A hollow square border number pattern visits every cell of an n×n grid, but prints consecutive numbers only on the boundary — clockwise from the top-left — and leaves the inside blank.
Remember
Rule: top → right → bottom ← left ↑ (inner = three spaces)
1 2 3 4 5
16 6
15 7
14 8
13 12 11 10 9 ← n = 5 (16 border cells)
Unlike Program 58 (diagonal diamond), this shape is a rectangle outline with side-specific counters and setw(3) alignment.
Approach
How to Solve It
Nested loops over rows and columns. Branch on side order (top, right, bottom, left). Use three counters for the non-top sides.
Method
Idea
Best for
Side counters
if / else if per side + k, l, m
Learning, interviews, demos
Compact trace
Same logic with n = 3
Quick dry-runs on paper
Pseudocode
Pseudocode
k = n + 1
l = 3 * n - 2
m = 4 * (n - 1)
for i from 1 to n:
for j from 1 to n:
if i == 1: print j (width 3)
else if j == n: print k; k = k + 1
else if i == n: print l; l = l - 1
else if j == 1: print m; m = m - 1
else: print three spaces
print newline
Cheat sheet
Goal
Pattern
Grid loops
for (i = 1; i <= n; i++) for (j = 1; j <= n; j++)
Top row
if (i == 1) cout << setw(3) << j;
Right column
else if (j == n) cout << setw(3) << k++;
Bottom row
else if (i == n) cout << setw(3) << l--;
Left column
else if (j == 1) cout << setw(3) << m--;
Inner cell
else cout << " ";
Counter starts
k = n+1, l = 3*n-2, m = 4*(n-1)
Printing Numbers vs Starting a New Line
API
Effect
Use for
cout << setw(3) << value / " "
Stays on the same line
Each cell
cout << "\n"
Ends the line
After the column loop
Print each cell without a newline, then end the row once.
Try it
Live Preview
Change the square size and the clockwise hollow border updates instantly.
Use size from 2 to 9. Border cells = 4×(n−1). Tap a chip or type a value — the preview redraws as you go.
Live result5×5 · 16 border
1 2 3 4 5
16 6
15 7
14 8
13 12 11 10 9
Trace
Worked Walkthrough
Trace the four sides for n = 5 — notice why j == n is checked before i == n.
Side
Condition
Values
Top
i == 1
1 2 3 4 5
Right
j == 5 (rows 2..5)
6 7 8 9
Bottom
i == 5 (cols 1..4)
13 12 11 10
Left
j == 1 (rows 2..4)
16 15 14
Bottom-right is claimed by the right branch (j == n), so the bottom loop never reprints corner 9.
Code
C++ Programs
Three complete programs: fixed 5×5, cin size with formulas, and a compact 3×3 demo. Use View Output to reveal sample results.
Example 1 — Fixed 5×5 Border
Hard-coded grid with classic counter starts k = 6, l = 13, m = 16.
C++
#include <iostream>
#include <iomanip>
using namespace std;
int main()
{
int i, j;
int k = 6, l = 13, m = 16;
for (i = 1; i <= 5; i++)
{
for (j = 1; j <= 5; j++)
{
if (i == 1)
cout << setw(3) << j;
else if (j == 5)
cout << setw(3) << k++;
else if (i == 5)
cout << setw(3) << l--;
else if (j == 1)
cout << setw(3) << m--;
else
cout << " ";
}
cout << "\n";
}
return 0;
}
Output
1 2 3 4 5
16 6
15 7
14 8
13 12 11 10 9
How It Works
1. Top row. When i == 1, print column index j with width 3.
2. Right, bottom, left.k++ fills the right side, l-- the bottom, m-- the left.
3. Hollow inside. Any non-border cell prints three spaces so columns stay lined up.
Example 2 — User Input Size
Read n and compute counter starts with the same formulas used in the live preview.
C++
#include <iostream>
#include <iomanip>
using namespace std;
int main()
{
int n, i, j;
int k, l, m;
cout << "Enter square size (n): ";
cin >> n;
k = n + 1;
l = 3 * n - 2;
m = 4 * (n - 1);
for (i = 1; i <= n; i++)
{
for (j = 1; j <= n; j++)
{
if (i == 1)
cout << setw(3) << j;
else if (j == n)
cout << setw(3) << k++;
else if (i == n)
cout << setw(3) << l--;
else if (j == 1)
cout << setw(3) << m--;
else
cout << " ";
}
cout << "\n";
}
return 0;
}
1. Same side logic. Only n and the three start formulas change.
2. Entering 4. Border cells = 4×3 = 12 — numbers run from 1 to 12.
3. Validate in real apps. Prefer checking cin failure and requiring n >= 2 (tip below).
Safer input tip
if (!(cin >> n) || n < 2)
{
cout << "Please enter an integer >= 2.\n";
return 1;
}
Example 3 — Compact 3×3 Border
Eight border cells and one hollow center — ideal for a paper dry-run.
C++
#include <iostream>
#include <iomanip>
using namespace std;
int main()
{
int n = 3;
int i, j;
int k = n + 1, l = 3 * n - 2, m = 4 * (n - 1);
for (i = 1; i <= n; i++)
{
for (j = 1; j <= n; j++)
{
if (i == 1)
cout << setw(3) << j;
else if (j == n)
cout << setw(3) << k++;
else if (i == n)
cout << setw(3) << l--;
else if (j == 1)
cout << setw(3) << m--;
else
cout << " ";
}
cout << "\n";
}
return 0;
}
Output
1 2 3
8 4
7 6 5
How It Works
1. Tiny grid. Easy to dry-run every if / else if branch on paper.
2. Same formulas.k = 4, l = 7, m = 8 — proving the counters scale.
3. One hollow cell. Center (2,2) prints three spaces; everything else is border.
Edge Cases & Pitfalls
Check these before calling the solution done.
Branch order
Test j == n before i == n
Bottom-right must come from the right counter — otherwise the clockwise sequence breaks.
Alignment
Match inner width to setw(3)
Inner cells need exactly three spaces. Two spaces skew the grid once numbers hit two digits.
Headers
Include <iomanip>
Without it, setw will not compile.
Bad cin
Require n >= 2
Size 1 is a single cell; size 0 or negative skips the loops. Validate before printing.
Analysis
Time and Space Complexity
Program
Time
Extra space
Hollow border (Examples 1–3)
O(n²)
O(1)
Every cell of the n×n grid is visited once — quadratic in n. Border length is only 4(n - 1), but the nested loops still walk the full square. Extra memory is a handful of counters.
Remember
Key Takeaways
Rule: top j, right k++, bottom l--, left m--, else spaces.
Align:setw(3) for numbers, three spaces for hollow cells.
Break the row: call cout << "\n" only after the column loop finishes.
Complexity:O(n²) time, O(1) extra space.
One line: walk the square cell by cell, number the border clockwise, leave the inside blank.
Frequently Asked Questions
Check sides in order: top (i == 1), right (j == n), bottom (i == n), left (j == 1). Everything else is an inner cell.
Fixed width 3 keeps columns aligned when border numbers have one or two digits. Include <iomanip>.
For n=5: a hollow 5×5 grid with numbers 1–16 clockwise on the boundary; the inside stays blank.
Program 58 prints a diagonal mirror diamond. Program 59 prints a rectangular hollow border with separate counters per side.
k tracks the right column, l the bottom row (descending), m the left column (descending). Top uses j directly.
Printing a padded number or three spaces stays on the same line. Printing a newline ends the current row.
O(n²) for an n×n grid because every cell is visited once.
A 2×2 grid has no inner cells — every position is on the border (4 cells = 4×(2-1)).
🤔
Did you know?
This pattern is a hollow n×n border: top 1..n, right continuing up, bottom descending, left descending — inner cells are three spaces so columns stay aligned with setw(3).
