Best Laptop for Under 600$?

Okay so my cousin is asking for help figuring out a good laptop for her to buy for under 600$.

She's going to using it for school work. But also to play some games. For example, Sims 3 and even possibly the upcoming Sims 4. Sooooo it needs some decent integrated graphics to play it but not on super high settings.

I would love everybody's opinion and recommendations. 

Also, are HP's laptops still known to overheat? 

the old HP that i have runs hot and it ran hot from day one. otherwise it was good. i'm not a fan of their printers and bloatware though.

$600 isnt much to play with for a new unit so would they be happy with 2nd hand? If so you could get quite a decent unit - sandybridge, gpu, etc etc. Or maybe even the apu route - amd a10 etc.

I think the best option is this only $30 over the price:

http://shop.lenovo.com/us/en/laptops/lenovo/z-series/z510/?sb=:000001C9:00011CC9:

or maybe this:

http://shop.lenovo.com/us/en/laptops/lenovo/z-series/z40/?sb=:000001C9:00011E09: