Definition
n² = n × n
If N = n², then n is the nonnegative square root of N.
Lesson 20.1 — Recognize, Count, and Locate Perfect Squares
第20讲 · 识别、计数并定位完全平方数
Build squares as arrays, locate numbers between neighboring squares, count squares in intervals, and discover why only square-numbered lamps remain on.
A square has a root, a place, and one unpaired factor.
Progress saves automatically.
Mission 1
Not completeKnow exactly what a perfect square is
A whole number is a perfect square when it equals one whole number multiplied by itself. The worked example includes zero: 0 = 0².
Square or root?
In 18² = 324:
- 18 is the square root.
- 324 is the perfect square.
original opening list
The worked example displays the ten squares below 100:
0, 1, 4, 9, 16, 25, 36, 49, 64, 81
The next square is 100 = 10².
Endpoint clarity
The worked example’s displayed list stops at 81. To avoid ambiguity, this lesson says “below 100” for that ten-number list and “from 0 through 100” when 100 is included.
Definition checkpoint
Need a hint? Start here
Multiply a whole number by itself, not by two. Zero also works.
Roots 0 through 9 give the squares below 100; root 10 reaches 100. Yes; root 0; ten squares; next square 100.
Mission 2
Not completeBuild a square as an equal-row array
A root of n creates an n by n array. Change the side length and watch the number of unit cells become n².
Predict first
Before using the array: a new square has 7 rows and 7 columns. Predict its number of cells.
Your prediction is checked with this mission. Try it before reading the worked steps below.
Root6
Rows6
Columns6
Unit cells36
6 × 6 = 36
Array checkpoint for the fixed 18 × 18 array
Answer for side length 18 even if the adjustable array shows a different size. Also complete the separate 7 × 7 prediction above.
Need a hint? Start here
Count rows and columns: the same root supplies both dimensions.
The prediction is 7×7=49; the fixed checkpoint uses 18×18. 324; 18; 18 rows; 324 unit cells.
Mission 3
Not completeLocate a number between neighboring squares
Find the largest square not above the target and the next square after it. This tells you whether the target is itself a square and where it sits on the square staircase.
Worked example 1
44² = 1936< 2016 <45² = 2025
There are exactly 44 positive square roots from 1 through 44, so there are 44 perfect squares from 1 through 2016.
Location checkpoint for 2016
Need a hint? Start here
Square the nearby whole-number roots and compare both results with the target.
44²=1936 and 45²=2025. Lower root 44; lower square 1936; upper square 2025; not a square.
Mission 4
Not completeCount perfect squares from 1 through N
Each positive root produces exactly one positive square. The count is therefore the greatest whole-number root whose square does not exceed N.
Counting rule
Count from 1 through N = ⌊√N⌋
The floor symbol means “take the greatest whole number not above the square root.”
Do not accidentally count 0
The interval starts at 1, so the square 0² = 0 is outside it.
Count roots 1, 2, 3, …, ⌊√N⌋.
Count checkpoint for the fixed interval 1 through 2016
Use this interval even if you have changed the laboratory endpoint.
Need a hint? Start here
Each positive root contributes exactly one square. Start counting at 1.
Positive roots 1–44 fit; root 45 gives 2025, which is too large. 44; 1936; 45; 44 squares.
Mission 5
Not completeCount squares inside any interval
Both interval endpoints are included. Find the first root whose square reaches the lower endpoint and the last root whose square stays within the upper endpoint.
First included root
⌈√L⌉
Move upward to the next whole root when L is not already a square.
Last included root
⌊√U⌋
Move downward to the greatest whole root whose square is at most U.
Interval checkpoint: 2024 through 2499
Need a hint? Start here
Find the first root that reaches the lower endpoint and the last root that stays within the upper endpoint. If the first is larger, the count is zero.
Roots 45–49 give 2025,2116,2209,2304,2401. First square 2025; last 2401; count 5; next square 2500.
Mission 6
Not completeSee the unpaired factor of a square
Factors usually come in pairs d and N ÷ d. In a perfect square, the square root pairs with itself, creating one unpaired divisor and an odd divisor count.
Why the parity changes
36: 1×36, 2×18, 3×12, 4×9, 6×6
The first four pairs contribute eight different divisors. The pair 6 × 6 contributes only one new divisor.
For positive integers: perfect square ⇔ odd number of positive divisors. Zero is a square, but every positive integer divides zero, so it is outside this rule.
Factor-pair checkpoint for 36
Need a hint? Start here
List factor pairs. Count both members unless they are the same number.
Pairs 1×36,2×18,3×12,4×9 each give two divisors; 6×6 gives one. Unpaired divisor 6; nine divisors; odd; yes.
Mission 7
Not completeRun the worked example’s 200-lamp experiment
All 200 lamps, numbered 1 through 200, start off at second 0. At each whole-number second s from 1 through 200, every lamp whose number is a multiple of s changes state. Lamp k is toggled once for every positive divisor of k.
State rule
To toggle means to switch off to on, or on to off. The square-number conclusion applies after all 200 rounds; intermediate rounds can leave other lamps on. A square outline marks the lamp’s number, not its current on/off state.
- An even number of toggles returns a lamp to off.
- An odd number of toggles leaves a lamp on.
- Only square numbers have an odd number of divisors.
Root boundary
14² = 196< 200 <15² = 225
So the possible on-lamps are numbered 1² through 14².
Second 0
offonsquare-numbered lamp
Lamp checkpoint — first jump to second 200
Need a hint? Start here
A lamp switches at its positive divisors. An even number of switches cancels in pairs; an odd number leaves one switch.
Each lamp began off and has now switched once for every positive divisor of its number. Fourteen lamps are on: 1² through 14². Largest 196; next square 225; square-numbered lamps.
Mission 8
Not completeIdentify squares efficiently
The worked example practice asks which numbers in a five-number list are perfect squares. Select every valid square, then use the general checker on new values.
original Guided Practice 1
Select every perfect square:
General square checker
A last digit of 0, 1, 4, 5, 6, or 9 does not prove that a number is a square. For example, 546 ends in 6 but is not a square. Locate it between consecutive squares or test its integer root.
Identification checkpoint
Need a hint? Start here
Test an integer root. An allowed final digit alone cannot confirm a square.
Only 324=18² works. 29²=841<897<900; 23²=529<546<576. 18; 841; 576; one square candidate.
Mission 9
Not completePerfect-square workshop
Correct all eight answers to complete the workshop.
0 / 8
Optional reflection — not automatically graded
Need a hint? Start here
Choose a method before calculating. Use the earlier local hint for that method, then verify the root, all stated conditions and the endpoints.
The odd-divisor rule applies to positive squares, excluding zero. Answers: yes; 196; 14; 44; 5; odd; 18; 81.
Mission 10
Not completeExit ticket
Complete all ten missions, including all five exit questions, to earn your certificate.
Fresh challenge: use the methods from this lesson on these new values. Check your calculations by squaring, listing a short cycle, or verifying every condition.
0 / 5
n²
Perfect-Square Locator
This certifies that a determined mathematician can build square arrays, locate neighboring squares, count squares in intervals, pair divisors, and explain the 200-lamp pattern.
Lesson 20.1 • Chapter 20
Need a hint? Start here
Choose a method before calculating. Use the earlier local hint for that method, then verify the root, all stated conditions and the endpoints.
22²=484≤500<529=23²; 26²=676<700<729=27². Answers: 361; 22; 729; odd; 10.
Learning notes and instructional additions
The original’s opening paragraph displays ten squares from 0 through 81 while using wording that can be read as including 100; this page states endpoints explicitly and treats that displayed list as squares below 100.The array builder, general locator, interval counter, factor-pair explorer, fully simulated lamp grid, square checker, workshop, and exit ticket are added instructional scaffolds.