7.3 Scenario Practice for Block Counting

7.3 Scenario Practice for Block Counting

Most figures are variations on a few position archetypes. If you memorize the neighbor count for each archetype inside a solid rectangular stack, you can answer many items instantly and save the slow coordinate sweep for irregular piles.

The four position archetypes (solid stack)

The pattern: neighbors = 6 minus the number of faces exposed to open air. Every face that points at empty space (or the floor) removes one neighbor.

Scenario 1 — Top corner of a solid stack

A block sits at the top-front-left corner of a 3x3x3 cube. Three of its faces point outward (left, front, top) and three point inward (right, back, bottom). Inward faces all have blocks -> 3 touching. This is the minimum possible answer for any block in a solid pile.

Scenario 2 — Edge block, not on the floor

A block on the top edge (but not a corner) of a solid stack exposes its top and one side. Two faces open -> 6 minus 2 = 4 touching. Quickly confirm: left/right or front/back along the edge are filled, plus the layer below.

Scenario 3 — The floor trap

Any block in the bottom layer has its downward face against the floor, not against a block. So a bottom-layer block that looks like a face-center block (one side exposed) actually has two open faces — the exposed side and the floor — giving 6 minus 2 = 4, not 5. Always check the bottom face explicitly; it is the most common miss.

Scenario 4 — Irregular (staircase) stacks

When blocks are stacked unevenly, the archetype shortcut fails and you must run the coordinate sweep. Trace each column's height from the visible front faces, then ask for the target block: is there a column beside me of equal or greater height (side neighbor)? Is there a block stacked directly on me (top neighbor)? A block at the top of a tall column next to short columns can have zero side neighbors above the short columns' height — count only what is physically there.

Predict-then-count habit

Before you tally, say the archetype: "this looks like an edge block, so I expect 4." Then run the sweep. If the sweep disagrees with your prediction, you caught either a hidden block or a floor/edge trap — investigate before answering. This two-pass habit costs one second and prevents the most expensive errors.

Scenario 5 — Hidden block confirmation

Consider a solid 2x2x2 cube of 8 blocks with the front-bottom-left block numbered. It is a corner -> predict 3. Sweep: right (filled), back (filled, hidden), top (filled). Left, front, and bottom all point to open air or floor. Three neighbors confirmed, two of them invisible in the drawing. This is the most reassuring scenario to drill, because it trains you to trust the coordinate prediction even when your eyes see nothing there. If you only counted visible blocks you might answer 1 or 2 and lose the point.

Scenario 6 — Block resting on a partial base

Irregular stacks often show a block perched on top of a shorter column with a taller column beside it. Walk the columns:

• Column to the left is two blocks tall; our block is on layer 2 -> a left neighbor exists.
• Column to the right is only one block tall -> no right neighbor at layer 2.
• Front is empty; back column is two tall -> a back neighbor exists.
• Below: a block in the same column -> bottom neighbor exists.
• Top: nothing stacked on it -> no top neighbor.

Total = 3 (left, back, bottom). The lesson: in irregular figures, side neighbors exist only up to the height of the adjacent column. A block can sit higher than its neighbors and lose side contacts it would have in a solid stack.

Quick-reference answer ranges

Use these ranges as a reasonableness filter: if your count lands outside the plausible range for the block's position, recount before bubbling. A buried-looking block that you tally as 2, or a lone corner you tally as 6, is almost always a counting slip, not a strange figure.

Drilling archetype recognition

The fastest way to internalize these archetypes is to label, not count. Open any practice figure and, before doing arithmetic, name the numbered block's position out loud: "corner," "top edge," "front face," "interior." Then state the expected count, then verify with one quick sweep. After a few dozen reps the label and the number fuse, and most solid-stack items become near-instant. Reserve full counting for the irregular figures that genuinely require it. This habit is what separates a candidate who finishes all 30 from one who stalls at question 20.