Origin of "the wrong end of the stick"
Solution 1:
According to The Phrase Finder, the two share the same origin, not really diverging in meaning until 1850 or so.
Solution 2:
Before toilet paper and Sears catalogs, there was a wooden spatula called the stick. If you were in the outhouse after dark and you had to find the stick in the dark, you had a good chance of finding the wrong, dirty, shitty end of the stick. Not everyone could afford candles or lanterns, and sometimes the wind would blow them out anyway.
Solution 3:
When harnessing a matched team, one uses a stick called an "evener" attached by a pin at its center to the wagon or farm implement. Each end of the evener is attached to the harness of each draft animal. When pulling with 2 animals of different size (am unmatched team) one moves the location of the center hole (fulcrum) towards the side near the larger animal. Each animal is then pulling an amount of weight corresponding to its size, thus the term "evener". The larger animal is pulling more of the load because he has the "short end of the stick".
Solution 4:
The wrong end of the stick is usually explained as having come from Roman culture. Toilet paper had not been invented in Roman times so, they usually used a sponge on a stick, like this http://2.bp.blogspot.com/-AKCoVbCisFw/UCnEu4eqv1I/AAAAAAAAAew/OzR_GuiOqDM/s1600/spongestick.jpg The end with the sponge, was used to clean themselves. If someone was not paying attention when it came time to use a stick, they could pick it up by the wrong end. There is an explanation here http://www.cracked.com/article_16108_the-bizarre-history-10-common-sayings_p2.html
That link also says:
There is, though, another origin that's widely held to be the true one. The origin pertains to walking sticks and accidentally grabbing the dirty, non-handled end, the "wrong end."
Solution 5:
Nay, the stick with the shitty end was not used to clean oneself; it was used to knock over the pile of shit lest it become too high and reach the hole. Also, you would want the pile to be relatively flat so that you could dust it with ashes, and thus limit the smell. (Have you people never used an outhouse?)