Work Backward with Square Roots, Gaps, and Shifts
第20讲 完全平方数 · Lesson 20.5
Turn a square clue into a root, factor the gap between two squares, and count every permitted shift without testing numbers one by one.
Build the square staircase
Growing an n × n square into an (n+1) × (n+1) square adds one row and one column. That makes every consecutive-square gap visible.
The gap formula
The new top row has n+1 cells. The new side column has n more cells. Together they add:
Adjust the lower root
Checkpoint
Use the two printed gaps below, even if the diagram shows a different root.
Need a hint? Start here
The new border has n+1 cells in one row and n more in one column.
For the two printed gaps, n=48 and n=24. 97;49;odd.
Work backward from a square year
A story can hide both the square root and a subtraction. First find the only square year that fits the stated century.
original problem
An older brother says: “In the year x² of the twenty-first century, I will be exactly x years old.” In what year was he born?
Search the century by roots
For this lesson, the twenty-first century means the calendar years 2001 through 2100.
Then subtract the age
Checkpoint
Answer the original 2001–2100 story, not an out-of-century slider example.
Need a hint? Start here
Find the square year within the stated century, then subtract the age, which is its root.
44²=1936,45²=2025,46²=2116. Subtract age45 from year2025. 2025;45;birth year1980.
Factor the gap between two squares
When two numbers are both squares, their difference factors into the difference and sum of their roots.
original problem
Can you find a positive whole number n such that both n and n+97 are perfect squares?
Translate into roots
Write:
Subtract:
Because 97 is prime, its only positive factor pair is:
Recover the roots
Checkpoint
Need a hint? Start here
Write the two squares as a² and b². Factor their difference as (b−a)(b+a).
b−a=1,b+a=97. Add or subtract to get b49,a48. 1;97;48;n=2304.
Explore every difference-of-squares case
The factor-pair method can produce one solution, several solutions, or no solution. This laboratory checks every factor pair, so no branch is missed.
Use the fresh gap 12 and factor pair 2×6. Predict the smaller root before using the laboratory.
Your prediction is checked with this mission. Try it before reading the worked steps below.
Difference-of-squares laboratory
Checkpoint
Answer the printed gaps 24, 2, and 105; also fill the separate prediction for gap 12.
Need a hint? Start here
List factor pairs of the gap. Keep pairs with the same parity so both recovered roots are whole numbers.
Gap24 has pairs2×12 and4×6. Gap105 has1×105,3×35,5×21,7×15. Prediction2. Checkpoint:two;impossible;root1;four.
Shift a fixed number onto a square
A bounded addend creates a bounded square target. Search the roots of that target interval instead of testing every addend.
Transfer problem
Add a two-digit prime to 2002 to obtain a perfect square. Find the prime.
Bound the target square
A two-digit prime is between 11 and 97, so the square must lie between:
Test a candidate addend
Checkpoint
Need a hint? Start here
Bound the target using the smallest and largest permitted addend; search only roots inside that interval.
2025−2002=23, and23 is prime. 2025;23;prime.
Count every allowed square shift
If B+x must be a square and L≤x≤U, then the square target lies in the shifted interval [B+L, B+U]. Root bounds count every solution at once.
Here B, L, U and x are nonnegative whole numbers; both endpoints are included. The symbol ⌈√A⌉ means the smallest whole-number root whose square is at least A; ⌊√A⌋ means the largest whole-number root whose square is at most A. Every allowed root gives exactly one addend x=r²−B.
Shift-range laboratory
Checkpoint
Use the printed three-digit and four-digit ranges below, even if you change the explorer.
Need a hint? Start here
Add the base to both endpoints, count the possible roots inclusively, then subtract the base from each square.
2010+[100,999]=[2110,3009], with roots46–54. 400+[1000,9999]=[1400,10399], with roots38–101. 46;54;nine;64.
Maximize a sum that must be square
An optimization proof needs an upper bound, the greatest square below that bound, and a construction that reaches it.
Transfer problem
Three different two-digit numbers have a perfect-square sum. What is the greatest possible sum?
Bound first
The greatest possible sum of three different two-digit numbers is:
The greatest perfect square not exceeding 294 is:
Now we still need a construction that reaches 289.
Build an optimal triple
Enter three different whole numbers from 10 through 99 in any order. Your construction is checked along with the checkpoint below. Also count every unordered optimal triple: changing the order of the same three numbers does not create another triple.
Checkpoint and construction audit
Need a hint? Start here
Prove an upper bound, find the greatest square below it, then construct distinct numbers that attain it.
The greatest square at most294 is289. The five unordered constructions are92,98,99;93,97,99;94,96,99;94,97,98;95,96,98. 294;289;five. Enter any one construction in the three boxes.
Find the least square in a consecutive-number story
Use the middle number to compress three consecutive integers into one expression, then use the square condition.
Transfer problem
Three consecutive four-digit positive integers have a perfect square as the middle number. Their sum is divisible by 15. Find the least possible middle number.
Compress the sum
Write the integers as:
Their sum is:
For 3n to be divisible by 15, n must be divisible by 5. If n is a square divisible by 5, its root is divisible by 5.
Test square roots, not all four-digit numbers
Checkpoint
Find the least valid middle number in the original story, not just any valid slider example.
Need a hint? Start here
Use the middle number n: the total is 3n. Apply divisibility before searching the square roots.
All three integers must be four-digit. Roots start at32; the first divisible by5 is35. 35;1225;sum3675;yes.
Square-roots, gaps, and shifts workshop
Correct all eight answers to complete the workshop.
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.
Count both endpoints in root intervals and count each square pair only once. 101;1980;2304;2;23;9;289;1225.
Exit ticket
Complete all ten missions, including all five exit questions, to earn the 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.
Optional reflection — not automatically graded
Square-Root, Gap & Shift Navigator
This certifies that Student completed all ten missions and can work backward from perfect-square clues.
Chapter 20 • Perfect Squares
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.
Its roots are34–100:100−34+1=67. A square middle divisible by5 needs a root divisible by5. 161;no;67;93;next root40.