Rebuild from front and side views

Reconstruct column heights from occupied footprint cells and two maximum-height profiles.

Adult planning suggestion: about 6–9 · Shape & spatial thinking · Logic & rule checking · Cooperation & communication

Players

Pairs

Time

12–18 minutes

Materials and quantities

A 3×3 footprint mat, front/right profile cards and eight large equal blocks or a height-record grid.

Thirty pupils: fifteen pairs; builder and profile checker swap with the labels left visible.

Prepare

Columns A–C run left to right; depth rows 1–3 run front to back. The occupied footprint cells are A1, B1, B2, C2, A3 and C3; other cells have height zero. Use eight equal blocks in total. Front profile heights at A/B/C are 2,2,1. The right-side profile is explicitly listed front-to-back for rows 1/2/3 as 2,1,2. These labels define the reading order; do not infer a mirrored order from a photograph.

Play

Assign a positive whole-number height to every occupied cell. The front profile is the maximum height along each column through all depth rows. The right-side profile is the maximum across each depth row. A profile does not tell you that every stack along that line has that height. Build with large blocks or write heights on the mat; adding several blocks to one cell is a stack, not a new footprint cell.

Worked example

One checked arrangement has front row heights 1,2,0; middle row 0,1,1; back row 2,0,1. It uses eight blocks. Column maxima are 2,2,1 and depth-row maxima 2,1,2, matching both cards. A flat layer of six blocks fits the footprint but fails the two profiles and total. The teacher drawing separates footprint occupancy from height, so a printed dark square is not itself a promise of one cube only.

Finish and check

Finish after counting eight blocks and auditing all three front and all three side profile values. Accept any arrangement satisfying every given condition rather than insisting it look like the answer drawing. If a model fails, name a specific maximum or total that differs before changing a stack. Stop after fifteen minutes and preserve a partial diagram for discussion. The task does not require memorizing a hidden model.

Adaptation and practical limits

For thirty pupils use fifteen pairs, builder and projection checker swapping halfway. Adult planning suggestion is around 6–9 with adult support for depth labels. The geometry article provides projection background; our coordinates, constraints and eight-block example are separate. A written height grid is the complete no-purchase alternative when suitable large blocks are unavailable. Do not use unstable tall stacks or small pieces with children who mouth objects.

Free alternatives: Use paper or clean reused cardboard and a pencil. Copy the supplied diagrams and tables by hand; no specialist kit or purchase is needed.

Printable materials

Preview at 100% scale, with browser headers and footers off. Cut with suitable help. Keep host keys separate. This is a manual tabletop kit, not an online opponent, automatic judge or solver.