In the funny elementary school cafeteria, lunch for 100 people is served. but it is topsy turvey, and the waiters are acting funny. the first waiter turned every plate upside down. then another waiter turned every other plate. then the third waiter turned every third plate (so if a plate was upside down it was placed normally, but if it was placed normally, it became upside down). the fourth waiter turned every fourth plate, and so on, until the hundredth waiter turned the hundredth plate. at the end, how many plates were placed normally?

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!

answer: 59

step-by-step explanation:

15 units use socratic

step-by-step explanation: 15 units