Question

How many ordered pairs of integers (x,y) are there that satisfy |x|+|y| ≤ 10?

Answer 1

160

Answer 2

from (0,10) to (0,-10) -- 21
from (1,9) to (1,-9) --- 19
from (2,8) to (2,-8) --- 17
..
from (9,1) to (9,-1) -- 3
and (10,0) ----- 1

same thing on the left side, but we can't count the centre line twice, so
sum = 2(1+3+5+...19) + 21
there are 10 terms in the series
sum = 2(5)(first + last) + 21
= 10(20) + 21 = 221

or 2(1+3+5+..+21) - 21
= 2(11/2)(1 + 21) - 21 = 221

There are 221 such ordered pairs.