Question

Now consider a Destroyer game where a single destroyer is placed (again on a 4×4 grid). The destroyer occupies 2 adjacent grid squares, and may be oriented horizontally or vertically.

Calculate the information content of a first-turn hit if the location is

• a corner square

• an edge square (other than a corner)

• an interior grid square

Answer 1

Consider the 4X4 grid with each square labelled with a number:

4 X 4 Grid - Destroyer

1	2	3	4
5	6	7	8
9	10	11	12
13	14	15	16

The possible positions that can be occupied by the destroyer are :

Possible Outcomes
(1,2)
(2,3)
(3,4)
(5,6)
(6,7)
(7,8)
(9,10)
(10,11)
(11,12)
(13,14)
(14,15)
(15,16)
(1,5)
(2,6)
(3,7)
(4,8)
(5,9)
(6,10)
(7,11)
(8,12)
(9,13)
(10,14)
(11,15)
(12,16)

• An interior grid Square : There are 4 times each for which all these interior grid numbers are selected - 6,7,10,11
• A Corner Square - There are 2 each times for which all these corner grid numbers are selected - 1,4,13,16
• An edge square (other than a corner) - There are 3 times each for which an edge square number are selected - 2,3,5,9,8,12,14,15

Now :-

P(An interior grid Square hit) = 16 / 24 = 2/3

P(A Corner Square hit ) = 8/24 = 1/3

P(An edge square hit) = 24/24 = 1

The information content of measuring X as outcome x is defined as :

I_x(x) = log(1/p(x))

considering the base to be exponential

I_x(x) = ln(1/p(x)) = -ln(p(x))

For the given question:

1. interior grid Square : I_x(x)= -ln(p(x))  = -ln(2/3) = 0.405465
2. Corner Square : I_x(x)= -ln(p(x))  = -ln(1/3) = 1.0986
3. An edge square hit: I_x(x)= -ln(p(x))  = -ln(1) = 0

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