Number and shape patterns Transitional Reasoning Download PDF

Questions

1. 1.

   An urban greening project installs modular living wall panels along a public walkway. Each panel is designed as a two-row grid of square planting pockets constructed from interlocking recycled plastic slats. Stage 1 is a 2-row by 1-column grid of 2 planting pockets assembled using 7 slats, Stage 2 is a 2-row by 2-column grid of 4 planting pockets assembled using 12 slats, and Stage 3 is a 2-row by 3-column grid of 6 planting pockets assembled using 17 slats, continuing this linear pattern. If a landscaping crew constructs a continuous living wall panel using exactly 142 plastic slats, how many square planting pockets does this completed panel provide?

   • A) 28
   • B) 54
   • C) 70
   • D) 56
2. 2.

   A conference center arranges identical trapezoidal tables end-to-end in a straight line for corporate events. A single freestanding table seats 5 guests, while joining them along their slanted sides creates a row of 2 tables that seats 8 guests, and a row of 3 tables that seats 11 guests, continuing this constant linear pattern. The catering manager sets up two separate banquet rows: a VIP row constructed from $k$ tables and a General row constructed from $2k - 1$ tables. If the two rows combined seat exactly 118 guests, how many guests are seated at the VIP row?

   • A) 38
   • B) 41
   • C) 77
   • D) 13
3. 3.

   An architectural lighting company designs modular triangular chandelier frames where each stage adds decorative light nodes along the frame structure. Stage 1 has 7 light nodes, Stage 2 has 11 light nodes, Stage 3 has 15 light nodes, and each consecutive stage continues to add 4 light nodes following this constant linear pattern. An event hall installs three consecutively sized frames—Stage $k$, Stage $k + 1$, and Stage $k + 2$—which together contain a combined total of 105 light nodes. How many light nodes are installed on the largest of these three chandelier frames?

   • A) 27
   • B) 31
   • C) 35
   • D) 39
4. 4.

   An interactive science museum designs geometric acoustic wall panels where Stage 1 uses 6 hexagonal tiles, Stage 2 uses 11 tiles, and Stage 3 uses 16 tiles, continuing this linear pattern for higher stages. The curator plans to construct two separate displays for an upcoming exhibition, one built to Stage 7 specifications and another built to Stage 10 specifications. If the hexagonal tiles are sold only in full boxes of 12 tiles each, what is the minimum number of boxes the curator must purchase to supply all the tiles needed for both displays?

   • A) 8
   • B) 7
   • C) 9
   • D) 11
5. 5.

   A civil engineering lab tests two modular pedestrian bridge designs constructed from prefabricated steel struts. Design X uses 14 struts for a single-span bridge and requires 9 additional struts for each consecutive span added to the structure. Design Y uses 38 struts for a single-span bridge and requires 5 additional struts for each consecutive span added to the structure. If both bridges are constructed with the same number of spans and use the exact same total number of struts, how many struts does each bridge contain?

   • A) 68
   • B) 59
   • C) 73
   • D) 77
6. 6.

   An automated packaging facility arranges shipping cartons on modular display pallets where each consecutive tier follows a consistent pattern. Tier 1 holds 16 cartons, Tier 2 holds 20 cartons, Tier 3 holds 24 cartons, and each successive tier continues to increase by the same fixed number of cartons. A quality-assurance sensor is installed at the specific tier whose carton capacity equals the combined carton capacity of Tier 1 and Tier 5. At which tier is the quality-assurance sensor installed?

   • A) Tier 6
   • B) Tier 9
   • C) Tier 10
   • D) Tier 12
7. 7.

   An artisan crafts geometric stained-glass decorative panels following a consistent growth pattern where Stage 1 consists of 2 rectangular clear panes and 5 triangular tinted panes. For each consecutive stage, the panel is enlarged by adding 1 rectangular clear pane and 4 triangular tinted panes. If an artisan creates a panel that contains exactly 33 more triangular tinted panes than rectangular clear panes, what is the total number of glass panes (clear and tinted combined) in this completed panel?

   • A) 47
   • B) 52
   • C) 57
   • D) 62
8. 8.

   A logistics warehouse assembles modular storage racks in a single continuous row. A freestanding initial bay requires 4 vertical support pillars and 5 horizontal shelf beams. Each additional bay joined directly to the end of the row shares 2 vertical support pillars with the preceding bay and adds 5 new horizontal shelf beams. If the warehouse team constructs a continuous row using a total of 58 vertical support pillars, how many horizontal shelf beams are installed in this completed row?

   • A) 116
   • B) 135
   • C) 140
   • D) 145
9. 9.

   A design studio manufactures symmetric plus-shaped mosaic signs where Stage 1 consists of 1 central square tile and 4 arm tiles, totaling 5 tiles. Each consecutive stage extends each of the 4 arms outward by 1 square tile, so Stage 2 has 9 tiles and Stage 3 has 13 tiles, continuing this linear pattern. Each tile in a finished sign is fitted with 2 miniature spotlights, and the entire sign is mounted to a frame using 6 structural brackets. If an installation requires a combined total of 80 spotlights and structural brackets, which stage sign is being installed?

   • A) Stage 8
   • B) Stage 9
   • C) Stage 10
   • D) Stage 11
10. 10.

    A robotics team builds a linear conveyor track using identical regular hexagonal plates connected edge-to-edge in a straight row. A single plate has 6 exposed sensor ports along its outer perimeter, a row of 2 plates has 10 exposed sensor ports, and a row of 3 plates has 14 exposed sensor ports, continuing this linear pattern. The team constructs a track that has exactly 86 exposed sensor ports. If every plate in the completed track requires 2 support legs and the entire track also requires 4 additional corner stabilizers, what is the total number of legs and stabilizers needed?

    • A) 42
    • B) 44
    • C) 48
    • D) 46
11. 11.

    An environmental club builds linear community garden beds where each stage adds one square soil plot surrounded completely by protective border tiles. Stage 1 contains 1 square plot surrounded by 8 border tiles, Stage 2 contains 2 adjacent square plots surrounded by 10 border tiles, and Stage 3 contains 3 adjacent square plots surrounded by 12 border tiles. If the club has exactly 54 border tiles available to construct one continuous linear garden bed following this exact geometric pattern, how many square soil plots will the bed contain?

    • A) 24
    • B) 23
    • C) 25
    • D) 27