An increasing number triangle from 0 uses zero-based loops: row $i prints $i + 1 values of $i + $j — so the first cell is 0.
Rule: for i from 0 to max
for j from 0 to i
print (i + j) and a space
0
1 2
2 3 4
3 4 5 6
4 5 6 7 8
5 6 7 8 9 10 ← max = 5Follows the triangle from 1 in Program 33; next is the right-aligned incremental triangle in Program 35.
Outer loop runs $i = 0..$max. Inner loop prints $j = 0..$i with the formula $i + $j, separated by spaces.
| Method | Idea | Best for |
|---|---|---|
Formula $i + $j | When $j = 0, value equals $i — row starts at its index | Learning, interviews, exams |
| Compare with Program 33 | Same shape; zero-based loops replace $i + $j - 1 | Seeing how indexing changes the start |
for i from 0 to max:
for j from 0 to i:
print (i + j) and a space
print newline| Goal | Pattern |
|---|---|
| Grow each row | for ($i = 0; $i <= $max; $i++) |
| Print i+1 values | for ($j = 0; $j <= $i; $j++) |
| Start at 0 | echo ($i + $j) . " "; |
| Row starts at i | When $j = 0, $i + $j = $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 max $i and the zero-based triangle updates instantly — capped at 9 for readable demos.
$max = 5Trace how each ($i, $j) pair maps to $i + $j.
$i | Values | Count | Prints |
|---|---|---|---|
0 | 0 | 1 | 0 |
1 | 1, 2 | 2 | 1 2 |
2 | 2, 3, 4 | 3 | 2 3 4 |
3 | 3..6 | 4 | 3 4 5 6 |
4 | 4..8 | 5 | 4 5 6 7 8 |
5 | 5..10 | 6 | 5 6 7 8 9 10 |
Total numbers = 1 + 2 + … + (max+1) = (max+1)(max+2)/2 → O(n²).
Three complete programs: fixed $max = 5, CLI input, and a compact $max = 2 demo. Use View Output to reveal sample results.
$max = 5Hard-coded max — zero-based loops print $i + $j with a trailing space.
<?php
for ($i = 0; $i <= 5; $i++) {
for ($j = 0; $j <= $i; $j++) {
echo ($i + $j) . " ";
}
echo PHP_EOL;
}
?>1. Outer grows from 0. Row $i prints exactly $i + 1 numbers.
2. Formula fills the cells. $i + $j makes row $i start at $i and count upward.
3. Newline once. Call echo PHP_EOL only after the inner loop finishes.
Read $max from the CLI, validate, then use the same formula core.
<?php
echo "Enter max i: ";
$input = trim(fgets(STDIN));
if (!is_numeric($input) || (int) $input < 0) {
echo "Please enter a non-negative integer." . PHP_EOL;
exit(1);
}
$max = (int) $input;
for ($i = 0; $i <= $max; $i++) {
for ($j = 0; $j <= $i; $j++) {
echo ($i + $j) . " ";
}
echo PHP_EOL;
}
?>1. Prompt and validate. Reject non-numeric and negative values before printing.
2. Same core. $i + $j matches Example 1 — only $max comes from the user.
3. Safer input tip. Cap demos for readable output:
if (!is_numeric($input) || (int) $input < 0 || (int) $input > 9) {
echo "Enter a whole number from 0 to 9." . PHP_EOL;
exit(1);
}$max = 2Same structure with only three rows — easy to confirm zero-based indexing on paper.
<?php
$max = 2;
for ($i = 0; $i <= $max; $i++) {
for ($j = 0; $j <= $i; $j++) {
echo ($i + $j) . " ";
}
echo PHP_EOL;
}
?>1. Three rows. $i = 0 → 0; $i = 1 → 1 2; $i = 2 → 2 3 4.
2. Trace on paper. If you start at $i = 1, the leading 0 disappears and the shape matches Program 33 less cleanly.
3. Scale up next. Once the small demo is clear, use Examples 1–2 for max 5 or user input.
Check these before calling the solution done.
Starting at $i = 1 or using $i + $j - 1 drops the leading 0. Keep $i = 0..$max with $i + $j.
Inner bound $j <= $max makes every row the same width. Use $j <= $i.
Without . " ", values run together (12). 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 0. 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 max = 2 (Example 3) | O(n²) | O(1) |
Row $i prints i + 1 numbers. Total = 1 + 2 + … + (n+1) = (n+1)(n+2)/2 for $max = n → O(n²) time. Only a few loop variables are needed.
$i + $j — first cell is 0 when both loops start at 0.$j = 0..$i so row $i has $i + 1 numbers.echo PHP_EOL after it finishes.O(n²) time from (n+1)(n+2)/2 prints; O(1) extra space.One line: for each $i from 0 to $max, print $i + $j for $j = 0..$i with spaces, then echo PHP_EOL.
Each printed value is computed as $i + $j. With $i = 0 and $j = 0 the first row prints 0; row $i = 2 prints 2, 3, 4 — a zero-based left-shifted increasing triangle.
Continue with a right-aligned triangle that counts continuously across rows.
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