Question

10 people are seating on chairs around a circular table. These chairs are marked
in a clockwise manner. There is a ball on the man’s hand who is seated on 0
marked chair, and the ball will be passed from one man to another in clockwise manner.
In first step, the ball goes to 1 marked chair with 1 turn. In second step, from there, the
ball goes to 5 marked chair with 2^2
turns. In third step, the ball goes to 2 marked chair by 3^3
turns from 5 marked chair. By this means, in which chair the ball will be in 2017^th
step?

Answer 1

Answer on Question #81427 – Math – Algebra

Question

10 people are seating on chairs around a circular table. These chairs are marked in a clockwise manner. There is a ball on the man’s hand who is seated on 0 marked chair, and the ball will be passed from one man to another in clockwise manner.

In first step, the ball goes to 1 marked chair with 1 turn.

In second step, from there, the ball goes to 5 marked chair with $2^2$ turns.

In third step, the ball goes to 2 marked chair by $3^3$ turns from 5 marked chair.

By this means, in which chair the ball will be in $2017^th$ step?

Solution

Write expressions for the first three turns:

$$\begin{array}{l} 0 + 1^{2} \cdot 10 + 1 = 11, \\ 1 + 2^{2} \cdot 10 + 4 = 45, \\ 5 + 3^{2} \cdot 10 + 7 = 102, \\ 2 + 4^{2} \cdot 10 + 10 = 172, \end{array}$$

[this problem must be solved by programming]

$$\begin{array}{l} 0 + 2016^{2} \cdot 10 + 6046 = 40648606, \\ 6 + 2017^{2} \cdot 10 + 6049 = 40688945. \end{array}$$

There is a pattern: the first term of the expression is the last number of the previous expression’s answer, $n^2$ – numbers of turns, 10 – quantity of chairs, the last term – term of arithmetic progression with the first term 1 and difference 3.

On the $2017^{\text{th}}$ turn the ball will be in the hands of the man sitting on the chair 5.

Answer:

5

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