You are about to leave on vacation and realize you forgot to pack socks. You race back to your room but the lights are off and you can't see the color of the socks in the drawer. The socks are all mixed up but you know that you have 10 pair of white socks, 10 pair of black socks, and 11 pair of blue socks. How many individual socks do you need to pull out of the drawer to be sure you have at least 1 matching pair?

4

Step-by-step explanation:

if you grab 4, then you might get one of 2 colors and 2 of the same color. If you grab 3 or 2, you could get all different colors.

there are lots of ways.   pick's theorem tells us if the vertices are lattice points the area a is

a = b/2 + i - 1

where b is the number of boundary lattice points and i is the number of interior lattice points.

in the figure i count b=16, i=50 so

a = 8 + 50 - 1 = 57

let's try the shoelace formula.   the area is half the sum of the cross products of the sides.  

a(-2,6)   -2(2)-6(-5) = 26

b(-5,2)   -5(-3) - 2(-2) = 19

c(-2,-3) -2(-3) - (-3)(4) = 18

d(4,-3)   4(3) - (-3)(5) = 27

e(5,3)   5(3) - 3(1) = 12

f(1,3)     1(6) - 3(-2) = 12

a(-2,6)

each of the cross products is twice the area of the triangle with the associated side and one vertex the origin. so the area of abo=26/2=13.

area = (1/2) (26 + 19 +   18 + 27 + 12 + 12) = 57

that checks!