Inquiry AI
🟦

Honor Guard Formation

Square numbers and the 4(n−1) ring

Drag a slider to grow an n × n honor guard, then hollow out the inside to discover why the outer ring holds 4(n − 1) people. Four Olympiad-style levels — build, predict, decide, and a three-layer parade-square story — turn the square-formation problem (方阵问题) into a hands-on visualizer.

What this game shows · Square Formation Problem (方阵问题)

A square formation is people or objects arranged in an n × n grid. This animated game lets you switch between solid and hollow formations and watch the outer ring light up edge by edge — so the counting formulas come from the picture, not from memorization.

Solid square
total = n² (a square number).
Outer ring
edge guards = 4 (n − 1) — four edges, four shared corners.
k-layer hollow
total = n² − (n − 2k)².

Aligned with CCSS 4.OA.C.5 (generate and analyze patterns) and 3.OA.D.9 (identify arithmetic patterns). Recommended for Grades 3–5.

Free formation

Drag the side, toggle hollow, change layer thickness — the equation panel writes itself.

Hint · Slide side from 1 to 12. Tap Hollow to dim the inner block. Layers controls how thick the outer band stays lit.

Side n = 4Total = 16Outer ring = 12

What the formation says

  • Solid n × n → n² = 4² = 16
  • Outer ring → 4 (n − 1) = 4 × 3 = 12

Pattern reasoning model

Who this demo helps, and where to practice next

Honor Guard Formation is built for students who need to see growth, differences, and rule structure before writing formulas. It gives the page a clear search purpose: learn the model, manipulate it, then continue into the matching grade-level practice.

Honor Guard Formation helps when a student can copy a procedure but cannot explain why it works. The demo slows the idea down into a visible model before sending the learner to guided missions.

Learning goals

  • A solid n × n formation has exactly n² guards — square numbers stop being abstract and become rectangles you can see.
  • The outer ring of an n × n square is 4 (n − 1), because four edges of length n share four corners and corners cannot be counted twice.
  • A k-layer hollow square holds n² − (n − 2k)² guards. For the parade square with side 13 and 3 layers: 169 − 49 = 120 — the same as 48 + 40 + 32 across the three concentric rings.

How to play

  1. 1 Compare two consecutive stages and focus on the new pieces.
  2. 2 Predict the next stage before revealing it.
  3. 3 Use the related pattern or expression topic when the rule can be explained without the animation.
FAQ

Square formation, answered.

01 What is the square-formation problem (方阵问题)? Olympiad model

A classic Olympiad model: arrange people or objects in an n × n grid. The math behind it is square numbers and boundary counting — total = n² for a solid square, and outer ring = 4(n − 1).

02 How many people are in the outer ring of an n × n square? 4 (n − 1)

4 × (n − 1). Four edges of length n share four corners, so corners cannot be counted twice. For n = 8, the outer ring has 4 × 7 = 28 people.

03 What is a hollow square formation? n² − (n − 2k)²

A square with the inside removed. If the wall is k layers thick on every side, the total is n² − (n − 2k)². For side 13 and 3 layers, that is 169 − 49 = 120.

04 Which grade is this game for? Grades 3–5

Designed for Grades 3–5, aligned with CCSS 4.OA.C.5 (generate and analyze patterns). Older students can use the decide and story levels as warm-ups for sequence and series problems.

Related Fun Math games

Explore more Fun Math →