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Tanish from Pate's Grammar School, Nishad from Rugby High School and Leona, all in the UK, found the transition matrix ${\bf M}$ and ${\bf M}^2$, ${\bf M}^3$ and ${\bf M}^4,$ as well as suggesting a formula for the general form ${\bf M}^n.$ This is Leona's work:
Tanish showed where the formula for ${\bf M}^n$ came from, and expressed it in a slightly different form:
Nishad used proof by induction to prove that this formula works for all values of $n.$ Click to see Nishad's proof.
Tanish used $\bf{M}^n$ to find the probability that, after $n$ whistle blasts, child A is holding the parcel:
If the game starts with child A holding the parcel, we can use the first column of the matrix above to find the probability that A, B, C or D are holding the parcel after $n$ whistles.
Leona and Nishad used matrix multiplication to get the same result. This is Leona's work:
To find these probabilities as $n\rightarrow\infty,$ Leona and Tanish used Tanish's formula. This is Leona's work:
Nishad used a different method: