somewhere between an [Arduino](https://en.wikipedia.org/wiki/Arduino
) and a [Raspberry Pi](https://en.wikipedia.org/wiki/Raspberry_Pi
), with the first not being able to run Doom and the second being able to do run it. Given that this came out in 2008, I'd say it is closer to an Arduino.
But there is actually [a short story showing what it looks like from inside](http://www.rainydaymagazine.com/RDM2006/UM2006/CubeWorld/RDMUM_CubeWorldInside.htm
). That doesn't go into details of the used chip, but we can see there that a [button cell](https://en.wikipedia.org/wiki/Button_cell
) is used to power that thing, something that isn't exactly used to power "powerful" stuff. The bigger ones hold up to 620mAh, which is nice, but have a maximum discharge current of ~~0.2mA~~ 2mA...which is...well, not a lot at all. From what I gather, that is hardly enough to run a very small variant of the Arduino. Which means that whatever is in there is actually not a general purpose chip, but the logic directly baked into circuits.
Which makes it rather unsuitable for running anything besides the logic it was designed for, unfortunately.
**Edit:** As /u/neotek pointed out in [their comment](https://www.reddit.com/r/theydidthemath/comments/6zxjy5/request_i_found_this_on_tumblr_in_theory_how_many/dmzeo6c/
), this is most likely not something baked as I assumed, but a simple, low tier chip. So, theoretically, you could string them together to get a network of those. But the limiting factor will be latency between them, it ain't going to be pretty.
Also /u/drummyfish is completely right in [their comment](https://www.reddit.com/r/theydidthemath/comments/6zxjy5/request_i_found_this_on_tumblr_in_theory_how_many/dmzjbcx/
), even if it does only a few things, you could still try to emulate logic gates with it and start building up from there...it ain't going to be pretty either and you will need a lot of those, and I'm talking *hundreds of thousands* at least here.
I couldn't find the actual specifications of a Cube World Cube. However, I think that Tamagochis will have fairly similar processing power to the Cube. But... I couldn't find the exact specifications of a Tamagochi either. What I did find however, is that they use a 4-bit microprocessor. A quick Google search led me to a PDF outlining the specifications of an EM6607 4-bit microprocessor, and so I am going to assume that the microprocessor has the same specifications and processing power as a cube. Let's do the math:
I'll start with the processor. Doom requires a single core processor running at 66Mhz to run. The Cube's processor runs at 32,768Hz. 66,000/32,768=2.01416016, but we'll round it up to 3, so we need 3 cubes minimum to have a fast enough processor to run Doom.
Now let's do the storage. Doom requires 40MB of ROM to install it. Were just going to assume that we don't need to install an operating system or a launcher, but that doom will launch when the cubes are turned on. Each cube has 32KB (2Kx16) of ROM. This means we need 40,000/32=1250 cubes minimum just to store the game. Let's move on to RAM.
The game requires 8MB of RAM, and each cube has 384 Bytes of RAM. 8,000,000/384=20833.3, but we'll round it to 20834 cubes. This is the new minimum amount of cubes so we have enough RAM for the game.
So assuming the cubes use the microprocessor I found online, we need a minimum of 20,834 cubes to run Doom. Whilst I was looking it up, I saw the average price for one cube was about £20 (~$26). To buy that many cubes, it would cost 20,834x20=£41,668 or $55,043. Hope this answers you question well OP.
Edit: There are 66MHz is 66,000,000Hz, not 66,000Hz. It's not 3 cubes to have enough processing power, it's 2015. Thanks u/ray_dog for pointing it out for me.
Edit 2: I forgot to multiply by 20, and just did it by 2 when working out the cost, so it would actually cost £416,680/$550,430.
They used to sell those in a store in Asheville in 2006. Well, they sold some. They were supposed to be this big thing, and I liked them, but the price point was bullshit. I see a lot of approximations of processing power on this thread. I'm here to tell you what this thing does and someone else can figure out the power.
So, it's a 24*24 screen with one of several semi unique stick characters per box. Each character randomly generates one of roughly 16 animations at random intervals for each side of the screen. If you flip them, the figure falls and rebalances, etcetera.
They link by the little magnets you see on the side. They also have them on top and on the bottom. When they are linked, the number of animations increases in magnitude. On top of that, it adds a semi unique set of animations based on which characters are linked, where. You can link up to four before they stop linking.
It couldn't be done with an unmodified version of Doom, according to [these requirements](http://gamesystemrequirements.com/game/doom
EDIT: **More processors doesn't mean more processing power. Especially not with just 3 pins to go with.**
For the speculations (since we don't know the exact structure of the cube) and math:
According to [this Wiki page](https://en.wikipedia.org/wiki/Radica_Games
), cubes were produced around 2005-2008, and even though they wouldn't be using most recent microcontrollers, let's assume they were running on 2005 parts.
) suggests that [these micro controllers](http://pdf1.alldatasheet.com/datasheet-pdf/view/142908/SILABS/C8051F330.html
) were amongs the most popular ones and they were produced around 2005. Considering relative simplicity of the cube, it is very probable that a chip similar to this one was it's processor. 8kB of storage, impressive.
It can theorethically do as much as 25MHz, but that's only the internal clock, also we would be limited by speed of data transfer in between the cubes. Okay, one won't do, memory- and speedwise how about connecting many of them?
If we assume, that all three of these metallic surfaces are pins connected directly to the MCU, we could assume, one would be able to use them to form a I^2 C connection. In its 2005 version, [our protocol](https://en.wikipedia.org/wiki/I%C2%B2C
) could go as high as 3,4 MHz (we have to keep in mind that this has to cover up data transfer for both sides, meaning that the MCUs would have to switch between going into slave and master mode, because with only three pins you can only send data one way (1 pin for ground, 1 for clock and 1 for data)).
This leaves us with less than 1,7MHz of clockspeed, with the biggest bottleneck being connection between cubes. Sooo... Unless you go real hard on engineering, you won't be able to launch Doom on them. Even if you did manage to get past this speed barrier, by making each and every cube do some other bit of computing, you'd still need around 1000 of them just to satisfy the need for RAM, not speaking of uncompressed game files (with the game files taking up around 140MB, you'd need additional 1750000 cubes).
EDIT: I also forgot to mention that producers could fit there as many memory chips as they wanted and the mcu could also be very different to one used as an example, so the only thing we know is that running Doom is nearly impossible and number of cubes to fit it in is ridicoulus.