Spatial

The standard method is a mental rotation task: two solid figures are shown, and you decide whether one is the other rotated or its mirror image. Accuracy and response time are both recorded, because speed alone rewards guessing and accuracy alone cannot separate someone who saw it instantly from someone who worked it out slowly.

The mental rotation test asks you to turn a solid figure in your head until it matches another and judge whether they are the same object. It comes from Shepard and Metzler's 1971 experiment in Science, is the most used paradigm in spatial ability research, and is the original of the solid-figure items on aptitude exams.

Yes, and this is the best-evidenced kind of cognitive training there is. The Uttal et al. meta-analysis of 206 training studies (Psychological Bulletin, 2013) found an average effect of g = 0.47, with gains transferring to untrained spatial tasks and surviving a delay before retesting. Memory and attention training have not shown transfer like that.

Pick one distinctive feature first -- the longest arm, the most unusual bend -- use it to fix the orientation, then check whether the rest lines up. That beats rotating the whole figure. Watch for mirror images: a mirrored figure can never be brought into alignment by rotation, and spotting that rules it out immediately.

Every solvable 8-puzzle -- the three-by-three sliding puzzle -- can be finished in at most 31 moves, which is the upper bound for that size. How far your move count sits above 31 says more about whether you were planning a route than your completion time does.

No. Only half of all possible tile arrangements on a three-by-three board are solvable; the other half cannot be restored by any sequence of legal moves, and the test is the parity of the permutation. Properly built sliding puzzles therefore scramble from a solved state rather than placing tiles at random.

Related but not the same. Spatial ability as tested here is figure manipulation; finding your way also requires remembering landmarks and integrating a route into an internal map. Someone strong at mental rotation can still get lost, and the reverse happens too.

On mental rotation specifically, a group-average difference is one of the more consistently reported findings in cognitive testing. Individual variation within either group is far larger than the gap between the group means, and training narrows the gap -- the Uttal et al. meta-analysis (2013) found both groups gain from it. Using a group average to predict any particular person's score tells you nothing useful.