Lamb and Kitten Chase Problem Solution

Problem Statement

Lamb and a kitten are playing a chase game with the following rules:

Lamb moves 3 steps per turn (fixed)

Kitten's movement per turn is random with probabilities:

Steps	Probability
1	1/12
2	1/12
3	1/6
4	1/6
5	1/4
6	1/4

They move toward each other (closing the distance)
Initial distance: 10 steps
Find the expected number of turns until the kitten catches Lamb

Solution Approach

We model this as a Markov process where the state is the current distance between them.

1. Calculate Expected Kitten Movement

First, compute the expected steps the kitten moves per turn:

E[X] = 1×(1/12) + 2×(1/12) + 3×(1/6) + 4×(1/6) + 5×(1/4) + 6×(1/4)
= 1/12 + 2/12 + 6/12 + 8/12 + 15/12 + 18/12
= 50/12 = 25/6 ≈ 4.1667 steps

2. Combined Expected Distance Reduction

Each turn, the expected distance reduction is:

E[reduction] = Lamb's steps + E[X] = 3 + 25/6 = 43/6 ≈ 7.1667 steps

3. Markov Process Calculation

Define E(d) as expected turns needed when distance is d:

E(d) = 1 + Σ P(X=k) × E(max(d - (3 + k), 0))

We compute E(d) recursively from d = 1 to d = 10:

Distance (d)	Calculation	E(d)
1-4	Any move (3+k ≥ 4) covers distance	1
5	1 + (1/12)×E(1)	13/12 ≈ 1.0833
6	1 + (1/12)×E(2) + (1/12)×E(1)	7/6 ≈ 1.1667
7	1 + (1/12)×E(3) + (1/12)×E(2) + (1/6)×E(1)	4/3 ≈ 1.3333
8	1 + (1/12)×E(4) + (1/12)×E(3) + (1/6)×E(2) + (1/6)×E(1)	3/2 = 1.5
9	1 + (1/12)×E(5) + (1/12)×E(4) + (1/6)×E(3) + (1/6)×E(2) + (1/4)×E(1)	253/144 ≈ 1.7569
10	1 + (1/12)×E(6) + (1/12)×E(5) + (1/6)×E(4) + (1/6)×E(3) + (1/4)×E(2) + (1/4)×E(1)	97/48 ≈ 2.0208

4. Detailed Calculation for E(10)

E(10) = 1 + (1/12)×(7/6) + (1/12)×(13/12) + (1/6)×1 + (1/6)×1 + (1/4)×1 + (1/4)×1
= 1 + 7/72 + 13/144 + 1/6 + 1/6 + 1/4 + 1/4
= 1 + (14/144 + 13/144 + 24/144 + 24/144 + 36/144 + 36/144)
= 1 + 147/144 = 1 + 49/48 = 97/48

Final Answer

The expected number of turns until the kitten catches Lamb is:

E = 97/48 turns ≈ 2.0208 turns