7.2 Core Workflows and Decision Points

7.2 Core Workflows and Decision Points

Under a nine-second pace you cannot afford to invent a method per question. You need one workflow you run identically every time. The reliable approach is a coordinate sweep: locate the numbered block in 3D, then check each of its six face positions for a neighbor.

Step 1 — Orient the stack with coordinates

Treat the pile as a grid with three axes:

• Column = left-to-right position (the X direction)
• Row = front-to-back depth (the Y direction, going up-and-right in the drawing)
• Layer = vertical level (the Z direction, the stack height)

Give the numbered block a label such as (column 2, row 1, layer 1). A face neighbor is any block that matches two of those coordinates and differs by exactly one in the third. That single rule replaces all guessing.

Step 2 — Sweep the six directions in a fixed order

Always check directions in the same sequence so you never lose your place:

Step 3 — Tally as you go

Keep a running count on your scratch paper or by tapping fingers — one tally per confirmed neighbor. Do not hold the number in your head while you keep scanning; that is where errors creep in. When the six-direction sweep ends, the tally is your answer.

Worked example

Imagine a 3-wide, 2-deep, 2-high stack (12 blocks). The numbered block sits at (column 2, row 1, layer 1) — middle column, front row, bottom layer. Run the sweep:

• Left (column 1): block present -> +1
• Right (column 3): block present -> +1
• Front (row 0): no row in front -> 0
• Back (row 2): block present (hidden) -> +1
• Top (layer 2): block present -> +1
• Bottom (layer 0): floor, no block -> 0

Total = 4 touching blocks. Notice two of the four (back and the hidden depth neighbor) are not fully visible — you found them only because the coordinate rule predicted them.

Decision point: visible vs. inferred

The core judgment in every item is deciding whether an unseen position is filled. Use the surrounding rows: if the stack is a solid rectangular block, every interior position is filled, so corner and edge blocks lose neighbors only where the stack ends. If the stack is irregular, trace each column's height from the visible faces and assume nothing beyond what the drawing supports.

Reading the isometric drawing correctly

In the standard isometric view, three directions map predictably onto the page, and learning this mapping is half the battle:

• Left/right (columns) run roughly horizontal across the page.
• Front/back (rows/depth) run up-and-to-the-right at about a 30-degree angle.
• Top/bottom (layers) run straight up and down.

A frequent error is confusing a depth neighbor (further back, drawn up-and-right) with a vertical neighbor (directly above, drawn straight up). They look similar in 2D but are different faces. Anchor yourself by finding the floor line first: blocks sitting on the floor define layer 1, and everything drawn higher than its column's floor block is a higher layer or a deeper row. Resolve that ambiguity before you count, or your sweep will mislabel directions.

When to use the sweep vs. the shortcut

You have two tools. The full six-direction sweep is exhaustive and never wrong, but it costs seconds. The archetype shortcut (covered next section) is instant but only valid for solid, gap-free stacks. Decision rule:

Building the routine into muscle memory

The goal is that, by test day, the six-direction order runs automatically: you should be able to whisper "left-right-front-back-top-bottom" without thinking, tapping a finger per confirmed neighbor. Practice the sweep aloud at first; the verbalization forces you to actually check the floor and the back rather than skipping straight to the visible faces. Once it is automatic, drop the words and keep only the finger taps so the whole process fits inside nine seconds. The decision points — where is the block, which faces are filled, is this the floor — never change; only your speed at executing them improves with reps.