Free guide · Spatial reasoning · 4 min read

How to solve block counting puzzles

Counting cubes you cannot see

Count column by column — every stack contributes its full height, including the hidden cubes holding up the visible ones.

Updated 30 September 2026

A stack of cubes, and a question about it — and the question varies: the total count, how many cubes are completely hidden, how many touch the ground, how many faces would be painted, how many cubes complete the box, or how many more one pile holds than another. All of them run on the same engine: read off each column's height, because a cube cannot float — anything visible at height 3 sits on two supporters.

Column accounting

Mentally slice the structure into vertical columns on its floor grid. For each column, read its height from the tallest visible evidence, then add height cubes for that column — all of them, seen or not. The total is the sum of column heights.

Hidden ≠ optional

A cube visible on top of a 3-stack proves cubes at levels 1 and 2 of that column. Structured items guarantee supporters; count them with confidence.

One grid, six questions

Once the height grid is in your head, each question is short arithmetic on it. Total: sum the heights. Hidden: count cubes with no visible face. Ground: count occupied columns — every column contributes exactly one floor cube. Painted faces: now faces are exactly what you count — walk the outside surfaces, remembering the base is NOT painted. Complete the box: the box is width × depth × the tallest height; subtract the cubes present. Compare two piles: total each pile separately, then subtract — never eyeball which 'looks bigger'.

The strategy

Read the question first: total, hidden, ground, painted faces, completing the box, or a comparison.
Establish the floor grid (how many columns exist), then read each column's height from its topmost visible cube.
On the TOTAL question, sum heights — never count faces; face-counting is only ever the method on the painted-faces question.
Sanity-check against the options: on totals the visible-only miscount is always offered, and your sum should exceed it.

Common traps

Counting visible cubes only on a total question (the classic undercount — it's always among the options).
Answering the total when the question asked for hidden cubes, ground cubes, or the difference between two piles.
Painting the base: the faces resting on the table are not painted.
Double-counting corner cubes seen on two faces.

Speed · ~45s/item: Column accounting is linear and calm: 30–60 seconds even for 4×4 structures.

Worked example

Made by the same generator and answer checker as the practice questions, at middling difficulty, but never served in a test. Commit to an option before you reveal — the pick is yours alone and is never recorded.

How many cubes are in this stack, counting the ones you cannot see?

Reveal the answer

The answer is D — 9.

Show why — the worked reasoning

Count column by column, not cube by cube: every stack you can locate contributes its FULL height, including cubes hidden behind or below the ones you see — a cube can't float, so anything supporting a visible cube must exist. The stacks here are 2 + 1 + 2 + 1 + 1 + 2 cubes tall, giving 9 in total. The classic mistake is counting only visible faces, which loses the hidden supporters.

Try it yourself: 4 practice questions

Pick an answer, then reveal it. Nothing is recorded, there is no clock, and none of these questions is ever used in a test.

Question 1 · Easy

How many cubes are touching the ground?

Reveal the answer

The answer is D — 8.

Show why — the worked reasoning

Each stack rests on the ground on exactly ONE square, however tall it is — so this asks for the size of the footprint, not the volume. Count the occupied squares of the base: 8. The stack holds 21 cubes altogether and its tallest column is 3 high, but height changes nothing here: a tower of four still touches the ground with a single cube. Squares of the base rectangle that carry no stack at all do not count either.

Question 2 · Easy

How many cubes are touching the ground?

Reveal the answer

The answer is D — 8.

Show why — the worked reasoning

Question 3 · Easy

How many cubes are in this stack, counting the ones you cannot see?

Reveal the answer

The answer is E — 15.

Show why — the worked reasoning

Count column by column, not cube by cube: every stack you can locate contributes its FULL height, including cubes hidden behind or below the ones you see — a cube can't float, so anything supporting a visible cube must exist. The stacks here are 2 + 2 + 2 + 2 + 1 + 2 + 2 + 2 cubes tall, giving 15 in total. The classic mistake is counting only visible faces, which loses the hidden supporters.

Question 4 · Easy

How many more cubes must be added to fill this out into a solid rectangular box, without making it any wider, deeper or taller?

Reveal the answer

The answer is A — 2.

Show why — the worked reasoning

The box must stay exactly as wide, deep and tall as the shape already is: 2 × 2 × 3, because 3 is the tallest column. A solid box that size holds 2 × 2 × 3 = 12 cubes, and 10 are already in place, so you must add 12 − 10 = 2. The same answer column by column — top each stack up to 3: 0 + 2 + 0 + 0 = 2. Answering the size of the finished box instead of the shortfall is the usual slip.

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Will block counting questions be on the free test?

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