Numerical answer
How many ordered pairs of integers (x, y) satisfy |x| + |y| < 6?
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Answer: 61
|x| + |y| < 6 with integers means |x| + |y| ≤ 5.
For each x, |y| ≤ 5 − |x|, giving 2(5 − |x|) + 1 values of y.
x = 0: 11; x = ±1: 9 each; x = ±2: 7 each; x = ±3: 5 each; x = ±4: 3 each; x = ±5: 1 each.
Total = 11 + 2(9 + 7 + 5 + 3 + 1) = 11 + 50 = 61.
Tip: The number of integer points with |x| + |y| ≤ n is 2n² + 2n + 1 (here 2·25 + 10 + 1 = 61).
For each x, |y| ≤ 5 − |x|, giving 2(5 − |x|) + 1 values of y.
x = 0: 11; x = ±1: 9 each; x = ±2: 7 each; x = ±3: 5 each; x = ±4: 3 each; x = ±5: 1 each.
Total = 11 + 2(9 + 7 + 5 + 3 + 1) = 11 + 50 = 61.
Tip: The number of integer points with |x| + |y| ≤ n is 2n² + 2n + 1 (here 2·25 + 10 + 1 = 61).
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