Intels four cores usually beat out AMDs eight, they are just that much faster and do more work per clock cycle (newer vs older technology so to speak, AMD are way behind intel these days and need something new and fast).
Example -
http://www.anandtech.com/bench/product/1289?vs=1261
That is AMDs top FX CPU vs Intels last gen Devils Canyon top i5.
These two are not a world apart in terms of pricing either,
My basket at Overclockers UK:
Total: £377.68
(includes shipping: £8.70)


Both these use DDR3
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Then we now have Skylake (DDR4 for the most part, there are some DDR3 boards as it has both types of memory controller) which is a bit quicker again,
http://www.anandtech.com/bench/product/1289?vs=1544
My basket at Overclockers UK:
Total: £388.68
(includes shipping: £8.70)


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These are OEM versions of the CPUs as you mention you have a NZXT cooler already (if all the mounting parts are still with it then it will fit skylake as socket 1151 is the same mounting holes as 1155/56/50).
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If you do want an AMD CPU and a suitable board that has m.2 and SLI support at 16x/16X,
My basket at Overclockers UK:
Total: £135.49
(includes shipping: £10.50)

http://www.gigabyte.com/products/product-page.aspx?pid=5655#ov
2 x PCI Express x16 slots, running at x16 (PCIEX16_1/PCIEX16_2)
* For optimum performance, if only one PCI Express graphics card is to be installed, be sure to install it in the PCIEX16_1 slot; if you are installing two PCI Express graphics cards, it is recommended that you install them in the PCIEX16_1 and PCIEX16_2 slots.
1 x PCI Express x16 slot, running at x4 (PCIEX4)
* The PCIEX4 slot shares bandwidth with the PCIEX1_3 slot. When PCIEX1_3 is populated with a PCI Express expansion card, the PCIEX4 slot will operate at up to x2 mode.
3 x PCI Express x1 slots
(All of the PCI Express slots conform to PCI Express 2.0 standard.)
This is a recently released motherboard (hard to imagine that new AM3+ boards are still being designed to be honest).
There is also a board that doesnt do SLI,
My basket at Overclockers UK:
Total: £90.49
(includes shipping: £10.50)

http://www.gigabyte.com/products/product-page.aspx?pid=5654#sp
1 x PCI Express x16 slot, running at x16 (PCIEX16)
* For optimum performance, if only one PCI Express graphics card is to be installed, be sure to install it in the PCIEX16 slot.
1 x PCI Express x16 slot, running at x4 (PCIEX4)
* The PCIEX4 slot shares bandwidth with the M2F_10G connector. The PCIEX4 slot will operate at up to x2 mode when a PCIe SSD is installed in the M2F_10G connector.
2 x PCI Express x1 slots
(All of the PCI Express slots conform to PCI Express 2.0 standard.)
2 x PCI slots
According to the specs both these boards m.2 run at a maximum of 20GB/S rather than 32GB/S of the new intel Z170 boards.
So still fast but not as fast as an intel board.
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As the CPU is what has the memory controller (for both intel and AMD) then you will not get a DDR4 version until AMD releases something new to try and compete with intel (and that will mean a new socket type+chipset and so a new bunch of boards that are not backwards compatible with current CPUs).
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If you want to make the jump to DDR4 and intel then there is a board+RAM bundle available for a short period of time (until stocks last).
https://www.overclockers.co.uk/giga...therboard-with-free-8gb-memory-mb-546-gi.html
In fact it does say not available anymore so I guess the offer has expired.
This board does not support SLI, review here -
http://hexus.net/tech/reviews/mainboard/90929-gigabyte-z170-gaming-k3/
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With a board such as that you would buy a Skylake i5 (again a four core CPU) that I am willing to bet even the slowest version intel offers is a better choice than the fastest AMD FX CPU,
My basket at Overclockers UK:
Total: £344.68
(includes shipping: £8.70)


£25 less but I reckon its as fast, runs cooler, uses less power and as it's the latest and greatest you have boards with DDR4 and m.2 32GB/S.
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There is a CPU+board skylake bundle,
My basket at Overclockers UK:
Total: £319.48
(includes shipping: £10.50)

That saves some money when bought together VS separate.
This board does support SLI at 8X/8X which is only a few % slower than 16X/16X.
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Skylake "K" CPUs (6600K and 6700K) do not include a heatsink as intel aim it at enthusiasts that should have/buy something superior to the intel stock heatsink as these two CPUs are aimed at overclocking as the K designates an unlocked multiplier.
All the rest of the Skylake family do include a stock intel heatsink, but again you do mention you have a NZXT cooler.
Once overclocked imagine what a difference there is between a Skylake CPU vs AMD FX considering the FX is mostly behind even at stock speeds.
I have done a review of the board included in that bundle above,
https://forums.overclockers.co.uk/showthread.php?t=18690667
Including how to overclock the CPU to a decent speed.
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If you dont want all the bells and whistles (SLI support, proper m.2 speed etc and dont mind owning Devils canyon and DDR3),
My basket at Overclockers UK:
Total: £379.49
(includes shipping: £10.50)

As that is an i7 CPU (4cores with Hyperthreading so 8 effective cores) it will lay waste to an AMD FX totally.