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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
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.

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