This thread has bothered me ever since I started reading it last month, and tonight, I finally figured out why. You can come up with a pathological example for any resizing ratio. It just depends on the arrangement of pixels in the image. Kiwi's example has evenly spaced bars, so naturally, they'll divide by two perfectly, and they won't divide evenly by three and they'll get smudged during the resize.
I created a different example below, using exactly the same dimensions as Kiwi's image. This one has 9-pixel black bars separated by 3-pixel white bars. If you're good at math, you'll realize that 9 and 3 are both divisible by 3, but they're not divisible by 2. Try to resize this to half size, and you'll see the bars smudge, but resize to a third (33.3%, or 321x220), and it'll look fine.
Here's the message: neither of these examples means anything if you're resizing a picture of a bird! (or anything else, for that matter). When you resize any real-life image, you lose information. There's no way around it, no matter if you do a half, a third, two thirds or whatever. The good news is that most image editors have a built-in error diffusion process that distributes the errors in a way that makes them far less noticeable. On a real-life image, you'll see pretty much the same amount of relative error, regardless of your ratio.
Feel free to resize your images as you see fit, and leave the math to the computer. That's what it's good at. 
