An increasing number triangle from 1 prints $i values on row $i, each computed as $i + $j - 1 — so each row starts with its own row number.
Rule: for i from 1 to rows
for j from 1 to i
print (i + j - 1) and a space
1
2 3
3 4 5
4 5 6 7
5 6 7 8 9 ← rows = 5Follows the triangle from 11 in Program 32; next is the variant from 0 in Program 34.
Outer loop grows $i. Inner loop prints $i values with the formula $i + $j - 1, separated by spaces.
| Method | Idea | Best for |
|---|---|---|
Formula $i + $j - 1 | When $j = 1, value equals $i — row starts at its index | Learning, interviews, exams |
| Compare offsets | Same shape as Program 32, but no base-9 shift | Seeing how formulas change the start |
for i from 1 to rows:
for j from 1 to i:
print (i + j - 1) and a space
print newline| Goal | Pattern |
|---|---|
| Grow each row | for ($i = 1; $i <= $rows; $i++) |
| Print i values | for ($j = 1; $j <= $i; $j++) |
| Start at 1 | echo ($i + $j - 1) . " "; |
| Row starts at i | When $j = 1, $i + $j - 1 = $i |
| End of row | echo PHP_EOL; |
| API | Effect | Use for |
|---|---|---|
echo value . " " | Stays on the same line | Each number on the row |
echo PHP_EOL | Ends the current line | After the inner loop finishes |
Print values without a newline, then end the row once.
Change the row count and the increasing triangle updates instantly — capped at 9 for readable demos.
$rows = 5Trace how each ($i, $j) pair maps to $i + $j - 1.
$i | Values | Count | Prints |
|---|---|---|---|
1 | 1 | 1 | 1 |
2 | 2, 3 | 2 | 2 3 |
3 | 3, 4, 5 | 3 | 3 4 5 |
4 | 4..7 | 4 | 4 5 6 7 |
5 | 5..9 | 5 | 5 6 7 8 9 |
Total numbers = 1 + 2 + … + n = n(n+1)/2 → O(n²).
Three complete programs: fixed $rows = 5, CLI input, and a compact $rows = 3 demo. Use View Output to reveal sample results.
$rows = 5Hard-coded height — print $i + $j - 1 with a trailing space on each cell.
<?php
for ($i = 1; $i <= 5; $i++) {
for ($j = 1; $j <= $i; $j++) {
echo ($i + $j - 1) . " ";
}
echo PHP_EOL;
}
?>1. Outer grows the width. Row $i prints exactly $i numbers.
2. Formula fills the cells. $i + $j - 1 makes row $i start at $i and count upward.
3. Newline once. Call echo PHP_EOL only after the inner loop finishes.
Read $rows from the CLI, validate, then use the same formula core.
<?php
echo "Enter rows: ";
$input = trim(fgets(STDIN));
if (!is_numeric($input) || (int) $input <= 0) {
echo "Please enter a positive integer." . PHP_EOL;
exit(1);
}
$rows = (int) $input;
for ($i = 1; $i <= $rows; $i++) {
for ($j = 1; $j <= $i; $j++) {
echo ($i + $j - 1) . " ";
}
echo PHP_EOL;
}
?>1. Prompt and validate. Reject non-numeric and non-positive values before printing.
2. Same core. $i + $j - 1 matches Example 1 — only $rows comes from the user.
3. Safer input tip. Cap demos for readable output:
if (!is_numeric($input) || (int) $input < 1 || (int) $input > 9) {
echo "Enter a whole number from 1 to 9." . PHP_EOL;
exit(1);
}$rows = 3Same structure with only three rows — easy to confirm the formula on paper.
<?php
$rows = 3;
for ($i = 1; $i <= $rows; $i++) {
for ($j = 1; $j <= $i; $j++) {
echo ($i + $j - 1) . " ";
}
echo PHP_EOL;
}
?>1. Three rows. $i = 1 → 1; $i = 2 → 2 3; $i = 3 → 3 4 5.
2. Trace on paper. If you use $i + $j instead of $i + $j - 1, every value shifts up by one.
3. Scale up next. Once the small demo is clear, use Examples 1–2 for five rows or user input.
Check these before calling the solution done.
Using $i + $j or 9 + $i + $j shifts every value — rows no longer start at the row number. Keep $i + $j - 1.
Inner bound $j <= $rows makes every row the same width. Use $j <= $i.
Without . " ", values run together (23). Always print a trailing space.
If echo PHP_EOL sits inside the inner loop, you get one number per line. Call it only after the loop.
Output is just 1. A good sanity check for input validation.
is_numericBare (int) casts turn non-numeric strings into 0 — prefer is_numeric before casting.
| Program | Time | Extra space |
|---|---|---|
| Fixed / input (Examples 1–2) | O(n²) | O(1) |
| Compact rows = 3 (Example 3) | O(n²) | O(1) |
Row $i prints i numbers. Total = 1 + 2 + … + n = n(n+1)/2 → O(n²) time. Only a few loop variables are needed.
$i + $j - 1 — row $i starts at $i.$j = 1..$i so row $i has $i numbers.echo PHP_EOL after it finishes.O(n²) time from n(n+1)/2 prints; O(1) extra space.One line: for each row $i, print $i + $j - 1 for $j = 1..$i with spaces, then echo PHP_EOL.
Each printed value is computed as $i + $j - 1. Row $i = 1 prints 1; row $i = 4 prints 4, 5, 6, 7 — a left-shifted increasing triangle starting at 1.
Continue with the same shape using $i + $j and an outer loop from 0.
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