Which is better for gaming?

The i7-7700K wins on raw clock speed and cache, and it edges out the Ryzen 3 4100 in most older games. The Ryzen 3 4100 is close in modern titles and leaves you on a platform you can upgrade later.

Can I put the i7-7700K in a modern motherboard?

No. It needs an LGA1151 board with a 100- or 200-series chipset. It will not fit 300-series boards or any current Intel socket.

The Ryzen 3 4100 at 65 W runs cooler and is easier to cool quietly. The i7-7700K at 91 W, especially overclocked, needs a larger air or liquid cooler.

Do I need a graphics card with either one?

The Ryzen 3 4100 has no integrated graphics, so you must add a discrete GPU. The i7-7700K has Intel HD Graphics 630 and can drive a display on its own.

Is the i7-7700K worth it over the Ryzen 3 4100?

Only if you already own a compatible LGA1151 board and find it cheap. For a new build, the Ryzen 3 4100's current platform, warranty and lower power make it the better buy.

Which has more upgrade room?

The Ryzen 3 4100. AM4 supports newer Ryzen chips and DDR4-3200, while LGA1151 100/200-series boards top out at 7th-gen Intel.

Intel Core i7-7700K vs AMD Ryzen 3 4100

For a new build, buy the Ryzen 3 4100. It is a current AM4 part with 8 threads, DDR4-3200 support and a 65 W TDP, and it sells new with a warranty. The i7-7700K is a 2017 LGA1151 chip that only fits 100/200-series boards, so it makes sense only as a cheap drop-in upgrade for a PC that already has that socket. No prices were given for either part, so compare live store prices before deciding.

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