Elementary Math Word Problems · Set Overlap & Inclusion-Exclusion
Chain Link Overlap & Inclusion-Exclusion Visualizer
Key pedagogical takeaway: When N links interlock and stretch tight, each joint consumes 2 wall thicknesses (2d). N links have (N-1) joints, so Total Length = N*L - 2d*(N-1).
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3D Simulation Stage● 60 FPS
Parameter Controls
cm
cm
Live Derivations & Metrics
Total Stretched Length
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Overlap Length Consumed
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Total Interlocking Joints
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Inner Cleared Length
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Real-Time Formula Derivation
Governing Formula:
--Substitution:
--Regime: --
Worked Examples & Exam Question Deep Dive
Worked Problem
Problem: 10 circular iron links (outer diameter 8cm, wall thickness 0.5cm) interlock into a straight chain. Find total length.
Step-by-Step Derivation:
- 9 joints. Each joint consumes 2 * 0.5cm = 1.0cm. Total length = 10*8 - 9*1 = 71cm.
Worked Problem
Problem: Class of 45: 28 join math, 25 join English, 12 join both. How many join neither?
Step-by-Step Derivation:
- At least one: |A ∪ B| = 28 + 25 - 12 = 41. Neither: 45 - 41 = 4 students.
Student Inquiry Worksheet & Practice Problems
Q1. Variant 1: Identical links (outer 12cm, wall 1cm) chain to total 92cm. How many links are there?
Answer Key: First link gives 12cm; each subsequent link adds inner diameter 10cm. 12 + 10*(N-1) = 92 => N-1 = 8 => N = 9 links.
Q2. Variant 2: 8 paper strips (each 20cm long) are glued with 3cm overlap joints. Find total assembled length.
Answer Key: 7 overlap joints. Total length = 8*20 - 7*3 = 160 - 21 = 139cm.
Q3. Variant 3: Three 80cm poles are spliced with 15cm overlap joints. Find total spliced length.
Answer Key: 2 joints. Total length = 3*80 - 2*15 = 240 - 30 = 210cm.
Frequently Asked Questions & Misconceptions
Why does each chain joint consume TWO wall thicknesses instead of one?
Because the right wall of the first link and left wall of the second link BOTH occupy the overlap zone! Neglecting the 2nd thickness is the classic error.
Why do closed loops have N joints while straight chains have N-1?
The closed-perimeter tree problem! In an open line, N items have N-1 gaps. When looped closed, the last link loops into the first, creating N joints.
What does Inclusion-Exclusion mean? What is the 3-set Venn formula?
Add individual sizes, subtract pairwise overlaps, and add back the 3-way intersection: |A∪B∪C| = A + B + C - (AB+BC+CA) + ABC.
When two overlapping rectangles lie on a table, what is the net covered area?
Total covered area = Area(A) + Area(B) - Overlap Area. Subtracting overlap removes the double-counted duplicate layer.