Q:

Biggest amount of cores on the smallest board

I'm looking for parallel computing in tiniest form possible. For example, having two 18 cores Intel Xeon parked on both sides of motherboard with size of credit card would be ideal solution.
Couldn't found any motherboards (compatible with CPUs with 10+ cores) without USBs, Wifi, PCIe and other ports, that makes them only bigger.
More specifically I need best Cores per Square Meter ratio possible: CPU and motherboard models.

1.5-2GHz is ok. More = better, but not critical
yes, it is to build high-dense CPU farm
no, it is not for mining, it is for math, non-GPU calculations
yes, amount of cores (multithreading) is critical
yes, huge 4x"Intel Broadwell-EP CPU"s server motherboards are ok until they bring best cores/square ratio

Updated (12.07):
Considering answers below which I have at the moment:

No AMD at all, Xeon processors it is, E5 or E7 is to be evaluated by cost/density/power factor, will update with calculations later.
I'm also in blade servers with extreme density WITH support of Xeon Phi - some of applications will run on it nicely with reuse of code and data, have spent a day looking for specifications and cost, will update as soon as will have nice configuration on hands
regarding CUDA and Nvidia Tesla, that is a separate question and it is solved already, will share specifications with you later, thank you, SEJPM!

What is already decided that configuration will have BOTH Xeon E5/E7 and Xeon Phi on same boards (not on all of them though). Cray supercomputers already use it
Updated (12.07)[2]:
To be clear: I've a lot of small binaries (cross-platform, C++ and Java) each of them work for seconds with parallel computing full-support. When it comes to situation where there are significantly less cores than amount of threads/processes started, overall efficiency drops due to a lot of context switching between processes. And there is no way to queue jobs, let's say they are to be done in real-time manner.
Best way is to distribute them among Xeon Phi (store code and preseed with data) + run others as a tiny-services (solving issue with binary loading overhead) waiting for jobs.
That is why I search for comparatively cheap solution with high cores density to not to maintain big amount of units, solving more problems with data-logistic.

A:

If you are willing to do a bit more work on integration with whatever you want to run, you can try a Parallella board. That gives you a 16 core RISC co processor plus a dual core main processor on a board the size of a credit card. 
They use very little energy and are specifically designed to be used in clusters or in parallel (hence the name) applications. 
As a bonus, they start at about $100 each and they run Linux.
https://www.parallella.org/

A:

I am going to assume you need x86 compatible cores/threads, and I am going to assume you don't want to use something like the Xeon Phi compute card, for some reason or another. I am also going to assume, because you didn't specify, that power requirements and heat dispersion considerations are not something you're worried about, since it wasn't mentioned.
That being said, there are some absolute monsters out there you can fit into pretty small builds. 
On the AMD side, you're stuck back in 2012 unless you get fancy with ARM, but you can still get very powerful 64 real-core systems built on quad-socket G34 server boards that are somehow crammed into 1U chassis by Supermicro and other companies.
On the Intel side, where I suspect you'll end up, you can get the formidable    Intel Xeon E5-2699 V4, with 22 cores and 44 threads PER PROCESSOR, onto large multi-CPU boards, or onto thin ITX boards like this one: http://www.anandtech.com/show/9221/asrock-rack-announces-epc612d4i, or this one http://www.asrockrack.com/general/productdetail.asp?Model=EP2C612D8HM#Specifications - whichever way nets you the most density. I'm not recommending AsRock as a brand per se either, I'm just showing you what's possible using their site.
If none of that will suffice, you're really gonna have to look into stuff like Xeon Phi, IBM Power8, or ARM architectures. Although more exotic and therefore more difficult to work with, these types of CPU feature very high compute densities, with different approaches each. More information/research would be necessary before anything in this area could be recommended.

