3D Reasoning: Cube Stacks, Hidden Faces & Tallest Stack

A drawing of cubes always hides some of them. The test expects you to count the hidden ones too, because a cube cannot float: if you can see a cube on the second layer, there must be one underneath it. The reliable method is to count by columns. Look at the top face of each column, work out how tall that column is, and add the heights. For a staircase two cubes deep, the front row is three plus two plus one, six cubes, and the back row has the same shape, so twelve in all — even though you can only see about nine of them. Counting the little squares you can see is the classic wrong answer, and it will be among the options.

Every cube has six faces. When two cubes are glued face to face, one face of each disappears — a join hides exactly two faces. So for any object made of glued cubes, the number of faces you can see is six times the number of cubes, minus two for every join, and the hidden faces are simply two per join. Three cubes in a row have two joins, so four faces are hidden and fourteen show. The test says the object can be picked up and looked at from any direction, which means the bottom faces count as visible. The whole skill is counting joins correctly: a cube that touches two neighbours contributes to two joins.

When a solid is broken into pieces, no face is created or destroyed; the only thing that changes is which faces are hidden. Every join you break uncovers two faces. A two-by-two-by-two cube has eight small cubes and twelve joins, so twenty-four faces hidden and twenty-four visible. Break it into two L-shaped pieces of four cubes with three joins each, and only six joins remain: forty-eight minus twelve is thirty-six visible. You can also add two per broken join to the old count — six broken joins, twelve more faces, twenty-four plus twelve. Both routes must agree; if they don't, recount the joins.

Another item shows two objects made of cubes, breaks them into loose cubes, and asks how many copies of a third object can be built. It is a division problem hiding behind a counting problem. Count the cubes in each object, hidden ones included, add them, then divide by the number of cubes in the target. Twelve loose cubes make three four-cube objects; fourteen loose cubes still make only three, because two spare cubes are not enough for a fourth. Once the cubes are loose the target's shape is irrelevant — only how many cubes it needs.

The tallest-stack question gives three blocks with their measurements and asks how tall a stack of all three can be. Each block can be turned any way, so its contribution is whichever edge points up — and for the tallest stack that is its longest edge. Pick the largest number on each block and add the three. The trap is adding the heights as the blocks happen to be drawn, which the picture invites you to do. The shortest possible stack is the same idea with the smallest edges. Unless the question mentions stability, a thin block standing on its end is fine. Quiz and exercises now!