The nodes do not modify the birational invariants of a surface. So if we blow-up the $k$ nodes of $X$ we obtain a smooth K3 surface $S$, containing $k$ $(-2)$-curves, whose topological Euler number is $24$. Coming back to $X$, we substitute each $(-2)$-curve (which is topologically a sphere, so has Euler number $2$) with a point. So the Euler number of $X$ is $24-k$.

Euler characteristic of nodal K3 surfaces (as in singular)
I'm having a very puzzling problem with my PC. Recently I have not been able to boot very consistently. The boot will hang during the Windows 7 splash screen and will not go further. The same thing happens when trying to run Startup Repair. At this point in time, I cannot boot, period. 

I've tried booting in safe mode. Safe mode boot hangs after loading disk.sys and will not go further. I've tried using LKGC, which also had no affect.

Normally in this situation, I would do some hardware testing (memtest, chkdsk, windows recovery), but for some reason I cannot boot to any disks whatsoever. The DVD drive I'm trying to boot with is only a few weeks old (my old one died recently), and I've used these disks to boot with before, so I know they are good. 

At this point, I'm a bit stymied as to what I should do next. I'm downloading Ubuntu now to try and backup some stuff, but again, I doubt the boot will be successful. If anyone has any advice on what to try now, I would really appreciate the help.
