Layouts and views: separate cells from volume
Review orthographic cells, full-stack counts, silhouette heights and observer-relative directions with complete worked examples and clear score limits.
This public guide explains one part of the original word-and-coordinate practice. Every complete attempt shows all five topic counts and all five full guides. It is not a private attempt, ranked outcome, percentile or general spatial-ability classification.
Name the view and the requested quantity
A three-dimensional cube arrangement can produce several different two-dimensional views. A front projection along the stated y direction keeps x and height z while dropping depth y; a top footprint keeps x and y while dropping height. Cubes at different depths may occupy the same projected cell and must then be counted once in that view. A question about total material is different from one about occupied footprint cells or a silhouette. Identify that requested quantity before adding stack heights or counting visible positions.
A fresh stack has three different counts
Consider two rows of three full stacks, with heights [2,1,0] and [1,4,3]. They contain 2+1+0+1+4+3=11 unit cubes and occupy five nonempty footprint cells. Viewed from the front along the row direction, the maximum heights at the three x columns are (2,4,3), producing nine occupied front-projection cells. Eleven, five and nine are all correct for their separate quantities. Choosing one because it appeared in an earlier item without naming the requested view confuses volume, footprint and projection.
Full columns justify hidden-cube counting
When a problem states every vertical column is filled from the floor without gaps, its height determines the number of cubes in that column. Hidden cubes still count toward material total. Without that condition or an explicit coordinate list, a view alone may not determine the interior arrangement: different solids can share a projection. Do not fill unseen spaces by habit. The bank states either the cube positions or the full-stack conditions needed for its particular question, so the requested count has a definite basis rather than an assumed picture.
Update heading and position separately on a route
For a fresh route, start at (2,-1) facing north. Move four units to (2,3), turn right and move two to (4,3), then turn right again and move one south to (4,2). Turning changes heading; movement changes position by the stated number of units. Write both after each step. Keeping the initial heading throughout would miss the second and third displacements, while treating a turn itself as an extra step would add movement the question never specified.
An observer’s left is not the map’s left
On a fixed map with east as positive x and north as positive y, a person facing south has east on their left. At (-2,3), one unit directly to that person’s left is therefore (-1,3). The map’s left side still points west, but the observer uses a body-relative frame. First establish facing direction, then rotate that heading left or right to find the relative direction, and only then update the fixed coordinate. Changing the observer’s orientation does not rename the map axes.
Read six item results as a bounded record
The layouts sub-count shows how many of these six written projection, stack and direction cases were answered correctly. It is not an architectural, navigation or workplace qualification, and it does not identify a person’s general spatial strength. A miss can involve the requested quantity, an occluded cell, a stated stack condition or a reference frame. The total alone cannot establish which process occurred. Read each explanation with the actual wording; all completed attempts receive the full answers and all five guides at every correct count.
Use fresh layouts and clearly stated frames
Draw a small new stack-height grid, explicitly state whether columns are solid, and calculate its material total, footprint and front projection as three separate questions. For a route example, change one turn while keeping the start and axis convention fixed. Sketches and calculators are allowed, and there is no speed adjustment. Retakes use the same original bank and may reward recognition. These text-and-coordinate counts have no norm, percentile or evidence of transfer to a different visual test, job task or real-world navigation setting.